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  • Articles  (42)
  • nonlinear programming  (42)
  • 2020-2022
  • 1975-1979  (42)
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  • Mathematics  (42)
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  • Articles  (42)
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  • Mathematics  (42)
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  • 1
    Electronic Resource
    Electronic Resource
    Optimality conditions and constraint qualifications in Banach space (1975)
    Springer
    Journal of optimization theory and applications 15 (1975), S. 667-684 
    Details
    ISSN: 1573-2878
    Keywords: Necessary conditions ; mathematical programming ; Banach spaces ; optimization theorems ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, necessary optimality conditions for nonlinear programs in Banach spaces and constraint qualifications for their applicability are considered. A new optimality condition is introduced, and a constraint qualification ensuring the validity of this condition is given. When the domain space is a reflexive space, it is shown that the qualification is the weakest possible. If a certain convexity assumption is made, then this optimality condition is shown to reduce to the well-known extension of the Kuhn-Tucker conditions to Banach spaces. In this case, the constraint qualification is weaker than those previously given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00935506
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  • 2
    Electronic Resource
    Electronic Resource
    Suboptimal control using a second-order parameter-optimization method (1975)
    Springer
    Journal of optimization theory and applications 17 (1975), S. 481-491 
    Details
    ISSN: 1573-2878
    Keywords: Parameter optimization ; suboptimal control ; trajectory optimization ; Newton-Raphson methods ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The optimal control problem is reduced to a suboptimal control problem by assuming the control histories to have particular functional forms involving a number of undetermined constants (Raleigh-Ritz method). A second-order parameter optimization method is discussed and applied to the suboptimal control problem. Also, it is shown that this approach can be used to obtain approximate Lagrange multiplier distributions for optimal control problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932785
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  • 3
    Electronic Resource
    Electronic Resource
    Quadratic multiplier method convergence (1976)
    Springer
    Journal of optimization theory and applications 20 (1976), S. 1-12 
    Details
    ISSN: 1573-2878
    Keywords: Methods of multipliers ; nonlinear programming ; numerical methods ; optimization theorems ; quadratically convergent algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In a recent paper (Ref. 1), the author briefly mentioned a variant of Hestenes' method of multipliers which would converge quadratically. This note examines that method in detail and provides some examples. In the quadratic-linear case, this algorithm converges in one iteration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933344
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  • 4
    Electronic Resource
    Electronic Resource
    On the rate of convergence of some feasible direction algorithms (1976)
    Springer
    Journal of optimization theory and applications 20 (1976), S. 297-313 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; inequality constraints ; numerical methods ; descent methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper is concerned with first-order methods of feasible directions. Pironneau and Polak have recently proved theorems which show that three of these methods have a linear rate of convergence for certain convex problems in which the objective functions have positive definite Hessians near the solutions. In the present note, it is shown that these theorems on rate of convergence can be extended to larger classes of problems. These larger classes are determined in part by certain second-order sufficiency conditions, and they include many nonconvex problems. The arguments used here are based on the finite-dimensional version of Hestenes' indirect sufficiency method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933625
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  • 5
    Electronic Resource
    Electronic Resource
    An accelerated multiplier method for nonlinear programming (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 137-174 
    Details
    ISSN: 1573-2878
    Keywords: Augmented penalty function ; method of multipliers ; penalty function methods ; nonlinear programming ; mathematical programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes an accelerated multiplier method for solving the general nonlinear programming problem. The algorithm poses a sequence of unconstrained optimization problems. The unconstrained problems are solved using a rank-one recursive algorithm described in an earlier paper. Multiplier estimates are obtained by minimizing the error in the Kuhn-Tucker conditions using a quadratic programming algorithm. The convergence of the sequence of unconstrained problems is accelerated by using a Newton-Raphson extrapolation process. The numerical effectiveness of the algorithm is demonstrated on a relatively large set of test problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932517
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  • 6
    Electronic Resource
    Electronic Resource
    A note on the complementarity problem (1977)
    Springer
    Journal of optimization theory and applications 21 (1977), S. 529-530 
    Details
    ISSN: 1573-2878
    Keywords: Complementarity ; mathematical programming ; monotone maps ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show by an example that, in a complementarity problem where the given map is continuous and monotone on the nonnegative orthant, the existence of a feasible solution is not sufficient to guarantee existence of a solution to the complementarity problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933096
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  • 7
    Electronic Resource
    Electronic Resource
    Diagonalized multiplier methods and quasi-Newton methods for constrained optimization (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 135-194 
    Details
    ISSN: 1573-2878
    Keywords: Newton-Raphson method ; quasi-Newton method ; nonlinear programming ; multiplier method ; augmented Lagrangian method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two approaches to quasi-Newton methods for constrained optimization problems inR n are presented. These approaches are based on a class of Lagrange multiplier approximation formulas used by the author in his previous work on Newton's method for constrained problems. The first approach is set in the framework of a diagonalized multiplier method. From this point of view, a new update rule for the Lagrange multipliers which depends on the particular quasi-Newton method employed is given. This update rule, in contrast to most other update rules, does not require exact minimization of the intermediate unconstrained problem. In fact, the optimal convergence rate is attained in the extreme case when only one step of a quasi-Newton method is taken on this intermediate problem. The second approach transforms the constrained optimization problem into an unconstrained problem of the same dimension.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933161
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  • 8
    Electronic Resource
    Electronic Resource
    Generalized reduced gradient method as an extension of feasible direction methods (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 209-226 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; gradient methods ; feasible direction methods ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper presents modifications of the generalized reduced gradient method which allows for a convergence proof. For that, a special construction of the basis is introduced, and some tools of the theory of feasible direction are used to modify the common choice of the direction at every step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933163
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  • 9
    Electronic Resource
    Electronic Resource
    On a characterization of convergence for the Hestenes method of multipliers (1977)
    Springer
    Journal of optimization theory and applications 22 (1977), S. 227-237 
    Details
    ISSN: 1573-2878
    Keywords: Mathematical programming ; nonlinear programming ; method of multipliers ; necessary conditions ; sufficient conditions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The connection between the convergence of the Hestenes method of multipliers and the existence of augmented Lagrange multipliers for the constrained minimum problem (P): minimizef(x), subject tog(x)=0, is investigated under very general assumptions onX,f, andg. In the first part, we use the existence of augmented Lagrange multipliers as a sufficient condition for the convergence of the algorithm. In the second part, we prove that this is also a necessary condition for the convergence of the method and the boundedness of the sequence of the multiplier estimates. Further, we give very simple examples to show that the existence of augmented Lagrange multipliers is independent of smoothness condition onf andg. Finally, an application to the linear-convex problem is given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933164
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  • 10
    Electronic Resource
    Electronic Resource
    Dual to primal conversion in geometric programming (1978)
    Springer
    Journal of optimization theory and applications 26 (1978), S. 243-252 
    Details
    ISSN: 1573-2878
    Keywords: Geometric programming ; duality ; subsidiary problems ; nonlinear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is not to derive new results, but rather to provide insight that will hopefully aid researchers involved in the design and coding of algorithms for geometric programs. The main contributions made here are: (i) a computationally useful interpretation of the Lagrange multipliers associated with the dual orthogonality constraints, (ii) a computationally useful interpretation of the Lagrange multiplier associated with the dual normality constraint, and (iii) an analysis of the much-avoided issue of subsidiary problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00933405
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