Geometry of optimality conditions and constraint qualifications: The convex case

Wolkowicz, Henry

doi:10.1007/bf01581627

Cited by 25 publications

(10 citation statements)

References 21 publications

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“…In fact, this closure has been referred to as a weakest constraint qualification, [21,37]. As an example of a closure condition, see, e.g., [23, pp.…”

Section: Homogenizationmentioning

confidence: 99%

Strong Duality for Semidefinite Programming

Ramana¹,

Tunçel²,

Wolkowicz³

1997

SIAM J. Optim.

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164

133

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Abstract.It is well known that the duality theory for linear programming (LP) is powerful and elegant and lies behind algorithms such as simplex and interior-point methods. However, the standard Lagrangian for nonlinear programs requires constraint qualifications to avoid duality gaps.Semidefinite linear programming (SDP) is a generalization of LP where the nonnegativity constraints are replaced by a semidefiniteness constraint on the matrix variables. There are many applications, e.g., in systems and control theory and combinatorial optimization. However, the Lagrangian dual for SDP can have a duality gap.We discuss the relationships among various duals and give a unified treatment for strong duality in semidefinite programming. These duals guarantee strong duality, i.e., a zero duality gap and dual attainment. This paper is motivated by the recent paper by Ramana where one of these duals is introduced.

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“…In fact, this closure has been referred to as a weakest constraint qualification, [21,37]. As an example of a closure condition, see, e.g., [23, pp.…”

Section: Homogenizationmentioning

confidence: 99%

Strong Duality for Semidefinite Programming

Ramana¹,

Tunçel²,

Wolkowicz³

1997

SIAM J. Optim.

Self Cite

164

133

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“…This is equal to the dimension of the feasible set S. Note that it has been shown in Ref. 16 that Slater's condition holds for (P) if and only if the interior of the feasible set, int S, is nonempty. This is due to the fact that we assume [k, k ~ ~ =, faithfully convex.…”

Section: ~=(E)3~ =mentioning

confidence: 98%

Method of reduction in convex programming

Wolkowicz

1983

J Optim Theory Appl

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Abstract. We present an algorithm which solves a convex program with faithfully convex (not necessarily differentiable) constraints. While finding a feasible starting point, the algorithm reduces the program to an equivalent program for which Slater's condition is satisfied. Included are algorithms for calculating various objects which have recently appeared in the literature. Stability of the algorithm is discussed.

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“…Some good early references on the geometry of the Slater condition, and weak-ened variants, are [57,93,94,144,19]. The concept of facial reduction for general convex programs was introduced in [23,24], while an early application to a semi-definite type best-approximation problem was given in [145].…”

Section: Related Workmentioning

confidence: 99%