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  • Articles  (113)
  • Nonlinear programming
  • Springer  (113)
  • Mathematics  (110)
  • Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics  (3)
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  • Articles  (113)
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  • Springer  (113)
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  • Mathematics  (110)
  • Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics  (3)
  • Computer Science  (23)
  • Economics  (6)
  • Technology  (1)
  • 1
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    Electronic Resource
    The use of augmented lagrangian functions for sensitivity analysis in nonlinear programming (1977)
    Springer
    Mathematical programming 12 (1977), S. 281-284 
    Details
    ISSN: 1436-4646
    Keywords: Nonlinear programming ; Augmented Lagrangian functions ; Sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01593794
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  • 2
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    A continuation method for monotone variational inequalities (1995)
    Springer
    Mathematical programming 69 (1995), S. 237-253 
    Details
    ISSN: 1436-4646
    Keywords: Variational inequality ; Nonlinear complementarity ; Nonlinear programming ; Continuation method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a continuation method for monotone variational inequality problems based on a new smooth equation formulation. The existence, uniqueness and limiting behavior of the path generated by the method are analyzed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585559
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  • 3
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    Pseudopower expansion of solutions of generalized equations and constrained optimization problems (1995)
    Springer
    Mathematical programming 70 (1995), S. 123-148 
    Details
    ISSN: 1436-4646
    Keywords: Generalized equations ; Variational inequalities ; Nonlinear programming ; Sensitivity analysis ; Power series ; Strong regularity ; Constrained optimization ; Perturbation theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We show that the solution of a strongly regular generalized equation subject to a scalar perturbation expands in pseudopower series in terms of the perturbation parameter, i.e., the expansion of orderk is the solution of generalized equations expanded to orderk and thus depends itself on the perturbation parameter. In the polyhedral case, this expansion reduces to a usual Taylor expansion. These results are applied to the problem of regular perturbation in constrained optimization. We show that, if the strong regularity condition is satisfied, the property of quadratic growth holds and, at least locally, the solutions of the optimization problem and of the associated optimality system coincide. If, in addition the number of inequality constraints is finite, the solution and the Lagrange multiplier can be expanded in Taylor series. If the data are analytic, the solution and the multiplier are analytic functions of the perturbation parameter.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585932
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  • 4
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    Duality of regularizations and hullfunctions for mathematical programming problems (1994)
    Springer
    OR spectrum 16 (1994), S. 187-191 
    Details
    ISSN: 1436-6304
    Keywords: Nonlinear programming ; duality ; solution methods ; parametric programming ; multicriteria optimization ; ill-posed problems ; Nichtlineare Optimierung ; Dualität ; Lösungsverfahren ; Parametrische Optimierung ; Vektor-Optimierung ; unlösbare Aufgaben
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Bei der Untersuchung von mathematischen Optimierungsproblemen und Lösungsmethoden liefert die Dualitätstheorie ein wichtiges Hilfsmittel. Die Konzepte derRegularisierung undStabilisierung des Ausgangsproblems erlauben eine Verbesserung des Verhaltens in praktischen Lösungsverfahren. Die nachfolgenden Untersuchungen behandeln die Dualität derartiger Regularisierungen sowie die Bildung vonHüllfunktionen. Die Bearbeitung sogenannter „unlösbarer Optimierungsprobleme“ (Eremin) durch Parametrisierung verdeutlicht die praktische Bedeutung dieses Konzeptes für numerische Verfahren. Darüber hinaus zeigen die Ergebnisse Anwendungsmöglichkeiten zur Lösung von Aufgaben der Parametrischen und Vektor-Optimierung.
    Notes: Abstract For the study of mathematical programming problems and solution methods the duality theory forms a powerful tool. There are also some concepts ofregularization andstabilization of a given problem for a better behavior in practical solution procedures. The aim of this paper is the investigation of duality aspects of such regularizations and the forming ofhullfunctions on the other hand. Applications for handling of so-calledill-posed problems (Eremin) using some parametrizations of the original problem will emphasize the importance for practical numerical methods, especially. This results will inspire some applications to solution methods for parametric and multicriteria optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01720706
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  • 5
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    A smooth method for the finite minimax problem (1993)
    Springer
    Mathematical programming 60 (1993), S. 187-214 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; unconstrained optimization ; nondifferentiable optimization ; minimax problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions. We prove that, under usual assumptions, it is possible to construct a continuously differentiable function, whose minimizers yield the minimizers of the max function and the corresponding minimum values. On this basis, we can define implementable algorithms for the solution of the minimax problem, which are globally convergent at a superlinear convergence rate. Preliminary numerical results are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580609
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  • 6
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    A trust-region strategy for minimization on arbitrary domains (1995)
    Springer
    Mathematical programming 68 (1995), S. 267-301 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; Trust-region methods ; Global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a trust-region method for minimizing a general differentiable function restricted to an arbitrary closed set. We prove a global convergence theorem. The trust-region method defines difficult subproblems that are solvable in some particular cases. We analyze in detail the case where the domain is a Euclidean ball. For this case we present numerical experiments where we consider different Hessian approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585768
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  • 7
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    Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities (1995)
    Springer
    Mathematical programming 69 (1995), S. 89-109 
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    ISSN: 1436-4646
    Keywords: Nondifferentiable (nonsmooth) optimization ; Convex programming ; Mathematical programming ; Nonlinear programming ; Saddle-points ; Variational inequalities ; Bundle methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value. We show that they enjoy almost optimal efficiency estimates. We give extensions for solving convex constrained problems, convex-concave saddle-point problems and variational inequalities with monotone operators. We present several variants, establish their efficiency estimates, and discuss possible implementations. In particular, our methods require bounded storage in contrast to the original level methods of Lemaréchal, Nemirovskii and Nesterov.
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    URL: http://dx.doi.org/10.1007/BF01585554
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  • 8
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    A unifying geometric solution framework and complexity analysis for variational inequalities (1995)
    Springer
    Mathematical programming 71 (1995), S. 327-351 
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    ISSN: 1436-4646
    Keywords: Variational inequalities ; Nonlinear programming ; Complexity analysis ; Monotone operators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result, we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs.
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    URL: http://dx.doi.org/10.1007/BF01590959
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  • 9
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    An SQP method for general nonlinear programs using only equality constrained subproblems (1998)
    Springer
    Mathematical programming 82 (1998), S. 413-448 
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    ISSN: 1436-4646
    Keywords: Sequential quadratic programming ; SQP method ; Nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we describe a new version of a sequential equality constrained quadratic programming method for general nonlinear programs with mixed equality and inequality constraints. Compared with an older version [P. Spellucci, Han's method without solving QP, in: A. Auslender, W. Oettli, J. Stoer (Eds), Optimization and Optimal Control, Lecture Notes in Control and Information Sciences, vol. 30, Springer, Berlin, 1981, pp. 123–141.] it is much simpler to implement and allows any kind of changes of the working set in every step. Our method relies on a strong regularity condition. As far as it is applicable the new approach is superior to conventional SQP-methods, as demonstrated by extensive numcrical tests. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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    URL: http://dx.doi.org/10.1007/BF01580078
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  • 10
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    A nonsmooth Newton method for variational inequalities, I: Theory (1994)
    Springer
    Mathematical programming 65 (1994), S. 151-194 
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    ISSN: 1436-4646
    Keywords: Nonsmooth equations ; Nonlinear complementarity ; Nonlinear programming ; Variational inequalities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a modified damped Newton algorithm for solving variational inequality problems based on formulating this problem as a system of equations using the Minty map. The proposed modified damped-Newton method insures convergence and locally quadratic convergence under the assumption of regularity. Under the assumption ofweak regularity and some mild conditions, the modified algorithm is shown to always create a descent direction and converge to the solution. Hence, this new algorithm is often suitable for many applications where regularity does not hold. Part II of this paper presents the results of extensive computational testing of this new method.
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    URL: http://dx.doi.org/10.1007/BF01581695
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  • 11
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    New algorithms for constrained minimax optimization (1977)
    Springer
    Mathematical programming 13 (1977), S. 140-155 
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    ISSN: 1436-4646
    Keywords: Minimax optimization ; Nonlinear programming ; Computer-aided network design
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A constrained minimax problem is converted to minimization of a sequence of unconstrained and continuously differentiable functions in a manner similar to Morrison's method for constrained optimization. One can thus apply any efficient gradient minimization technique to do the unconstrained minimization at each step of the sequence. Based on this approach, two algorithms are proposed, where the first one is simpler to program, and the second one is faster in general. To show the efficiency of the algorithms even for unconstrained problems, examples are taken to compare the two algorithms with recent methods in the literature. It is found that the second algorithm converges faster with respect to the other methods. Several constrained examples are also tried and the results are presented.
