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  • Articles  (37)
  • Articles: DFG German National Licenses  (37)
  • nonlinear programming
  • 2010-2014
  • 1985-1989  (22)
  • 1980-1984  (15)
  • 1950-1954
  • Mathematics  (37)
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  • Articles  (37)
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  • Articles: DFG German National Licenses  (37)
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  • 1
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    Springer
    Mathematical programming 43 (1989), S. 87-95 
    ISSN: 1436-4646
    Keywords: Ellipsoids ; nonlinear programming ; rank-two updates
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study the performance of some rank-two ellipsoid algorithms when used to solve nonlinear programming problems. Experiments are reported which show that the rank-two algorithms studied are slightly less efficient than the usual rank-one (center-cut) algorithm. Some results are also presented concerning the growth of ellipsoid asphericity in rank-one and rank-two algorithms.
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  • 2
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    Annals of operations research 5 (1985), S. 557-573 
    ISSN: 1572-9338
    Keywords: Modeling ; micro-computers ; integer programming ; nonlinear programming ; heuristics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We describe the development and successful implementation of a decision support system now being used by several leading firms in the architecture and space planning industries. The system, which we call SPDS (spatialprogrammingdesignsystem) has the following characteristics: (i) user-friendly convenience features permitting architects and space planners to operate the system without being experienced programmers; (ii) interactive capabilities allowing the user to control and to manipulate relevant parameters, orchestrating conditions to which his or her intuition provides valuable input; (iii) informative and understandable graphics, providing visual displays of interconnections that the computer itself treats in a more abstract methematical form; (iv) convenient ways to change configurations, and to carry out ‘what if’ analyses calling on the system's decision support capabilities; (v) a collection of new methods, invisible to the user, capable of generating good solutions to the mathematical programming problems that underlie each major design component. These new methods succeed in generating high quality solutions to a collection of complex discrete, highly nonlinear problems. While these problems could only be solved in hours, or not at all, with previously existing software, the new methods obtain answers in seconds to minutes on a minicomputer. Major users, including Dalton, Dalton, Newport, and Marshal Erdwin, report numerous advantages of the system over traditional architectural design methods.
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  • 3
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    Annals of operations research 5 (1986), S. 517-538 
    ISSN: 1572-9338
    Keywords: Networks ; nonlinear programming ; relaxation ; dual coordinate descent
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions. The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative internal tactics for the dual problem are examined. Computational experiments — aimed at the improved efficiency of the algorithm — are presented.
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  • 4
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    Mathematical programming 44 (1989), S. 221-234 
    ISSN: 1436-4646
    Keywords: Composite nonsmooth functions ; nonlinear programming ; penalty functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We describe an inexact version of Fletcher's QL algorithm with second-order corrections for minimizing composite nonsmooth functions. The method is shown to retain the global and local convergence properties of the original version, if the parameters are chosen appropriately. It is shown how the inexact method can be implemented, for the case in which the function to be minimized is an exact penalty function arising from the standard nonlinear programming problem. The method can also be applied to the problems of nonlinearl 1 - andl ∞-approximation.
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  • 5
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    Mathematical programming 43 (1989), S. 277-303 
    ISSN: 1436-4646
    Keywords: Sequential quadratic programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible, or if the associated sequence of search directions is unbounded. This paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems. Furthermore, questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed.
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  • 6
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    Applied mathematics & optimization 6 (1980), S. 335-360 
    ISSN: 1432-0606
    Keywords: nonlinear programming ; multiplier methods ; penalty methods ; global convergence ; penalty limitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with penalty function and multiplier methods for the solution of constrained nonconvex nonlinear programming problems. Starting from an idea introduced several years ago by Polak, we develop a class of implementable methods which, under suitable assumptions, produce a sequence of points converging to a strong local minimum for the problem, regardless of the location of the initial guess. In addition, for sequential minimization type multiplier methods, we make use of a rate of convergence result due to Bertsekas and Polyak, to develop a test for limiting the growth of the penalty parameter and thereby prevent ill-conditioning in the resulting sequence of unconstrained optimization problems.
