An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

Klep, Igor; Schweighofer, Markus

doi:10.1287/moor.1120.0584

Cited by 40 publications

(65 citation statements)

References 37 publications

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“…See the comments after Theorem 3.5 in Sturm [24] and also Theorem 7.5.1 of [12] for another certificate of infeasibility. Klep and Schweighofer [6] also developed certificates for infeasibility and a hierarchy of infeasibility in which 0-infeasibility corresponds to strong infeasibility and k-infeasibility to weak infeasibility, when k > 0. Liu and Pataki [7] also introduced an infeasibility certificate for semidefinite programming.…”

Section: C) Is Feasible If and Only If Rd(k N L C) Is Infeasiblementioning

confidence: 99%

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Lourenço

Muramatsu

Tsuchiya

2016

JORSJ

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In this article, we develop a detailed analysis of semidefinite feasibility problems (SDFPs) to understand how weak infeasibility arises in semidefinite programming. This is done by decomposing a SDFP into smaller problems, in a way that preserves most feasibility properties of the original problem. The decomposition utilizes a set of vectors (computed in the primal space) which arises when we apply the facial reduction algorithm to the dual feasible region. In particular, we show that for a weakly infeasible problem over n × n matrices, at most n − 1 directions are required to approach the positive semidefinite cone. We also present a discussion on feasibility certificates for SDFPs and related complexity results.

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Section: C) Is Feasible If and Only If Rd(k N L C) Is Infeasiblementioning

confidence: 99%

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Lourenço

Muramatsu

Tsuchiya

2016

JORSJ

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“…Correspondingly we refer to thin and thick linear pencils L as those for which D L (1) is thin, or respectively thick. A paper of Klep and Schweighofer gives an iterative process for finding a set of linear polynomials in R[x] whose zero set defines the affine subspace in which a spectrahedron lies [KS13,§3].…”

Section: Operations On Setsmentioning

confidence: 99%

Noncommutative polynomials nonnegative on a variety intersect a convex set

Helton¹,

Klep²,

Nelson³

2014

Journal of Functional Analysis

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Abstract. By a result of Helton and McCullough [HM12], open bounded convex free semialgebraic sets are exactly open (matricial) solution sets D • L of a linear matrix inequality (LMI) L(X) ≻ 0. This paper gives a precise algebraic certificate for a polynomial being nonnegative on a convex semialgebraic set intersect a variety, a so-called "Perfect" Positivstellensatz.For example, given a generic convex free semialgebraic set D • L we determine all "(strong sense) defining polynomials" p for D • L . Such polynomials must have the formwhere q i , r j are matrices of polynomials, and C j are real matrices satisfying C j L = LC j . This follows from our general result for a given linear pencil L and a finite set I of rows of polynomials. A matrix polynomial p is positive where L is positive and all ι ∈ I vanish if and only ifwhere each p i , q j and r k are matrices of polynomials of appropriate dimension, and each ι k is an element of the "L-real radical" of I. In this representation, we can restrict p i , q i , ι k and r k to be elements of a low-dimensional subspace of matrices of polynomials, and in particular, their degrees depend in a very tame way only on the degree of p and the degrees of the elements of I. Further, this paper gives an efficient algorithm for computing the L-real radical of I. This Positivstellensatz extends and unifies two different lines of results: (1) the free real Nullstellensatz of [CHMN13, Nel] which gives an algebraic certificate corresponding to one polynomial being zero on the free variety where others are zero; this is (P) with L = 1; (2) the convex Positivstellensatz of [HKM12, KS11] which is (P) without I; i.e., I = {0}. The representation (P) has a number of additional consequences which will be presented.

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“…This statement is equivalent to Sturm's Farkas' Lemma for semidefinite programming (see Lemmas 3.1.1 and 3.1.2 in [17]) and a variation of Theorem 2.21 in [9], and its proof is provided for completeness.…”

Section: Infeasible Systems and Block Structurementioning

confidence: 94%

Irreducible Infeasible Subsystems of Semidefinite Systems

Kellner¹,

Pfetsch

Theobald

2019

J Optim Theory Appl

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Farkas' lemma for semidefinite programming characterizes semidefinite feasibility of linear matrix pencils in terms of an alternative spectrahedron. In the wellstudied special case of linear programming, a theorem by Gleeson and Ryan states that the index sets of irreducible infeasible subsystems are exactly the supports of the vertices of the corresponding alternative polyhedron.We show that one direction of this theorem can be generalized to the nonlinear situation of extreme points of general spectrahedra. The reverse direction, however, is not true in general, which we show by means of counterexamples. On the positive side, an irreducible infeasible block subsystem is obtained whenever the extreme point has minimal block support. Motivated by results from sparse recovery, we provide a criterion for the uniqueness of solutions of semidefinite block systems.

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An Exact Duality Theory for Semidefinite Programming Based on Sums of Squares

Cited by 40 publications

References 37 publications

A Structural Geometrical Analysis of Weakly Infeasible SDPS

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Noncommutative polynomials nonnegative on a variety intersect a convex set

Irreducible Infeasible Subsystems of Semidefinite Systems

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