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    URL: http://dx.doi.org/10.1007/BF01584333
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  • 12
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    Some computational issues in optimal control by nonlinear programming (1993)
    Springer
    Annals of operations research 43 (1993), S. 249-257 
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    ISSN: 1572-9338
    Keywords: Nonlinear programming ; multivariable control systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The Roppenecker [11] parameterization of multi-input eigenvalue assignment, which allows for common open- and closed-loop eigenvalues, provides a platform for the investigation of several issues of current interest in robust control. Based on this parameterization, a numerical optimization method for designing a constant gain feedback matrix which assigns the closed-loop eigenvalues to desired locations such that these eigenvalues have low sensitivity to variations in the open-loop state space model was presented in Owens and O'Reilly [8]. In the present paper, two closely related numerical optimization methods are presented. The methods utilize standard (NAG library) unconstrained optimization routines. The first is for designing a minimum gain state feedback matrix which assigns the closed-loop eigenvalues to desired locations, where the measure of gain taken is the Frobenius norm. The second is for designing a state feedback matrix which results in the closed-loop system state matrix having minimum condition number. These algorithms have been shown to give results which are comparable to other available algorithms of far greater conceptual complexity.
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    URL: http://dx.doi.org/10.1007/BF02025454
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  • 13
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    NLPQL: A fortran subroutine solving constrained nonlinear programming problems (1986)
    Springer
    Annals of operations research 5 (1986), S. 485-500 
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    ISSN: 1572-9338
    Keywords: Nonlinear programming ; sequential quadratic programming method ; numerical implementation ; test results
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract NLPQL is a FORTRAN implementation of a sequential quadratic programming method for solving nonlinearly constrained optimization problems with differentiable objective and constraint functions. At each iteration, the search direction is the solution of a quadratic programming subproblem. This paper discusses the organization of NLPQL, including the formulation of the subproblem and the information that must be provided by a user. A summary is given of the performance of different algorithmic options of NLPQL on a collection of test problems (115 hand-selected or application problems, 320 randomly generated problems). The performance of NLPQL is compared with that of some other available codes.
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    URL: http://dx.doi.org/10.1007/BF02022087
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  • 14
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    Integrating modeling, algorithm design, and computational implementation to solve a large-scale non-linear mixed integer programming problem (1986)
    Springer
    Annals of operations research 5 (1986), S. 395-411 
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    ISSN: 1572-9338
    Keywords: Nonlinear programming ; decomposition ; branch and bound ; network ; transportation ; mixed integer programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This paper describes the formulation of a nonlinear mixed integer programming model for a large-scale product development and distribution problem and the design and computational implementation of a special purpose algorithm to solve the model. The results described demonstrate that integrating the art of modeling with the sciences of solution methodology and computer implementation provides a powerful approach for attacking difficult problems. The efforts described here were successful because they capitalized on the wealth of existing modeling technology and algorithm technology, the availability of efficient and reliable optimization, matrix generation and graphics software, and the speed of large-scale computer hardware. The model permitted the combined use of decomposition, general linear programming and network optimization within a branch and bound algorithm to overcome mathematical complexity. The computer system reliably found solutions with considerably better objective function values 30 to 50 times faster than had been achieved using general purpose optimization software alone. Throughout twenty months of daily use, the system was credited with providing insights and suggesting strategies that led to very large dollar savings.
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    URL: http://dx.doi.org/10.1007/BF02022082
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  • 15
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    Optimising tolerance allocation for mechanical components correlated by selective assembly (1996)
    Springer
    The international journal of advanced manufacturing technology 12 (1996), S. 349-355 
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    ISSN: 1433-3015
    Keywords: Kuhn-Tucker theorem ; Lagrange multiplier method ; Nonlinear programming ; Optimal tolerance allocation ; Selective assembly ; Statistical model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract Selective assembly can enlarge the tolerances of mechanical components for easier manufacturing. However, the non-independent dimensions of correlated components make it difficult to optimise tolerance allocation for an assembly. This paper proposes a solution for this constrained optimisation problem consisting of tolerances and non-independent dimensions as design variables. The approach is to develop a simplified algorithm applying a Lagrange multiplier method to evaluate the optimal tolerances efficiently. The solution is shown to be a global optimum at the given correlation coefficients. The correlation coefficients are key elements in determining the optimal solution, which is demonstrated in the given examples. The results are helpful in designing tolerances for selective assembly.
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    URL: http://dx.doi.org/10.1007/BF01179810
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  • 16
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    A successive quadratic programming algorithm with global and superlinear convergence properties (1986)
    Springer
    Mathematical programming 35 (1986), S. 253-264 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; successive quadratic programming ; Maratos effect ; exact penalty function ; superlinear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a successive quadratic programming algorithm for solving general nonlinear programming problems. In order to avoid the Maratos effect, direction-finding subproblems are derived by modifying the second-order approximations to both objective and constraint functions of the problem. We prove that the algorithm possesses global and superlinear convergence properties.
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    URL: http://dx.doi.org/10.1007/BF01580879
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  • 17
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    Simplicial decomposition in nonlinear programming algorithms (1977)
    Springer
    Mathematical programming 13 (1977), S. 49-68 
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    ISSN: 1436-4646
    Keywords: Algorithm ; APL-code ; Barycentric representation ; Decomposition ; Nonlinear programming ; Pseudo-concave objective ; Simplex
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Carathéodory's theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it. The paper presents the theory for this class of algorithms, the APL-code of a “demonstration” method and some computational experience with Colville's test problems.
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    URL: http://dx.doi.org/10.1007/BF01584323
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  • 18
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    Nonlinear programming and nonsmooth optimization by successive linear programming (1989)
    Springer
    Mathematical programming 43 (1989), S. 235-256 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; nonsmooth optimization ; global convergence ; superlinear convergence ; trust region method ; Coleman-Conn method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Methods are considered for solving nonlinear programming problems using an exactl 1 penalty function. LP-like subproblems incorporating a trust region constraint are solved successively both to estimate the active set and to provide a foundation for proving global convergence. In one particular method, second order information is represented by approximating the reduced Hessian matrix, and Coleman-Conn steps are taken. A criterion for accepting these steps is given which enables the superlinear convergence properties of the Coleman-Conn method to be retained whilst preserving global convergence and avoiding the Maratos effect. The methods generalize to solve a wide range of composite nonsmooth optimization problems and the theory is presented in this general setting. A range of numerical experiments on small test problems is described.
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    URL: http://dx.doi.org/10.1007/BF01582292
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  • 19
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    Large scale nonlinear deterministic and stochastic optimization: Formulations involving simulation of subsurface contamination (1990)
    Springer
    Mathematical programming 48 (1990), S. 19-39 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; numerical simulation ; chance constraints ; Monte Carlo methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Once subsurface water supplies become contaminated, designing cost-effective and reliable remediation schemes becomes a difficult task. The combination of finite element simulation of groundwater contaminant transport with nonlinear optimization is one approach to determine the best well selection and optimal fluid withdrawal and injection rates to contain and remove the contaminated water. Both deterministic and stochastic programming problems have been formulated and solved. These tend to be large scale problems, owing to the simulation component which serves as a portion of the constraint set. The overall problem of combined groundwater process simulation and nonlinear optimization is discussed along with example problems. Because the contaminant transport simulation models give highly uncertain results, quantifying their uncertainty and incorporating reliability into the remediation design results in a class of large stochastic nonlinear problems. The reliability problem is beginning to be addressed, and some strategies and formulations involving chance constraints and Monte Carlo methods are presented.
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    URL: http://dx.doi.org/10.1007/BF01582250
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  • 20
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    Computable bounds on parametric solutions of convex problems (1988)
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    Mathematical programming 40 (1988), S. 213-221 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; convex programming ; sensitivity and stability analysis ; parametric solutions ; optimal solution bounds
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We present a new method for computing bounds on parametric solutions of convex problems. The approach is based on a uniform quadratic underestimation of the objective function and a simple technique for the calculation of bounds on the optimal value function.
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    URL: http://dx.doi.org/10.1007/BF01580732
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    A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems (1991)
    Springer
    Mathematical programming 51 (1991), S. 101-131 
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    ISSN: 1436-4646
    Keywords: Nonlinear programming ; variational inequality/complementarity problems ; Maratos effect ; damped-Newton method ; nonsmooth equations ; B-differentiable function
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.
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    URL: http://dx.doi.org/10.1007/BF01586928
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    A nonsmooth Newton method for variational inequalities, II: Numerical results (1994)
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    Mathematical programming 65 (1994), S. 195-216 
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    ISSN: 1436-4646
    Keywords: Nonsmooth equations ; Nonlinear complementarity ; Nonlinear programming ; Variational inequalities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents the results of extensive computational testing of the modified damped Newton algorithm for solving variational inequality problems presented in Part I [8].
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    URL: http://dx.doi.org/10.1007/BF01581696
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    The complexity of approximating a nonlinear program (1995)
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    Mathematical programming 69 (1995), S. 429-441 
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    ISSN: 1436-4646
    Keywords: Approximation ; Optimization ; Probabilistically checkable proofs ; Quadratic programming ; Nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time. Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.