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  • 7
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    Journal of optimization theory and applications 33 (1981), S. 479-495 
    ISSN: 1573-2878
    Keywords: Lagrangians ; nonlinear programming ; Kuhn-Tucker theory ; convex optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For convex optimization inR n,we show how a minor modification of the usual Lagrangian function (unlike that of the augmented Lagrangians), plus a limiting operation, allows one to close duality gaps even in the absence of a Kuhn-Tucker vector [see the introductory discussion, and see the discussion in Section 4 regarding Eq. (2)]. The cardinality of the convex constraining functions can be arbitrary (finite, countable, or uncountable). In fact, our main result (Theorem 4.3) reveals much finer detail concerning our limiting Lagrangian. There are affine minorants (for any value 0〈θ≤1 of the limiting parameter θ) of the given convex functions, plus an affine form nonpositive onK, for which a general linear inequality holds onR nAfter substantial weakening, this inequality leads to the conclusions of the previous paragraph. This work is motivated by, and is a direct outgrowth of, research carried out jointly with R. J. Duffin.
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  • 8
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    Journal of optimization theory and applications 36 (1982), S. 495-519 
    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; Numerical methods ; computational methods ; augmented Lagrangian functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new augmented Lagrangian function is introduced for solving nonlinear programming problems with inequality constraints. The relevant feature of the proposed approach is that, under suitable assumptions, it enables one to obtain the solution of the constrained problem by a single unconstrained minimization of a continuously differentiable function, so that standard unconstrained minimization techniques can be employed. Numerical examples are reported.
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  • 9
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    Journal of optimization theory and applications 30 (1980), S. 161-179 
    ISSN: 1573-2878
    Keywords: Optimization techniques ; nonlinear programming ; direct methods ; numerical methods ; conjugate directions ; nongradient methods ; ridge-path methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A modification based on a linearization of a ridge-path optimization method is presented. The linearized ridge-path method is a nongradient, conjugate direction method which converges quadratically in half the number of search directions required for Powell's method of conjugate directions. The ridge-path method and its modification are compared with some basic algorithms, namely, univariate method, steepest descent method, Powell's conjugate direction method, conjugate gradient method, and variable-metric method. The assessment indicates that the ridge-path method, with modifications, could present a promising technique for optimization.
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  • 10
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    Journal of optimization theory and applications 31 (1980), S. 27-39 
    ISSN: 1573-2878
    Keywords: Least-square methods ; variable-metric methods ; Levenberg-Marquardt methods ; nonlinear programming ; testing algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Computational results are presented for Davidon's new least-square algorithm. Computational experience with this algorithm is reported which motivated the development of a production code version of the algorithm. Several heuristic modifications, which have been added, are described. Fifteen zero-residual test problems have been used in comparing the new production code version with two established versions of the Levenberg-Marquardt algorithm. The production code version of Davidon's least-square algorithm performed faster and used less function evaluations than the Levenberg-Marquardt algorithm in almost every case of the test problems.
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  • 11
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    Journal of optimization theory and applications 31 (1980), S. 361-371 
    ISSN: 1573-2878
    Keywords: Nash-equilibrium solutions ; partially controllable strategies ; nonlinear programming ; complementary eigenvalue problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper deals with a class of nonzero-sum, two-person games with finite strategies when there are constraints on the strategies selected by the players. The constraints arise due to the subjective difficulty that each player often has in assigning to the states probabilities with which he is completely satisfied, and the model specifies how much each player must perturb his initial probability estimate in order to change his maximum utility alternative from the alternative originally best under the initial estimate. It is shown that the Nash-equilibrium solution of this class of nonzero-sum games can be characterized by an equivalent nonlinear program which leads in some cases to a pair of complementary eigenvalue problems. Applications to normal or approximate solutions of linear programming problems are also indicated.
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  • 12
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    Journal of optimization theory and applications 32 (1980), S. 407-425 
    ISSN: 1573-2878
    Keywords: Generalized convexity ; global minimality ; nonlinear programming ; nonconvex programming ; optimization theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.
