Method of reduction in convex programming

Wolkowicz, Henry

doi:10.1007/bf00933505

Cited by 9 publications

(11 citation statements)

References 15 publications

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“…In a special case (fixed O), we recover characterizations of optimal solutions for convex models. The results, presented here on optimal inputs, are quite recent (see [73,75,76,88,89]), while those on optimal solutions of convex programs were formulated in the mid-and late seventies and they are a part of the theory formulated by Ben-Israel, Ben-Tal and Zlobec, and their colleagues, see [6,10,80,81] (abbreviated: BBZ). In the presence of extraneous conditions (such as Slater's condition) the BBZ theory recovers the KKT theory of Karush [23,48,55] and Kuhn and Tucker [49] for convex programs.…”

Section: Input Optimizationmentioning

confidence: 99%

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Characterizing optimality in mathematical programming models

Zlobec

1988

Acta Appl Math

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This paper is a survey of basic results that characterize optimality in single-and multiobjective mathematical programming models. Many people believe, or want to believe, that the underlying behavioural structure of management, economic, and many other systems, generates basically 'continuous' processes. This belief motivates our definition and study of optimality, termed 'structural' optimality. Roughly speaking, we say that a feasible point of a mathematical programming model is structurally optimal if every improvement of the optimal value function, with respect to parameters, results in discontinuity of the. corresponding feasible set of decision variables. This definition appears to be more suitable for many applications and it is also more general than the usual one: every optimum is a structural optimum but not necessarily vice versa. By characterizing structural optima, we obtain some new, and recover the familiar, optimality conditions in nonlinear programming.The paper is self-contained. Our approach is geometric and inductive: we develop intiution by studying finite-dimensional models before moving on to abstract situations.AMS subject clmsilieatiom (1980). 90C25, 90C48, 90C31, 54C10. 54C60, 49B27, 49B50.

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Section: Input Optimizationmentioning

confidence: 99%

“…The above objects are computable, see [1,6,81,90]. We will list various regions of stability in the next section.…”

Section: Definition (I)mentioning

confidence: 99%

Characterizing optimality in mathematical programming models

Zlobec

1988

Acta Appl Math

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“…This problem is a Nonlinear Programming (NLP) problem when the index set T is finite, and is a SIP problem otherwise. It was noticed by many authors that CQ may fail for problem (1) in the presence of the constraints that vanish for any feasible solution (see [2,7,8,15,21] et al ). In different papers these constraints are referred to differently.…”

Section: Introductionmentioning

confidence: 99%

“…In different papers these constraints are referred to differently. For example, for NLP problems with finite set T , they are called "always binding constraints" in [2], and implicit equality constraints in [8], while in [21] such constraints are said to form the equality set of constraints. In [7], where convex semi-infinite systems in the form of the constraints of problem (1) are studied, the indices of the inequalities that vanish for all feasible values of the decision variable, are called carrier indices.…”

Section: Introductionmentioning

confidence: 99%

Optimality criteria without constraint qualifications for linear semidefinite problems

Kostyukova

Tchemisova

2012

J Math Sci

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We consider two closely related optimization problems: a problem of convex SemiInfinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions.

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“…The correct calculation of Ds(x) proves to be vital in solving these programs (see Ref. 3 for a discussion). However, the calculation of Ds(x) is itself an ill-posed problem, i.e., it is discontinuous with respect to small perturbations in the data.…”

Section: Introductionmentioning

confidence: 99%

Numerical decomposition of a convex function

Lamoureux

Wolkowicz

1985

J Optim Theory Appl

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Abstract. Given the n x p orthogonal matrix A and the convex function f : R"-~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an orthogonal direct sum decomposition of the column space of A. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a point x, of a convex function. Applications to convex programming are discussed.

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Method of reduction in convex programming

Cited by 9 publications

References 15 publications

Characterizing optimality in mathematical programming models

Characterizing optimality in mathematical programming models

Optimality criteria without constraint qualifications for linear semidefinite problems

Numerical decomposition of a convex function

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