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    URL: http://dx.doi.org/10.1007/BF01585569
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    A simulation-based approach to two-stage stochastic programming with recourse (1998)
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    Mathematical programming 81 (1998), S. 301-325 
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    ISSN: 1436-4646
    Keywords: Two-stage stochastic programming with recourse ; Monte Carlo simulation ; Likelihood ratios ; Variance reduction techniques ; Confidence intervals ; Hypotheses testing ; Validation analysis ; Nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we consider stochastic programming problems where the objective function is given as an expected value function. We discuss Monte Carlo simulation based approaches to a numerical solution of such problems. In particular, we discuss in detail and present numerical results for two-stage stochastic programming with recourse where the random data have a continuous (multivariate normal) distribution. We think that the novelty of the numerical approach developed in this paper is twofold. First, various variance reduction techniques are applied in order to enhance the rate of convergence. Successful application of those techniques is what makes the whole approach numerically feasible. Second, a statistical inference is developed and applied to estimation of the error, validation of optimality of a calculated solution and statistically based stopping criteria for an iterative alogrithm. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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    URL: http://dx.doi.org/10.1007/BF01580086
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    Homotopy techniques in linear programming (1986)
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    Algorithmica 1 (1986), S. 529-535 
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    ISSN: 1432-0541
    Keywords: Homotopy ; Linear programming ; Interior point methods ; Nonlinear programming ; Karmarkar's method ; Quadratic regularization ; Path-following
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this note, we consider the solution of a linear program, using suitably adapted homotopy techniques of nonlinear programming and equation solving that move through the interior of the polytope of feasible solutions. The homotopy is defined by means of a quadratic regularizing term in an appropriate metric. We also briefly discuss algorithmic implications and connections with the affine variant of Karmarkar's method.
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    Convergence of a Tuy-type algorithm for concave minimization subject to linear inequality constraints (1981)
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    Applied mathematics & optimization 7 (1981), S. 1-9 
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    ISSN: 1432-0606
    Keywords: Nonlinear programming ; nonconvex programming ; concave minimization ; convergence
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    Topics: Mathematics
    Notes: Abstract A modification of Tuy's cone splitting algorithm for minimizing a concave function subject to linear inequality constraints is shown to be convergent by demonstrating that the limit of a sequence of constructed convex polytopes contains the feasible region. No geometric tolerance parameters are required.
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    Local analysis of Newton-type methods for variational inequalities and nonlinear programming (1994)
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    Applied mathematics & optimization 29 (1994), S. 161-186 
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    ISSN: 1432-0606
    Keywords: Variational inequalities ; Nonlinear programming ; Successive ; quadratic programming superlinear convergence ; Primary 90C30 ; Secondary 65K05 ; 90C26
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents some new results in the theory of Newton-type methods for variational inequalities, and their application to nonlinear programming. A condition of semistability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth functions is given. The second part of the paper considers some particular variational inequalities with unknowns (x, λ), generalizing optimality systems. Here only the question of superlinear convergence of {x k } is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow us to obtain the superlinear convergence of {x k }. Application of the previous results to nonlinear programming allows us to strengthen the known results, the main point being a characterization of the superlinear convergence of {x k } assuming a weak second-order condition without strict complementarity.
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    A modified Frank-Wolfe algorithm and its convergence properties (1995)
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    Acta mathematicae applicatae sinica 11 (1995), S. 285-291 
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    ISSN: 1618-3932
    Keywords: Nonlinear programming ; Frank-Wolfe algorithm ; convergence properties
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    Topics: Mathematics
    Notes: Abstract This paper modifies the Frank-Wolfe's algorithm. Under weaker conditions it proves that the modified algorithm is convergent, and specially under the assumption of convexity of the objective function that $$\mathop {\lim }\limits_{k \to \infty } f(x^k ) = \mathop {\inf }\limits_{x \in R} f(x)$$ without assuming {x k } is bounded.
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    Tolerance synthesis adopting a nonlinear programming approach (1996)
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    The international journal of advanced manufacturing technology 11 (1996), S. 387-393 
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    ISSN: 1433-3015
    Keywords: Nonlinear programming ; Tolerance synthesis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract One of the main objectives in tolerance charting is to determine the working dimensions and tolerances at the lowest cost without violating the blueprint specifications. In view of this, this paper presents a simple dimensional chains identification method. After establishing all necessary equations and constraints, a nonlinear objective function is formulated. Subsequently, all these relationships are submitted to an optimisation software, OPTIVAR written in FORTRAN for the determination of the unknown variables. In addition, an example is incorporated to demonstrate the effectiveness of this proposed methodology.
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    A comparative analysis of genetic algorithms and directed grid search for parametric optimization (1998)
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    Engineering with computers 14 (1998), S. 197-205 
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    ISSN: 1435-5663
    Keywords: Genetic algorithms ; Grid search ; Nonlinear programming ; Optimal design ; Parametric optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Technology
    Notes: Abstract In this paper, a comparative analysis of the performance of the Genetic Algorithm (GA) and Directed Grid Search (DGS) methods for optimal parametric design is presented. A genetic algorithm is a guided random search mechanism based on the principle of natural selection and population genetics. The Directed Grid Search method uses a selective directed search of grid points in the direction of descent to find the minimum of a real function, when the initial estimate of the location of the minimum and the bounds of the design variables are specified. An experimental comparison and a discussion on the performance of these two methods in solving a set of eight test functions is presented.
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    Inexact-Restoration Algorithm for Constrained Optimization1 (2000)
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    Journal of optimization theory and applications 104 (2000), S. 135-163 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; trust regions ; feasible methods ; global convergence ; numerical experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We introduce a new model algorithm for solving nonlinear programming problems. No slack variables are introduced for dealing with inequality constraints. Each iteration of the method proceeds in two phases. In the first phase, feasibility of the current iterate is improved; in second phase, the objective function value is reduced in an approximate feasible set. The point that results from the second phase is compared with the current point using a nonsmooth merit function that combines feasibility and optimality. This merit function includes a penalty parameter that changes between consecutive iterations. A suitable updating procedure for this penalty parameter is included by means of which it can be increased or decreased along consecutive iterations. The conditions for feasibility improvement at the first phase and for optimality improvement at the second phase are mild, and large-scale implementation of the resulting method is possible. We prove that, under suitable conditions, which do not include regularity or existence of second derivatives, all the limit points of an infinite sequence generated by the algorithm are feasible, and that a suitable optimality measure can be made as small as desired. The algorithm is implemented and tested against the LANCELOT algorithm using a set of hard-spheres problems.
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    A nonlinear programming approach to space shuttle trajectory optimization (1974)
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    Journal of optimization theory and applications 13 (1974), S. 94-118 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; penalty-function methods ; trajectories ; state variable constraints ; flight mechanics
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    Notes: Abstract Space shuttle trajectory design requires the optimization of highly-constrained, branched, atmospheric trajectories. This paper presents a general mathematical model for trajectory optimization incorporating multiple simulation sections, state variable discontinuities, subarc elimination, branching, and six types of equality and inequality constraints. A solution is sought using a nonlinear programming approach based on SUMT, the sequential unconstrained minimization technique. Results of the numerical solution of a highly constrained space shuttle reentry problem are presented.
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    Nonlinear programming using minimax techniques (1974)
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    Journal of optimization theory and applications 13 (1974), S. 607-619 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; minimax approximation ; least-square methods ; function minimization ; penalty function methods
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    Topics: Mathematics
    Notes: Abstract A minimax approach to nonlinear programming is presented. The original nonlinear programming problem is formulated as an unconstrained minimax problem. Under reasonable restrictions, it is shown that a point satisfying the necessary conditions for a minimax optimum also satisfies the Kuhn-Tucker necessary conditions for the original problem. A leastpth type of objective function for minimization with extremely large values ofp is proposed to solve the problem. Several numerical examples compare the present approach with the well-known SUMT method of Fiacco and McCormick. In both cases, a recent minimization algorithm by Fletcher is used.
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    New approach to comparison of search methods used in nonlinear programming problems (1974)
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    Journal of optimization theory and applications 13 (1974), S. 635-659 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; numerical methods ; unconstrained minimization ; function minimization
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    Topics: Mathematics
    Notes: Abstract This paper is concerned with the problem of investigating the properties and comparing the methods of nonlinear programming. The steepest-descent method, the method of Davidon, the method of conjugate gradients, and other methods are investigated for the class of essentially nonlinear valley functions.
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    On the existence and nonexistence of Lagrange multipliers in Banach spaces (1976)
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    Journal of optimization theory and applications 20 (1976), S. 81-110 
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    Keywords: Nonlinear programming ; Lagrange multipliers ; existence theorems ; Banach spaces
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    Topics: Mathematics
    Notes: Abstract The paper deals with the existence of Lagrange multipliers for a general nonlinear programming problem. Some regularity conditions are formulated which are, in a sense, the weakest to assure the existence of multipliers. A number of related conditions are discussed. The connection between the choice of suitable function spaces and the existence of multipliers is analyzed.
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    Generalized Lagrange multiplier technique for nonlinear programming (1977)
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    Journal of optimization theory and applications 21 (1977), S. 121-135 
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    Keywords: Nonlinear programming ; max-min problems ; Lagrange multiplier technique ; Newton's method
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    Notes: Abstract Our aim here is to present numerical methods for solving a general nonlinear programming problem. These methods are based on transformation of a given constrained minimization problem into an unconstrained maximin problem. This transformation is done by using a generalized Lagrange multiplier technique. Such an approach permits us to use Newton's and gradient methods for nonlinear programming. Convergence proofs are provided, and some numerical results are given.