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  • 13
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    Journal of optimization theory and applications 43 (1984), S. 237-263 
    ISSN: 1573-2878
    Keywords: Geometric programming ; computational comparisons ; nonlinear programming ; ellipsoid algorithm ; generalized reduced gradient algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study the performance of four general-purpose nonlinear programming algorithms and one special-purpose geometric programming algorithm when used to solve geometric programming problems. Experiments are reported which show that the special-purpose algorithm GGP often finds approximate solutions more quickly than the general-purpose algorithm GRG2, but is usually not significantly more efficient than GRG2 when greater accuracy is required. However, for some of the most difficult test problems attempted, GGP was dramatically superior to all of the other algorithms. The other algorithms are usually not as efficient as GGP or GRG2. The ellipsoid algorithm is most robust.
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  • 14
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    Journal of optimization theory and applications 43 (1984), S. 527-541 
    ISSN: 1573-2878
    Keywords: Linear complementarity ; nonlinear programming ; gradient projection method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Levitin-Poljak gradient-projection method is applied to solve the linear complementarity problem with a nonsymmetric matrixM, which is either a positive-semidefinite matrix or aP-matrix. Further-more, if the quadratic functionx T(Mx + q) is pseudoconvex on the feasible region {x ∈R n |Mx + q ≥ 0,x≥0}, then the gradient-projection method generates a sequence converging to a solution, provided that the problem has a solution. For the case when the matrixM is aP-matrix and the solution is nondegenerate, the gradient-projection method is finite.
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  • 15
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    Journal of optimization theory and applications 53 (1987), S. 407-427 
    ISSN: 1573-2878
    Keywords: Sensitivity analysis ; nonlinear programming ; Lipschitzian mappings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the dependence of the solutions and the associated multipliers of a nonlinear programming problem when the data of the problem are subjected to small perturbations. Sufficient conditions are given which imply that the solutions and the multipliers of a perturbed nonlinear programming problem are Lipschitzian with respect to the perturbations.
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  • 16
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    Journal of optimization theory and applications 35 (1981), S. 517-533 
    ISSN: 1573-2878
    Keywords: Two-level planning ; multi-objective systems ; decentralized systems ; resource allocation ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider optimization methods for hierarchical power-decentralized systems composed of a coordinating central system and plural semi-autonomous local systems in the lower level, each of which possesses a decision making unit. Such a decentralized system where both central and local systems possess their own objective function and decision variables is a multi-objective system. The central system allocates resources so as to optimize its own objective, while the local systems optimize their own objectives using the given resources. The lower level composes a multi-objective programming problem, where local decision makers minimize a vector objective function in cooperation. Thus, the lower level generates a set of noninferior solutions, parametric with respect to the given resources. The central decision maker, then, parametric with respect to the given resources. The central decision maker, then, chooses an optimal resource allocation and the best corresponding noninferior solution from among a set of resource-parametric noninferior solutions. A computational method is obtained based on parametric nonlinear mathematical programming using directional derivatives. This paper is concerned with a combined theory for the multi-objective decision problem and the general resource allocation problem.
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  • 17
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    Journal of optimization theory and applications 37 (1982), S. 1-21 
    ISSN: 1573-2878
    Keywords: Sensitivity analysis ; geometric programming ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A unified approach to computing first, second, or higher-order derivatives of any of the primal and dual variables or multipliers of a geometric programming problem, with respect to any of the problem parameters (term coefficients, exponents, and constraint right-hand sides) is presented. Conditions under which the sensitivity equations possess a unique solution are developed, and ranging results are also derived. The analysis for approximating second and higher-order sensitivity generalizes to any sufficiently smooth nonlinear program.
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  • 18
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    Journal of optimization theory and applications 40 (1983), S. 333-348 
    ISSN: 1573-2878
    Keywords: Numerical optimization ; global search ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper describes a new version, known as CRS2, of the author's controlled random search procedure for global optimization (CRS). The new procedure is simpler and requires less computer storage than the original version, yet it has a comparable performance. The results of comparative trials of the two procedures, using a set of standard test problems, are given. These test problems are examples of unconstrained optimization. The controlled random search procedure can also be effective in the presence of constraints. The technique of constrained optimization using CRS is illustrated by means of examples taken from the field of electrical engineering.
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  • 19
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    Journal of optimization theory and applications 44 (1984), S. 701-721 
    ISSN: 1573-2878
    Keywords: Kuhn-Tucker points ; local and global minima ; nonlinear programming ; Morse functions ; convex transformable programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Consider minimizingf onD which is diffeomorphic to a disk. Under a genericity assumption, the number of points onD satisfying the Kuhn-Tucker necessary conditions for minimum is odd. We give conditions which imply that a local minimum is global and a necessary and sufficient condition that a Kuhn-Tucker point is the solution. Convex transformable problems satisfy the latter condition.D may be of full dimension or be embedded on a manifold or it may be given by a system of concave inequalities.