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    Square-root variable-metric methods for minimization (1977)
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    Journal of optimization theory and applications 21 (1977), S. 251-259 
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    Keywords: Nonlinear programming ; variable-metric methods ; parameter optimization ; function minimization ; mathematical programming
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    Topics: Mathematics
    Notes: Abstract Variable-metric methods are presented which do not need an accurate one-dimensional search and eliminate roundoff error problems which can occur in updating the metric for large-dimension systems. The methods are based on updating the square root of the metric, so that a positive-definite metric always results. The disadvantage of intentionally relaxing the accuracy of the one-dimensional search is that the number of iterations (and hence, gradient evaluations) increases. For problems involving a large number of variables, the square-root method is presented in a triangular form to reduce the amount of computation. Also, for usual optimization problems, the square-root procedure can be carried out entirely in terms of the metric, eliminating storage and computer time associated with computations of the square root of the metric.
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    An economical method of calculating Lagrange multiplier estimates for nonlinear programming problems when many of the constraints are bounds on the variables (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 235-239 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; computing methods ; Lagrange multiplier estimates ; bounded variables ; large-scale problems
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    Topics: Mathematics
    Notes: Abstract Some recent methods for solving nonlinear programming problems make use of estimates of the Lagrange multipliers. These estimates are usually calculated by solving a system oft linear equations, wheret is the number of active constraints. It is shown that, when a large proportion of the active constraints consists of simple upper or lower bounds on the variables, then computational effort can be saved by means of a reorganization of this linear system.
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    A comparison of computational strategies for geometric programs (1978)
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    Journal of optimization theory and applications 26 (1978), S. 185-203 
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    Keywords: Nonlinear programming ; geometric programming ; computation ; comparisons
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    Topics: Mathematics
    Notes: Abstract Numerous algorithms for the solution of geometric programs have been reported in the literature. Nearly all are based on the use of conventional programming techniques specialized to exploit the characteristic structure of either the primal or the dual or a transformed primal problem. This paper attempts to elucidate, via computational comparisons, whether a primal, a dual, or a transformed primal solution approach is to be preferred.
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    Numerical study of some feasible direction methods in mathematical programming (1983)
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    Journal of optimization theory and applications 40 (1983), S. 1-23 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; feasible directions ; linear least squares ; Householder orthogonal factorization ; Gauss-Jordan factorization
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    Topics: Mathematics
    Notes: Abstract Some feasible direction methods for the minimization of a linearly constrained convex function are studied. Special emphasis is placed on the analysis of the procedures which find the search direction, by developing active set methods which use orthogonal or Gauss-Jordan-like transformations. Numerical experiments are performed on a class of quadratic problems depending on two parameters, related to the conditioning of the matrix associated with the quadratic form and the matrix of active constraints at the optimal point. Results are given for the rate of convergence and the average iteration time.
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    On the convergence rate for a penalty function method of exponential type (1979)
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    Journal of optimization theory and applications 27 (1979), S. 495-508 
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    Keywords: Nonlinear programming ; penalty function methods ; Lagrange multipliers ; superlinearly convergent algorithms
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    Topics: Mathematics
    Notes: Abstract Recently, Kort and Bertsekas (Ref. 1) and Hartman (Ref. 2) presented independently a new penalty function algorithm of exponential type for solving inequality-constrained minimization problems. The main purpose of this work is to give a proof on the rate of convergence of a modification of the exponential penalty method proposed by these authors. We show that the sequence of points generated by the modified algorithm converges to the solution of the original nonconvex problem linearly and that the sequence of estimates of the optimal Lagrange multiplier converges to this multiplier superlinearly. The question of convergence of the modified method is discussed. The present paper hinges on ideas of Mangasarian (Ref. 3), but the case considered here is not covered by Mangasarian's theory.
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    Properties of updating methods for the multipliers in augmented Lagrangians (1979)
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    Journal of optimization theory and applications 28 (1979), S. 135-156 
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    Keywords: Nonlinear programming ; constrained optimization ; augmented Lagrangians ; quasi-Newton methods ; rate of convergence ; penalty functions ; Lagrange multipliers
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    Notes: Abstract The convergence properties of different updating methods for the multipliers in augmented Lagrangians are considered. It is assumed that the updating of the multipliers takes place after each line search of a quasi-Newton method. Two of the updating methods are shown to be linearly convergent locally, while a third method has superlinear convergence locally. Modifications of the algorithms to ensure global convergence are considered. The results of a computational comparison with other methods are presented.
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    An outer approximations algorithm for computer-aided design problems (1979)
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    Journal of optimization theory and applications 28 (1979), S. 331-352 
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    Keywords: Nonlinear programming ; mathematical programming ; outer approximation algorithm ; computer-aided design ; design problems
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    Topics: Mathematics
    Notes: Abstract This paper presents an implementable algorithm of the outer approximations type for solving nonlinear programming problems with functional inequality constraints. The algorithm was motivated by engineering design problems in circuit tolerancing, multivariable control, and shock-resistant structures.
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    Nonlinear programming, approximation, and optimization on infinitely differentiable functions (1979)
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    Journal of optimization theory and applications 29 (1979), S. 199-213 
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    Keywords: Nonlinear programming ; approximation of functions ; Tchebycheff polynomials ; infinitely differentiable functions
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    Notes: Abstract A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ 0 1 φ(t)dt=1. In this article, the following problem is considered. Determine Δ k =inf∫ 0 1 |φ(k)(t)|dt,k=1, 2, ..., where φ(k) denotes thekth derivative of φ and the infimum is taken over the set of all mollifier functions φ, which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that Δ k =k!22k−1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.
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    Nonlinear programming and an optimization problem on infinitely differentiable functions (1979)
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    Journal of optimization theory and applications 29 (1979), S. 345-367 
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    Keywords: Nonlinear programming ; convexity ; optimization on function spaces ; monotone approximation ; infinitely differentiable functions
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    Notes: Abstract A nonnegative, infinitely differentiable function ø defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ 0 1 ø(t)dt=1. In this article the following problem is considered. Determine Δ k =inf∫ 0 1 ∣vø(k)(t)∣dt, k=1,..., where ø(k) denotes thekth derivative of ø and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining Δ k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k−1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., Δ k =k~22k−1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.
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    Theoretical properties and numerical tests of an efficient nonlinear decomposition algorithm (1980)
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    Journal of optimization theory and applications 30 (1980), S. 211-227 
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    Keywords: Nonlinear programming ; decomposition algorithm ; convexity ; convergence ; SUMT algorithm
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    Topics: Mathematics
    Notes: Abstract Kronsjö's nonlinear generalization (Ref. 1) of Benders' algorithm (Ref. 2) is reviewed. At each iteration, this algorithm produces upper and lower bounds to the true optimum, and the sequence of lower bounds is increasing. The algorithm is modified, so that the sequence of upper bounds is ε-decreasing as well. The two versions are tested numerically using an ALGOL program originally written by Wong (Ref. 3), incorporating the SUMT method (Fiacco and McCormick, Refs. 4 and 5). The two versions are compared against each other, and the problem of the optimal degree of decomposition is considered. Finally, an attempt is made to express the computer time required to solve the test problems as a function of master problem size, number of subproblems, and average subproblem size.
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    Second-order and related extremality conditions in nonlinear programming (1980)
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    Journal of optimization theory and applications 31 (1980), S. 143-165 
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    Keywords: Nonlinear programming ; local extrema second-order conditions ; constraint qualification ; extremality conditions
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    Notes: Abstract This paper is concerned with the problem of characterizing a local minimum of a mathematical programming problem with equality and inequality constraints. The main object is to derive second-order conditions, involving the Hessians of the functions, or related results where some other curvature information is used. The necessary conditions are of the Fritz John type and do not require a constraint qualification. Both the necessary conditions and the sufficient conditions are given in equivalent pairs of primal and dual formulations.
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    A trace minimization problem with applications in joint estimation and control under nonclassical information (1980)
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    Journal of optimization theory and applications 31 (1980), S. 343-359 
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    Keywords: Nonlinear programming ; quadratic inequality constraints ; team problems ; signaling strategies ; nonclassical information
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    Notes: Abstract Let Λ and Σ be positive-definite matrices of dimensionsn×n andm×n. Then, this paper considers the problem of minimizing Tr[Λ(I+C′C)−1] over allm×n real matrices and under the constraint Tr[ΣCC′]≥1. The solution is obtained rigorously and withouta priori employing the Lagrange multipliers technique. An application of this result to a decentralized team problem which involves joint estimation and control and with signaling strategies is also discussed.
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    Some numerical experience with a globally convergent algorithm for nonlinearly constrained optimization (1980)
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    Journal of optimization theory and applications 32 (1980), S. 1-16 
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    Keywords: Nonlinear programming ; global convergence ; numerical experience
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    Notes: Abstract Global convergence properties are established for a quite general form of algorithms for solving nonlinearly constrained minimization problems. A useful feature of the methods considered is that they can be implemented easily either with or without using quadratic programming techniques. A particular implementation, designed to be both efficient and robust, is described in detail. Numerical results are presented and discussed.
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    A least-distance programming procedure for minimization problems under linear constraints (1983)
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    Journal of optimization theory and applications 40 (1983), S. 489-514 
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    Keywords: Nonlinear programming ; linear constraints ; algorithms ; global convergence
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract In this paper, an algorithm is developed for solving a nonlinear programming problem with linear contraints. The algorithm performs two major computations. First, the search vector is determined by projecting the negative gradient of the objective function on a polyhedral set defined in terms of the gradients of the equality constraints and the near binding inequality constraints. This least-distance program is solved by Lemke's complementary pivoting algorithm after eliminating the equality constraints using Cholesky's factorization. The second major calculation determines a stepsize by first computing an estimate based on quadratic approximation of the function and then finalizing the stepsize using Armijo's inexact line search. It is shown that any accumulation point of the algorithm is a Kuhn-Tucker point. Furthermore, it is shown that, if an accumulation point satisfies the second-order sufficiency optimality conditions, then the whole sequence of iterates converges to that point. Computational testing of the algorithm is presented.
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    Differential dynamic programming and Newton's method for discrete optimal control problems (1984)
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    Journal of optimization theory and applications 43 (1984), S. 395-414 
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    Keywords: Nonlinear programming ; optimal control ; optimal control algorithms ; nonlinear dynamics ; quadratic convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The purpose of this paper is to draw a detailed comparison between Newton's method, as applied to discrete-time, unconstrained optimal control problems, and the second-order method known as differential dynamic programming (DDP). The main outcomes of the comparison are: (i) DDP does not coincide with Newton's method, but (ii) the methods are close enough that they have the same convergence rate, namely, quadratic. The comparison also reveals some other facts of theoretical and computational interest. For example, the methods differ only in that Newton's method operates on a linear approximation of the state at a certain point at which DDP operates on the exact value. This would suggest that DDP ought to be more accurate, an anticipation borne out in our computational example. Also, the positive definiteness of the Hessian of the objective function is easy to check within the framework of DDP. This enables one to propose a modification of DDP, so that a descent direction is produced at each iteration, regardless of the Hessian.
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    Perturbation theory for abstract optimization problems (1981)
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    Journal of optimization theory and applications 35 (1981), S. 417-441 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; perturbation theory ; multipliers ; abnormality ; abstract optimization problems
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    Topics: Mathematics
    Notes: Abstract A general perturbation theory is given for optimization problems in locally convex, linear spaces. Neither differentiability of the constraints nor regularity of the solutions of the unperturbed problem are assumed. Without reference to a particular multiplier rule, multipliers of the unperturbed problem are defined and used for characterizing solutions of a perturbed problem. In case of differentiable constraints or finite-dimensional spaces, the results exceed those known so far.
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    Discretization methods for the solution of semi-infinite programming problems (1991)
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    Journal of optimization theory and applications 71 (1991), S. 85-103 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; semi-infinite programming ; discretization of semi-infinite programming problems ; Chebyshev approximation
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    Topics: Mathematics
    Notes: Abstract In the first part of this paper, we prove the convergence of a class of discretization methods for the solution of nonlinear semi-infinite programming problems, which includes known methods for linear problems as special cases. In the second part, we modify and study this type of algorithms for linear problems and suggest a specific method which requires the solution of a quadratic programming problem at each iteration. With this algorithm, satisfactory results can also be obtained for a number of singular problems. We demonstrate the performance of the algorithm by several numerical examples of multivariate Chebyshev approximation problems.
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    A quasi-discrete Newton algorithm with a nonmonotone stabilization technique (1990)
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    Journal of optimization theory and applications 64 (1990), S. 495-510 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unconstrained minimization ; Newton-type methods ; line search techniques
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    Topics: Mathematics
    Notes: Abstract In this paper, we define an unconstrained optimization algorithm employing only first-order derivatives, in which a nonmonotone stabilization technique is used in conjunction with a quasidiscrete Newton method for the computation of the search direction. Global and superlinear convergence is proved, and numerical results are reported.
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    Using two successive subgradients in the ellipsoid method for nonlinear programming (1994)
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    Journal of optimization theory and applications 82 (1994), S. 543-554 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; ellipsoid algorithm ; alternative cuts ; ellipsoids of smaller volume
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    Topics: Mathematics
    Notes: Abstract A variant of the ellipsoid method for nonlinear programming is introduced to enhance the speed of convergence. This variant is based on a new simple scheme to reduce the ellipsoid volume by using two center cuts generated in two consecutive iterations of the ellipsoid method. Computational tests show a significant improvement in computational efficiency. The tests show that the improvement is more significant for larger-size problems.
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    Convergence analysis of norm-relaxed method of feasible directions (1996)
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    Journal of optimization theory and applications 91 (1996), S. 389-418 
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    Keywords: Nonlinear programming ; method of feasible directions
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    Notes: Abstract This paper gives a complete treatment of the asymptotic rate of convergence for a class of feasible directions methods, including those studied by Pironneau and Polak and by Cawood and Kostreva. Rate estimates of Pironneau and Polak are sharpened in an analysis which shows the dependence on certain parameters of the direction-finding subproblem and the problem functions. Special cases of interior optimal solution, linear constraints, and fixed matrix norm are analyzed in detail. Numerical verification is provided.
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    Two-Phase Model Algorithm with Global Convergence for Nonlinear Programming (1998)
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    Journal of optimization theory and applications 96 (1998), S. 397-436 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; trust regions ; GRG methods ; SGRA methods ; projected gradient methods ; SQP methods ; global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The family of feasible methods for minimization with nonlinear constraints includes the nonlinear projected gradient method, the generalized reduced gradient method (GRG), and many variants of the sequential gradient restoration algorithm (SGRA). Generally speaking, a particular iteration of any of these methods proceeds in two phases. In the restoration phase, feasibility is restored by means of the resolution of an auxiliary nonlinear problem, generally a nonlinear system of equations. In the minimization phase, optimality is improved by means of the consideration of the objective function, or its Lagrangian, on the tangent subspace to the constraints. In this paper, minimal assumptions are stated on the restoration phase and the minimization phase that ensure that the resulting algorithm is globally convergent. The key point is the possibility of comparing two successive nonfeasible iterates by means of a suitable merit function that combines feasibility and optimality. The merit function allows one to work with a high degree of infeasibility at the first iterations of the algorithm. Global convergence is proved and a particular implementation of the model algorithm is described.
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    Nonlinear Programming Analysis to Estimate Implicit Inventory Backorder Costs (1998)
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    Journal of optimization theory and applications 97 (1998), S. 71-92 
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    Keywords: Nonlinear programming ; inventory theory ; backorders costs
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    Topics: Mathematics
    Notes: Abstract In this paper, we use nonlinear programming to provide an alternative treatment of the economic order quantity problem with planned backorders. Many businesses, such as capital-goods firms that deal with expensive products and some service industries that cannot store their services, operate with substantial backlogs. In practical problems, it is usually very difficult to estimate accurately the values of the two types of backorder costs, i.e., the time-dependent unit backorder cost and the unit backorder cost. We redefine the original problem without including these backorder costs and construct a nonlinear programming problem with two service measure constraints which may be easier to specify than the backorder costs. We find that, with this different formulation of our new problem, we obtain results which give implicit estimates of the backorder costs. The alternative formulation provides an easier-to-use model and managerially meaningful results. Next, we show that, for a wide range of parameter values, it usually suffices to consider only one type of backorder cost, or equivalently, only one type of service measure constraint. Finally, we develop expressions which bracket the optimal values of the decision variables in a narrow range and provide a simple method for computing the optimal solution. In the most complicated case, this method requires finding the unique root of a polynomial.
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    A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem (1995)
    Springer
    Journal of global optimization 6 (1995), S. 87-105 
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    ISSN: 1573-2916
    Keywords: Nonlinear programming ; global optimization ; d.c. programming ; outer approximation ; system of d.c. inequalities ; (ɛ, η)-solution ; fuel mixture problem
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract We present a numerical method for solving the d.c. programming problem $$c^* = \min \{ \langle c,x\rangle s.t. f_i (x) \leqslant 0, i = 1,...,m, x \in D\} $$ wherefi, i=1,...,m are d.c. (difference of two convex functions) and D is a convex set in ℝn. An (ɛ, η)-solutionx(ɛ, η) satisfying $$x(\varepsilon ,\eta ) \in D, \langle c,x(\varepsilon ,\eta )\rangle \leqslant c^* + \varepsilon , f_i (x(\varepsilon ,\eta )) \leqslant \eta , i = 1,...,m,$$ can be found after a finite number of iterations. This algorithm combines an outer approximation procedure for solving a system of d.c. inequalities with a simple general scheme for minimizing a linear function over a compact set. As an application we discuss the numerical solution of a fuel mixture problem (encountered in the oil industry).
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    On solving a D.C. programming problem by a sequence of linear programs (1991)
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    Journal of global optimization 1 (1991), S. 183-203 
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    ISSN: 1573-2916
    Keywords: Nonlinear programming ; global optimization ; d.c. programming ; branch and bound ; outer approximation ; prismatic partition
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    Notes: Abstract We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ℝ n . This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.