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  • 20
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    Journal of optimization theory and applications 54 (1987), S. 253-271 
    ISSN: 1573-2878
    Keywords: Global optimization ; multiextremal optimization ; optimization algorithms ; nonlinear programming ; branch-and-bound methods ; Lipschitzian optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general class of derivative-free optimization procedures is presented including the corresponding convergence theory. This theory turns out to be very constructive, in the sense that the convergence conditions not only can be verified easily for many existing algorithms, but also allow one to construct new procedures. It is shown that popular methods such as branch-and-bound concepts, Pintér's general class of procedures, the algorithms of Pijavskii, Shubert, and Mladineo, and the approach of Zheng and Galperin can not only be subsumed under this class of methods, but also partly be improved by regarding them within the framework presented.
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  • 21
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    Journal of optimization theory and applications 45 (1985), S. 533-543 
    ISSN: 1573-2878
    Keywords: Dynamic optimization ; Ritz parametrization of the control functions ; nonlinear systems ; nonlinear programming ; projection method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Using Ritz's procedure of representing the control functions of an optimal control problem by a function series with parameters to be optimized, it is shown that, from the well-known gradient procedure for dynamic problems, a simple iteration formula for the optimization of these parameters can be derived. Using an example with a technical background, the effectiveness of the program realization of this approach is demonstrated and is compared with the results of unrestricted dynamic optimization.
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  • 22
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    Journal of optimization theory and applications 57 (1988), S. 399-410 
    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; penalty functions ; exact penalty functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we consider a class of nondifferentiable penalty functions for the solution of nonlinear programming problems without convexity assumptions. Preliminarily, we introduce a notion of exactness which appears to be of relevance in connection with the solution of the constrained problem by means of unconstrained minimization methods. Then, we show that the class of penalty functions considered is exact, according to this notion.
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  • 23
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    Journal of optimization theory and applications 36 (1982), S. 477-494 
    ISSN: 1573-2878
    Keywords: Unconstrained optimization ; variable-metric methods ; quasi-Newton methods ; numerical algorithms ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Quasi-Newton algorithms minimize a functionF(x),x ∈R n, searching at any iterationk along the directions k=−H kgk, whereg k=∇F(x k) andH k approximates in some sense the inverse Hessian ofF(x) atx k. When the matrixH is updated according to the formulas in Broyden's family and when an exact line search is performed at any iteration, a compact algorithm (free from the Broyden's family parameter) can be conceived in terms of the followingn ×n matrix: $$H{_R} = H - Hgg{^T} H/g{^T} Hg,$$ which can be viewed as an approximating reduced inverse Hessian. In this paper, a new algorithm is proposed which uses at any iteration an (n−1)×(n−1) matrixK related toH R by $$H_R = Q\left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & K \\ \end{array} } \right]Q$$ whereQ is a suitable orthogonaln×n matrix. The updating formula in terms of the matrixK incorporated in this algorithm is only moderately more complicated than the standard updating formulas for variable-metric methods, but, at the same time, it updates at any iteration a positive definite matrixK, instead of a singular matrixH R. Other than the compactness with respect to the algorithms with updating formulas in Broyden's class, a further noticeable feature of the reduced Hessian algorithm is that the downhill condition can be stated in a simple way, and thus efficient line searches may be implemented.
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  • 24
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    Journal of optimization theory and applications 35 (1981), S. 159-182 
    ISSN: 1573-2878
    Keywords: Variable penalty methods ; nonlinear programming ; sequential unconstrained minimization technique ; approximations ; Hessian matrix ; penalty methods ; ill-conditioning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of generalized variable penalty formulations for solving nonlinear programming problems is presented. The method poses a sequence of unconstrained optimization problems with mechanisms to control the quality of the approximation for the Hessian matrix, which is expressed in terms of the constraint functions and their first derivatives. The unconstrained problems are solved using a modified Newton's algorithm. The method is particularly applicable to solution techniques where an approximate analysis step has to be used (e.g., constraint approximations, etc.), which often results in the violation of the constraints. The generalized penalty formulation contains two floating parameters, which are used to meet the penalty requirements and to control the errors in the approximation of the Hessian matrix. A third parameter is used to vary the class of standard barrier or quasibarrier functions, forming a branch of the variable penalty formulation. Several possibilities for choosing such floating parameters are discussed. The numerical effectiveness of this algorithm is demonstrated on a relatively large set of test examples.