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    On stability concepts in nonlinear programming (1995)
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    Mathematical methods of operations research 41 (1995), S. 153-160 
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    ISSN: 1432-5217
    Keywords: Nonlinear programming ; stability concepts ; strong stability
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    Topics: Mathematics , Economics
    Notes: Abstract Kojima's strong stability of stationary solutions can be characterized by means of first and second order terms. We treat the problem whether there is a characterization of the stability concept allowing perturbations of the objective function only, keeping the feasible set unchanged. If the feasible set is a convex polyhedron, then there exists a characterization which is in fact weaker than that one of strong stability. However, in general it appears that data of first and second order do not characterize that kind of stability. As an interpretation we have that the strong stability is the only concept of stability which both admits a characterization and works for large problem classes.
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    On the solution of the unidimensional local minimization problem (1976)
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    Journal of optimization theory and applications 18 (1976), S. 425-428 
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    Keywords: Nonlinear programming ; unidimensional search ; numerical methods
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    Topics: Mathematics
    Notes: Abstract If the functionf to be minimized is not unimodal, the Fibonacci search and the golden section search can determine final search intervals where the functionf takes greater values than at the borders of the first search interval. This can be avoided by small modifications for sequential search procedures where, in each iteration step,f is evaluated at two interior points of the present search interval. The properties of the point of concentration of the search intervals are given.
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    A globally convergent method for nonlinear programming (1977)
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    Journal of optimization theory and applications 22 (1977), S. 297-309 
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    Keywords: Nonlinear programming ; global convergence ; exact penalty function
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    Notes: Abstract Recently developed Newton and quasi-Newton methods for nonlinear programming possess only local convergence properties. Adopting the concept of the damped Newton method in unconstrained optimization, we propose a stepsize procedure to maintain the monotone decrease of an exact penalty function. In so doing, the convergence of the method is globalized.
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    On the solution of the unidimensional local minimization problem (1977)
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    Journal of optimization theory and applications 23 (1977), S. 471-471 
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    Keywords: Nonlinear programming ; unidemensional search ; numerical methods
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    Notes: Abstract A correction of the procedure of Ref. 1 is given.
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    Feasible directions in economic policy (1978)
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    Journal of optimization theory and applications 25 (1978), S. 383-406 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; mathematical programming ; computational methods ; applications to economics ; objective functions ; inequality constraints ; choice of endpoints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Zoutendijk's method of feasible directions is used in this paper to derive numerical control strategies for the United Kingdom economy. The way in which the algorithm permits an examination of the sensitivity of the optimum short-term economic policy to changes in various assumptions demonstrates the versatility of the algorithm. Examined are the implications of different forms for the social welfare function; altering the length of the planning horizon, varying the magnitude of the terminal capital constraint, reducing the maximum permitted level of unemployment, changing the initial endowment of foreign currency reserves, fixing the interest rate for the whole planning period, and imposing a minimum growth rate for public expenditure.
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    A finite steepest-ascent algorithm for maximizing piecewise-linear concave functions (1978)
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    Journal of optimization theory and applications 25 (1978), S. 437-442 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; nondifferentiable optimization ; subgradient search ; steepest-ascent algorithm
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    Topics: Mathematics
    Notes: Abstract In this note, we describe a finitely convergent steepest-ascent scheme for maximizing piecewise-linear concave functions. Given any point, the algorithm moves along the direction of steepest ascent, that is, along the shortest subgradient, until a new ridge is reached. The overall process is then repeated by moving along the new steepest-ascent direction.
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    Algorithms for a class of nondifferentiable problems (1981)
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    Journal of optimization theory and applications 34 (1981), S. 41-82 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; nondifferentiable optimization ; algorithms ; min-max problems ; duality
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    Notes: Abstract A nonlinear programming problem with nondifferentiabilities is considered. The nondifferentiabilities are due to terms of the form min(f 1(x),...,f n(x)), which may enter nonlinearly in the cost and the constraints. Necessary and sufficient conditions are developed. Two algorithms for solving this problem are described, and their convergence is studied. A duality framework for interpretation of the algorithms is also developed.
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    Iterative determination of parameters for an exact penalty function (1975)
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    Journal of optimization theory and applications 16 (1975), S. 49-66 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; penalty-function methods ; Lagrange multipliers ; mathematical programming
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    Topics: Mathematics
    Notes: Abstract As an approach to solving nonlinear programs, we study a class of functions known to be exact penalty functions for a proper choice of the parameters. The goal is to iteratively determine the correct parameters. A basic algorithm has been developed. We have proved that this algorithm converges to a global solution for concave programs and, in the limited computational tests performed to date, it has always converged to at least a local solution for nonconcave programs also.
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    A direct method of linearization for continuous minimax problems (1987)
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    Journal of optimization theory and applications 55 (1987), S. 271-287 
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    Keywords: Nonlinear programming ; nondifferentiable optimization ; minimax problems ; semi-infinite programming ; recursive quadratic programming
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    Topics: Mathematics
    Notes: Abstract We consider the problem of minimizing a nondifferentiable function that is the pointwise maximum over a compact family of continuously differentiable functions. We suppose that a certain convex approximation to the objective function can be evaluated. An iterative method is given which uses as successive search directions approximate solutions of semi-infinite quadratic programming problems calculated via a new generalized proximity algorithm. Inexact line searches ensure global convergence of the method to stationary points.
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    Experiments with successive quadratic programming algorithms (1988)
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    Journal of optimization theory and applications 56 (1988), S. 359-383 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; SQP algorithms ; computational comparisons ; trust regions ; line searches
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    Topics: Mathematics
    Notes: Abstract There are many variants of successive quadratic programming (SQP) algorithms. Important issues include: the choice of either line search or trust region strategies; the QP formulation to be used; and how the QP is to be solved. Here, we consider the QP's proposed by Fletcher and Powell and discuss a specialized reduced-gradient procedure for solving them. A computer implementation is described, and the various options are compared on some well-known test problems. Factors influencing robustness and speed are identified.
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    An exterior point method for the convex programming problem (1992)
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    Journal of optimization theory and applications 72 (1992), S. 37-63 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; convex programming ; gradient projections ; optimization
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    Topics: Mathematics
    Notes: Abstract This paper gives a proof of convergence of an iterative method for maximizing a concave function subject to inequality constraints involving convex functions. The linear programming problem is an important special case. The primary feature is that each iteration is very simple computationally, involving only one of the constraints. Although the paper is theoretical in nature, some numerical results are included.
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    Optimality conditions in smooth nonlinear programming (1996)
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    Journal of optimization theory and applications 90 (1996), S. 483-515 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; necessary and sufficient optimality conditions ; strong and strict local minimizers ; isolated minimizers
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    Topics: Mathematics
    Notes: Abstract This survey is concerned with necessary and sufficient optimality conditions for smooth nonlinear programming problems with inequality and equality constraints. These conditions deal with strict local minimizers of order one and two and with isolated minimizers. In most results, no constraint qualification is required. The optimality conditions are formulated in such a way that the gaps between the necessary and sufficient conditions are small and even vanish completely under mild constraint qualifications.
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    Self-Tuning Norm-Relaxed Method of Feasible Directions (1997)
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    Journal of optimization theory and applications 92 (1997), S. 311-330 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; methods of feasible directions ; numerical methods ; computational experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents three updating techniques for the scaling matrix or the scalar weight used in the norm-relaxed method of feasible directions, a generalization of the popular Pironneau–Polak algorithm. These techniques include variable metric updates and tuning of a scalar weight in a way characteristic of trust-region methods, and also techniques based on the idea of multiple directions, where the update decision is made by comparing results of searching along several directions determined by distinct values of weights. Numerical results obtained on a standard test set are provided. These results indicate that the updating techniques allow considerable computational savings when compared with the original Pironneau-Polak method.
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    Smooth Transformation of the Generalized Minimax Problem (1997)
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    Journal of optimization theory and applications 95 (1997), S. 1-24 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unconstrained optimization ; nondifferentiable optimization ; generalized minimax problems ; minimax problems
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    Notes: Abstract We consider the generalized minimax problem, that is, the problem of minimizing a function φ(x)=F(g 1(x),...,g m(x)), where F is a smooth function and each g i is the maximum of a finite number of smooth functions. We prove that, under suitable assumptions, it is possible to construct a continuously differentiable exact barrier function, whose minimizers yield the minimizers of the function φ. In this way, the nonsmooth original problem can be solved by usual minimization techniques for unconstrained differentiable functions.
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    Pivoting in an Outcome Polyhedron (2000)
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    Journal of global optimization 16 (2000), S. 301-323 
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    ISSN: 1573-2916
    Keywords: Outcome polyhedron ; Linear programming ; Pivoting ; Nonlinear programming ; Global optimization ; Extreme point mathematical programming ; Neighborhood problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In many types of linear, convex and nonconvex optimization problems over polyhedra, a global optimal solution can be found by searching the extreme points of the outcome polyhedron Y instead of the extreme points of the decision set polyhedron Z. Since the dimension of Y is often significantly smaller than the dimension of Z, and since the structure of Y is often much simpler than the structure of Z, such an approach has the potential to often yield significant computational savings. This article seeks to motivate these potential savings through both general theory and concrete examples. The article then develops two new procedures. The first procedure is linear-programming based and finds an initial extreme point of an outcome polyhedron Y. The second procedure provides a mechanism for moving from a given extreme point y of Y along any chosen edge of Y emanating from y until a neighboring extreme point to y is reached. As a by-product of the second procedure, as in the pivoting process of the simplex method, a complete algebraic description of the chosen edge can also be easily obtained.
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    Integral global optimization method for solution of nonlinear complementarity problems (1994)
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    Journal of global optimization 5 (1994), S. 181-193 
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    ISSN: 1573-2916
    Keywords: Nonlinear programming ; nonlinear complementarity problem ; integral global optimization
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    Topics: Mathematics
    Notes: Abstract The mapping in a nonlinear complementarity problem may be discontinuous. The integral global optimization algorithm is proposed to solve a nonlinear complementarity problem with a robust piecewise continuous mapping. Numerical examples are given to illustrate the effectiveness of the algorithm.
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    A Framework for Globally Convergent Algorithms Using Gradient Bounding Functions (1999)
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    Journal of optimization theory and applications 100 (1999), S. 661-697 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; linear bounding functions ; global convergence
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    Topics: Mathematics
    Notes: Abstract This paper presents a novel framework for developing globally convergent algorithms without evaluating the value of a given function. One way to implement such a framework for a twice continuously differentiable function is to apply linear bounding functions (LBFs) to its gradient function. The algorithm thus obtained can get a better point in each iteration without using a line search. Under certain conditions, it can achieve at least superlinear convergent rate 1.618 without calculating explicitly the Hessian. Furthermore, the strategy of switching from the negative gradient direction to the Newton-alike direction is derived in a natural manner and is computationally effective. Numerical examples are used to show how the algorithm works.
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    Refinements of necessary conditions for optimality in nonlinear programming (1992)
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    Journal of optimization theory and applications 73 (1992), S. 65-74 
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    Keywords: Nonlinear programming ; lipschitz programming ; necessary optimality conditions ; constraint qualifications
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    Topics: Mathematics
    Notes: Abstract In this paper, a new set of necessary conditions for optimality is introduced with reference to the differentiable nonlinear programming problem. It is shown that these necessary conditions are sharper than the usual Fritz John ones. A constraint qualification relevant to the new necessary conditions is defined and extensions to the locally Lipschitz case are presented.
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    Nonanticipativity in stochastic programming (1985)
    Springer
    Journal of optimization theory and applications 46 (1985), S. 23-30 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; stochastic optimization ; duality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper gives strong duality results in multistage stochastic programming without assuming compactness and without applying induction arguments.
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    URL: http://dx.doi.org/10.1007/BF00938756
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    A comparison between a primal and a dual cutting plane algorithm for posynomial geometric programming problems (1985)
    Springer
    Journal of optimization theory and applications 47 (1985), S. 159-180 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; geometric programming ; posynomial geometric programs ; cutting planes ; subgradients
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, primal and dual cutting plane algorithms for the solution of posynomial geometric programming problems are presented. It is shown that these cuts are deepest, in the sense that they cut off as much of the infeasible set as possible. Problems of nondifferentiability in the dual cutting plane are circumvented by the use of a subgradient. Although the resulting dual problem seems easier to solve, the computational experience seems to show that the primal cutting plane outperforms the dual.
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    URL: http://dx.doi.org/10.1007/BF00940767
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    Necessary conditions for upper semicontinuity in parametric semi-infinite programming (1986)
    Springer
    Journal of optimization theory and applications 48 (1986), S. 65-79 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; semi-infinite programming ; parametric programming ; set-valued mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We treat semi-infinite optimization problems: Minimizep(x) subject tox ∈ ℝ m , anda(t,x) ≦b(t) for allt ∈T, whereT is a σ-compact topological space, andp,a,b are suitable (−∞, ∞]-valued functions on R m , respectively. Linear, convex, and quasi-convex semi-infinite programming are included in this concept. The main results of this paper are on the necessity of the compactness of the set of feasible points for (a,b), and the set of ϕ-optimal solutions for (p,a,b) for the (Hausdorff) upper semicontinuity of the feasible set-mapping in (a,b), and the ϕ-optimal solution-mapping in (p,a,b), respectively (where the parameter sets are provided with a suitable topology). Some more special results complete the paper.
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    URL: http://dx.doi.org/10.1007/BF00938590
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    Convexity and concavity properties of the optimal value function in parametric nonlinear programming (1986)
    Springer
    Journal of optimization theory and applications 48 (1986), S. 95-126 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; parametric analysis ; convexity ; optimal value function ; point-to-set mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Convexity and concavity properties of the optimal value functionf* are considered for the general parametric optimization problemP(ɛ) of the form min x f(x, ɛ), s.t.x εR(ɛ). Such properties off* and the solution set mapS* form an important part of the theoretical basis for sensitivity, stability, and parametric analysis in mathematical optimization. Sufficient conditions are given for several standard types of convexity and concavity off*, in terms of respective convexity and concavity assumptions onf and the feasible region point-to-set mapR. Specializations of these results to the general parametric inequality-equality constrained nonlinear programming problem and its right-hand-side version are provided. To the authors' knowledge, this is the most comprehensive compendium of such results to date. Many new results are given.
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    URL: http://dx.doi.org/10.1007/BF00938592
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    Numerical examples of the behavior of REQP on nonlinear programming problems involving linear dependence among the constraint normals (1986)
    Springer
    Journal of optimization theory and applications 48 (1986), S. 215-227 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; quadratic programming subproblems ; active constraints ; linearly dependent constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This report illustrates, by means of numerical examples, the behavior of the constrained minimization algorithm REQP in situations where the active constraint normals are not linearly independent. The examples are intended to demonstrate that the presence of the penalty parameter in the equations for calculating the Lagrange multiplier estimates enables a useful search direction to be computed. This is shown to be true, whether the dependence among the constraint normals occurs at the solution or in some other region.
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    URL: http://dx.doi.org/10.1007/BF00940670
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    A new constraint qualification condition (1986)
    Springer
    Journal of optimization theory and applications 48 (1986), S. 459-468 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; constraints ; qualification conditions ; linearization ; rank theorem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We introduce a new constraint qualification condition in mathematical programming which encompasses the Mangasarian-Fromovitz's condition and the constant rank condition of Janin. Contrarily to the Mangasarian-Fromovitz's condition, our condition is still satisfied when one translates equalities as double inequalities. It relies on the fact that linearization stability is easier to check with equalities than with inequalities.
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    URL: http://dx.doi.org/10.1007/BF00940572
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    Convergent stepsizes for constrained optimization algorithms (1986)
    Springer
    Journal of optimization theory and applications 49 (1986), S. 135-160 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; stepsize strategies ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A fundamental problem in constrained nonlinear optimization algorithms is the design of a satisfactory stepsize strategy which converges to unity. In this paper, we discuss stepsize strategies for Newton or quasi-Newton algorithms which require the solution of quadratic optimization subproblems. Five stepsize strategies are considered for three different subproblems, and the conditions under which the stepsizes will converge to unity are established. It is shown that these conditions depend critically on the convergence of the Hessian approximations used in the algorithms. The stepsize strategies are constructed using basic principles from which the conditions to unit stepsizes follow. Numerical results are discussed in an Appendix.
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    URL: http://dx.doi.org/10.1007/BF00939251
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    A finite algorithm for finding the projection of a point onto the canonical simplex of ∝ n (1986)
    Springer
    Journal of optimization theory and applications 50 (1986), S. 195-200 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; quadratic programming ; projection onto a simplex ; optimality conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An algorithm of successive location of the solution is developed for the problem of finding the projection of a point onto the canonical simplex in the Euclidean space ℝ n . This algorithm converges in a finite number of steps. Each iteration consists in finding the projection of a point onto an affine subspace and requires only explicit and very simple computations.
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    URL: http://dx.doi.org/10.1007/BF00938486
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    On convergence properties of a least-distance programming procedure for minimization problems under linear constraints (1986)
    Springer
    Journal of optimization theory and applications 50 (1986), S. 365-370 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; linear constraints ; algorithms ; quasiconvex functions ; pseudoconvex functions ; global convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Ref. 1, Bazaraa and Goode provided an algorithm for solving a nonlinear programming problem with linear constraints. In this paper, we show that this algorithm possesses good convergence properties.
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    URL: http://dx.doi.org/10.1007/BF00939279
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    Enhanced methods of feasible directions for engineering design problems (1986)
    Springer
    Journal of optimization theory and applications 51 (1986), S. 475-504 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; methods of feasible directions ; computer-aided design ; engineering design ; linear constraints ; scaling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract After the advantages of methods of feasible directions in an engineering design environment are pointed out, several modifications to the classical scheme are proposed, aimed at improving computational efficiency while preserving convergence properties. First, an abstract algorithm model is set up and a set of sufficient conditions to ensure convergence is given. Several modifications are proposed, inspired by difficulties arising in an engineering design environment, and it is shown that the resulting algorithms satisfy the sufficient conditions just mentioned. An integrated-circuit bipolar operational amplifier is optimized in order to show the improvement in computation efficiency that the proposed enhancements can provide.
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    URL: http://dx.doi.org/10.1007/BF00940286
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    Global convergence and stabilization of unconstrained minimization methods without derivatives (1988)
    Springer
    Journal of optimization theory and applications 56 (1988), S. 385-406 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unconstrained minimization ; nonderivative methods ; line search techniques
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, acceptability criteria for the stepsize and global convergence conditions are established for unconstrained minimization methods employing only function values. On the basis of these results, the convergence of an implementable line search algorithm is proved and some global stabilization schemes are described.