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  • 25
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    Journal of optimization theory and applications 59 (1988), S. 165-172 
    ISSN: 1573-2878
    Keywords: Cluster theory ; artificial intelligence ; decision rules ; optical lens systems ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Optical lens design problems are solved by numerical optimization. If we describe an optical lens system by constructive parameters, we may identify such a system by a vectorx ∈ ℝ n . To find an adequate system for a given problem, we have to organize the optimization process in a parameter space with a fixed dimensionn. Often, such an optimization leads to inadequate results. For this reason, we have to alter the number of lenses contained in the system. Consequently, we have to change the dimension of the space for the parametric description of the lens system. This paper describes a special sequential cluster algorithm to realize the move from one parameter space to another by computation.
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  • 26
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    Journal of optimization theory and applications 50 (1986), S. 479-493 
    ISSN: 1573-2878
    Keywords: Constrained optimization ; nonlinear programming ; differential exact penalty functions ; computational methods ; augmented Lagrangian functions
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    Topics: Mathematics
    Notes: Abstract In this work, we study a differentiable exact penalty function for solving twice continuously differentiable inequality constrained optimization problems. Under certain assumptions on the parameters of the penalty function, we show the equivalence of the stationary points of this function and the Kuhn-Tucker points of the restricted problem as well as their extreme points. Numerical experiments are presented that corroborate the theory, and a rule is given for choosing the parameters of the penalty function.
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  • 27
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    Journal of optimization theory and applications 57 (1988), S. 323-339 
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; nonlinear least squares ; quasi-Newton methods
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    Notes: Abstract In this paper, the classical Gauss-Newton method for the unconstrained least squares problem is modified by introducing a quasi-Newton approximation to the second-order term of the Hessian. Various quasi-Newton formulas are considered, and numerical experiments show that most of them are more efficient on large residual problems than the Gauss-Newton method and a general purpose minimization algorithm based upon the BFGS formula. A particular quasi-Newton formula is shown numerically to be superior. Further improvements are obtained by using a line search that exploits the special form of the function.
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  • 28
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    Journal of optimization theory and applications 58 (1988), S. 259-282 
    ISSN: 1573-2878
    Keywords: Optimization ; nonlinear programming ; numerical methods ; computational methods ; augmented Lagrangian functions
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    Topics: Mathematics
    Notes: Abstract In this paper, a new continuously differentiable exact augmented Lagrangian is introduced for the solution of nonlinear programming problems with compact feasible set. The distinguishing features of this augmented Lagrangian are that it is radially unbounded with respect to the multiplier and that it goes to infinity on the boundary of a compact set containing the feasible region. This allows one to establish a complete equivalence between the unconstrained minimization of the augmented Lagrangian on the product space of problem variables and multipliers and the solution of the constrained problem.
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  • 29
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    Journal of optimization theory and applications 60 (1989), S. 453-473 
    ISSN: 1573-2878
    Keywords: Sequential quadratic programming ; large-scale programming ; nonlinear programming ; incomplete Cholesky factorization ; quadratic programming ; dual methods
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    Topics: Mathematics
    Notes: Abstract Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.
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    Journal of optimization theory and applications 62 (1989), S. 77-98 
    ISSN: 1573-2878
    Keywords: Sequential gradient-restoration algorithm ; nonlinear programming ; gradient-type optimization ; feasibility restoration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 1, the original SGRA algorithm is described and is compared with two other related methods: the gradient projection and the generalized reduced gradient methods. Next, the special case of linear equalities is analyzed. It is shown that, in this case, only the gradient-type minimization phase is needed, and the SGRA becomes identical to the steepest-descent method. Convergence proofs for the nonlinearly constrained case are given in Part 2.