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    URL: http://dx.doi.org/10.1007/BF00939550
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    Some remarks on computer-aided design of optical lens systems (1988)
    Springer
    Journal of optimization theory and applications 59 (1988), S. 173-181 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; global optimization ; optical lens systems ; minimization without derivatives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Computer-aided design of optical lens systems implies the solution of an optimization problem of large dimension. To apply a local algorithm, we want to know a starting point within a small neighborhood of the desired system. For a given task, the designer is not able to define a suitable starting point. In this paper, we describe a global optimization algorithm which uses the principles of biological evolution with the aim of locating an adequate starting point for a local correction routine.
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    A semistrong sufficiency condition for optimality in nonconvex programming and its connection to the perturbation problem (1989)
    Springer
    Journal of optimization theory and applications 60 (1989), S. 7-18 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; nonconvex optimization ; perturbation of nonlinear programs ; sufficiency conditions for optimality
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract What happens when a nonconvex program, having a local solutionx 0 at which the gradients of the binding constraints are linearly independent, but without strict complementarity hypothesis, is perturbed? Under a relatively weak second-order assumption (some nonnegative second-order terms are supposed to be strictly positive), the perturbed problem has, in the neighborhood ofx 0, a finite number of local minima, situated on curves that are connected to some pseudo-solutions of the tangent quadratic problem.
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    A truncated Newton method with nonmonotone line search for unconstrained optimization (1989)
    Springer
    Journal of optimization theory and applications 60 (1989), S. 401-419 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; unconstrained optimization ; Newton's method ; line search techniques ; large-scale optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, an unconstrained minimization algorithm is defined in which a nonmonotone line search technique is employed in association with a truncated Newton algorithm. Numerical results obtained for a set of standard test problems are reported which indicate that the proposed algorithm is highly effective in the solution of illconditioned as well as of large dimensional problems.
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    Parallel algorithms for nonlinear programming problems (1989)
    Springer
    Journal of optimization theory and applications 61 (1989), S. 23-46 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; parallel optimization ; penalty methods ; dual methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes several parallel algorithms for solving nonlinear programming problems. Two approaches where parallelism can successfully be introduced have been explored: a quadratic approximation method based on penalty function and a dual method. These methods are improved by using two algorithms originally proposed for solving unconstrained problems: the parallel variable metric algorithm and the parallel Jacobson-Oksman algorithm. Even though general problems are dealt with, particular emphasis is placed on the potential of these parallel methods for separable programming problems. The numerical effectiveness of the algorithms is demonstrated on a set of test problems using a Cray-1S vector computer and serial computers (with respect to sequential versions of the same methods).
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    URL: http://dx.doi.org/10.1007/BF00940841
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    Multilinear programming: Duality theories (1992)
    Springer
    Journal of optimization theory and applications 72 (1992), S. 459-486 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; duality ; game theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A type of nonlinear programming problem, called multilinear, whose objective function and constraints involve the variables through sums of products is treated. It is a rather straightforward generalization of the linear programming problem. This, and the fact that such problems have recently been encountered in several fields of application, suggested their study, with particular emphasis on the analogies between them and linear problems. This paper develops one such analogy, namely a duality concept which includes its linear counterpart as a special case and also retains essentially all of the desirable characteristics of linear duality theory. It is, however, found that a primal then has several duals. The duals are arrived at by way of a game which is closely associated with a multilinear programming problem, but which differs in some respects from those generally treated in game theory. Its generalizations may in fact be of interest in their own right.
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    URL: http://dx.doi.org/10.1007/BF00939837
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    Trajectory-following algorithms for min-max optimization problems (1992)
    Springer
    Journal of optimization theory and applications 75 (1992), S. 501-519 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; numerical min-max solutions ; trajectory-following algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider a class of nonlinear minimum-maximum optimization problems subject to boundedness constraints on the decision vectors. Three algorithms are developed for finding the min-max point using the concept of solving an associated dynamical system. In the first and third algorithms, solutions are obtained by solving systems of differential equations. The second algorithm is a discrete version of the first algorithm. The trajectories generated by the first and second algorithms may move inside or on the boundary of the constraint set, while the third algorithm ensures that any trajectory that begins inside the constraint region remains in its interior. Sufficient conditions for global convergence of the two algorithms are also established. For illustration, four numerical examples are solved.
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    Equality and inequality constrained optimization algorithms with convergent stepsizes (1993)
    Springer
    Journal of optimization theory and applications 76 (1993), S. 429-453 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; sequential quadratic programming ; nonlinear equality and inequality constraints ; augmented Lagrangians ; adaptive penalty parameter ; convergence analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider two algorithms for nonlinear equality and inequality constrained optimization. Both algorithms utilize stepsize strategies based on differentiable penalty functions and quadratic programming subproblems. The essential difference between the algorithms is in the stepsize strategies used. The objective function in the quadratic subproblem includes a linear term that is dependent on the penalty functions. The quadratic objective function utilizes an approximate Hessian of the Lagrangian augmented by the penalty functions. In this approximation, it is possible to ignore the second-derivative terms arising from the constraints in the penalty functions. The penalty parameter is determined using a strategy, slightly different for each algorithm, that ensures boundedness as well as a descent property. In particular, the boundedness follows as the strategy is always satisfied for finite values of the parameter. These properties are utilized to establish global convergence and the condition under which unit stepsizes are achieved. There is also a compatibility between the quadratic objective function and the stepsize strategy to ensure the consistency of the properties for unit steps and subsequent convergence rates.
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    URL: http://dx.doi.org/10.1007/BF00939376
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    Extended convergence results for the method of multipliers for nonstrictly binding inequality constraints (1993)
    Springer
    Journal of optimization theory and applications 79 (1993), S. 273-310 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; method of multipliers ; penalty Lagrangian methods ; second-order optimality conditions ; nonstrict complementary slackness ; asymptotical linear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we extend the classical convergence and rate of convergence results for the method of multipliers for equality constrained problems to general inequality constrained problems, without assuming the strict complementarity hypothesis at the local optimal solution. Instead, we consider an alternative second-order sufficient condition for a strict local minimum, which coincides with the standard one in the case of strict complementary slackness. As a consequence, new stopping rules are derived in order to guarantee a local linear rate of convergence for the method, even if the current Lagrangian is only asymptotically minimized in this more general setting. These extended results allow us to broaden the scope of applicability of the method of multipliers, in order to cover all those problems admitting loosely binding constraints at some optimal solution. This fact is not meaningless, since in practice this kind of problem seems to be more the rule rather than the exception. In proving the different results, we follow the classical primaldual approach to the method of multipliers, considering the approximate minimizers for the original augmented Lagrangian as the exact solutions for some adequate approximate augmented Lagrangian. In particular, we prove a general uniform continuity property concerning both their primal and their dual optimal solution set maps, a property that could be useful beyond the scope of this paper. This approach leads to very simple proofs of the preliminary results and to a straight-forward proof of the main results.
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    Multilinear programming: Duality theories (1994)
    Springer
    Journal of optimization theory and applications 81 (1994), S. 421-422 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; duality ; game theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It has been pointed out that the text and proof of a proposition in Ref. 1 are worded ambiguously. The version below is intended to be clearer.
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    A Modified SQP Method and Its Global Convergence (1997)
    Springer
    Journal of global optimization 11 (1997), S. 193-205 
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    ISSN: 1573-2916
    Keywords: Nonlinear programming ; SQP method ; pseudo directional derivatives
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The sequential quadratic programming method developed by Wilson, Han andPowell may fail if the quadratic programming subproblems become infeasibleor if the associated sequence of search directions is unbounded. In [1], Hanand Burke give a modification to this method wherein the QP subproblem isaltered in a way which guarantees that the associated constraint region isnonempty and for which a robust convergence theory is established. In thispaper, we give a modification to the QP subproblem and provide a modifiedSQP method. Under some conditions, we prove that the algorithm eitherterminates at a Kuhn–Tucker point within finite steps or generates aninfinite sequence whose every cluster is a Kuhn–Tucker point.Finally, we give some numerical examples.
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    URL: http://dx.doi.org/10.1023/A:1008255227457
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    Convergence properties of local solutions of sequences of mathematical programming problems in general spaces (1974)
    Springer
    Journal of optimization theory and applications 13 (1974), S. 1-12 
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    ISSN: 1573-2878
    Keywords: Nonlinear programming ; approximation theory ; nonconvex programming ; perturbation methods ; sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper gives several sets of sufficient conditions that alocal solutionx k exists of the problem $$\min _{R^k } f^k (x)$$ ,k=1, 2,..., such that {x k } has cluster points that arelocal solutions of a problem of the form min R f(x). The results are based on a well-known concept of topological, orpoint-wise convergence of the sets {R k } toR. Such results have been used to construct and validate large classes of mathematical programming methods based on successive approximations of the problem functions. They are also directly applicable to parametric and sensitivity analysis and provide additional characterizations of optimality for large classes of nonlinear programming problems.
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    URL: http://dx.doi.org/10.1007/BF00935606
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