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    Journal of optimization theory and applications 62 (1989), S. 99-125 
    ISSN: 1573-2878
    Keywords: Sequential gradient-restoration algorithm ; nonlinear programming ; gradient-type optimization ; feasibility restoration
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    Topics: Mathematics
    Notes: Abstract The sequential gradient-restoration algorithm (SGRA) was developed in the late 1960s for the solution of equality-constrained nonlinear programs and has been successfully implemented by Miele and coworkers (Refs. 2 and 3) on many large-scale problems. The algorithm consists of two major sequentially applied phases. The first is a gradient-type minimization in a subspace tangent to the constraint surface, and the second is a feasibility restoration procedure. In Part 2, the convergence properties of the SGRA for the general case of nonlinear constraints are analyzed. It is shown that, for analytical convergence purposes, the feasibility restoration phase plays a crucial role. A slight modification of the original restoration algorithm is proposed, and global convergence of the modified version is proven. Finally, a slightly modified version of the complete algorithm is presented and global convergence is proven. The asymptotic rate of convergence of the SGRA is also analyzed. The reader is assumed to be familiar with the problem statement and the description of the SGRA, presented in Part 1 (Ref. 1).
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  • 32
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    Journal of optimization theory and applications 53 (1987), S. 509-515 
    ISSN: 1573-2878
    Keywords: Duality theorems ; Mond-Weir duals ; duality ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Recently, Gulati and Craven and Mond and Egudo established strict converse duality theorems for some of Mond-Weir duals for nonlinear programming problems. Here, we establish various duality theorems under weaker convexity conditions that are different from those of Gulati and Craven, Mond and Weir, and Mond and Egudo.
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  • 33
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    Journal of optimization theory and applications 54 (1987), S. 121-141 
    ISSN: 1573-2878
    Keywords: Global minimization ; global optimization ; search trajectories ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new multi-start algorithm for global unconstrained minimization is presented in which the search trajectories are derived from the equation of motion of a particle in a conservative force field, where the function to be minimized represents the potential energy. The trajectories are modified to increase the probability of convergence to a comparatively low local minimum, thus increasing the region of convergence of the global minimum. A Bayesian argument is adopted by which, under mild assumptions, the confidence level that the global minimum has been attained may be computed. When applied to standard and other test functions, the algorithm never failed to yield the global minimum.
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  • 34
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    Journal of optimization theory and applications 57 (1988), S. 85-121 
    ISSN: 1573-2878
    Keywords: Sensitivity analysis ; geometric programming ; duality ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Sensitivity analysis results for general parametric posynomial geometric programs are obtained by utilizing recent results from nonlinear programming. Duality theory of geometric programming is exploited to relate the sensitivity results derived for primal and dual geometric programs. The computational aspects of sensitivity calculations are also considered.
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  • 35
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    Journal of optimization theory and applications 55 (1987), S. 133-146 
    ISSN: 1573-2878
    Keywords: Numerical optimization ; global search ; nonlinear programming ; parallel processing ; concurrent algorithms ; computer-aided design
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes two new versions of the controlled random search procedure for global optimization (CRS). Designed primarily to suit the user of a CAD workstation, these algorithms can also be used effectively in other contexts. The first, known as CRS3, speeds the final convergence of the optimization by combining a local optimization algorithm with the global search procedure. The second, called CCRS, is a concurrent version of CRS3. This algorithm is intended to drive an optimizing accelerator, based on a concurrent processing architecture, which can be attached to a workstation to achieve a significant increase in speed. The results are given of comparative trials which involve both unconstrained and constrained optimization.
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  • 36
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    Journal of optimization theory and applications 57 (1988), S. 223-238 
    ISSN: 1573-2878
    Keywords: Derivative-free optimization ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The cubic algorithm (Ref. 1) is a nongradient method for the solution of multi-extremal, nonconvex Lipschitzian optimization problems. The precision and complexity of this algorithm are studied, and improved computational schemes are proposed.
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  • 37
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    Journal of optimization theory and applications 59 (1988), S. 353-367 
    ISSN: 1573-2878
    Keywords: Semi-infinite programming ; discretization ; approximation ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, a new method for semi-infinite programming problems with convex constraints is presented. The method generates a sequence of feasible points whose cluster points are solutions of the original problem. No maximization over the index set is required. Some computational results are also presented.
    Type of Medium: Electronic Resource
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