On the Closedness of the Linear Image of a Closed Convex Cone

Pataki, Gábor

doi:10.1287/moor.1060.0242

Cited by 53 publications

(67 citation statements)

References 22 publications

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“…A comprehensive treatment of the subject was given by Pataki [14]. We will discuss the connection between Pataki's results and weak infeasibility in Section 2.…”

Section: Given a Weakly Infeasible Sdfp How Can We Generate A Weaklymentioning

confidence: 98%

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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“…A comprehensive treatment of the subject was given by Pataki [14]. We will discuss the connection between Pataki's results and weak infeasibility in Section 2.…”

Section: Given a Weakly Infeasible Sdfp How Can We Generate A Weaklymentioning

confidence: 98%

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Lourenço

Muramatsu

Tsuchiya

2016

JORSJ

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“…Since z * is a KKT point of max z Qz : z ∈ 1 m Λ , there are vectors v ∈ R m and u ∈ R M + such that −Qz * = H v + u and u z * = 0 . Now for X := −m Diag (v) we obtain (H X H − Q)z * = H X 1 m e − Qz * = −H v + H v + u = u , so that z * is indeed a KKT point of (12). Now for z ∈ R M + we have z u ≥ 0 = u z * , thus also (z − z * ) u ≥ 0, wherefrom the result (13) follows immediately.…”

Section: Lemma 32mentioning

confidence: 99%

“…In some cases, even if the duality gap is zero and the primal problem has a compact feasible set, the primally optimal solution value need not be attained in the dual problem; see, e.g., Boyd and Vandenberghe [3]. Among the more recent extensions of the Frank-Wolfe theorem are Belousov and Klatte [1], Luo and Zhang [9], Ozdaglar and Tseng [11], and Pataki [12]. However, none of the results there seem to be directly applicable to our problem.…”

Section: Introductionmentioning

confidence: 99%

A Conic Duality Frank–Wolfe-Type Theorem via Exact Penalization in Quadratic Optimization

Schachinger

Bomze

2009

Mathematics of OR

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Abstract. The famous Frank-Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive-semidefiniteness or copositivity conditions, despite the fact that the objective can be even linear. This paper studies exact penalizations of (classical) quadratic programs, i.e. optimization of quadratic functions over a polyhedron, and applies the results to establish a Frank-Wolfe type theorem for the primal-dual pair of a class of conic programs which frequently arises in applications. One result is that uniqueness of the solution of the primal ensures dual attainability, i.e., existence of the solution of the dual.

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“…This result has been shown by Auslender and Teboulle, but using a more specialized argument (see [1], Theorems 2.3.1 and 2.3.3). A very different condition for closedness of AC was given by Pataki [23], who assumes C to be a "nice" closed convex cone (in particular, a polyhedral cone, the second-order cone, or the cone of symmetric positive semidefinite matrices) in addition to a technical constraint qualification involving the range of A , the dual cone C * , and the closure of the set of feasible directions of C * . Pataki's result is different in nature from ours, since it can be seen that neither the second-order cone, which is the epigraph of the Euclidean norm function, nor the cone of n × n symmetric positive semidefinite matrices are retractive.…”

Section: Proposition 6 Let {C K } Be a Nested Sequence Of Nonempty Cmentioning

confidence: 99%

Set Intersection Theorems and Existence of Optimal Solutions

Bertsekas

Tseng

2006

Math. Program.

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Abstract. The question of nonemptiness of the intersection of a nested sequence of closed sets is fundamental in a number of important optimization topics, including the existence of optimal solutions, the validity of the minimax inequality in zero sum games, and the absence of a duality gap in constrained optimization. We consider asymptotic directions of a sequence of closed sets, and introduce associated notions of retractive, horizon, and critical directions, based on which we provide new conditions that guarantee the nonemptiness of the corresponding intersection. We show how these conditions can be used to obtain simple and unified proofs of some known results on existence of optimal solutions, and to derive some new results, including a new extension of the Frank-Wolfe Theorem for (nonconvex) quadratic programming.

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On the Closedness of the Linear Image of a Closed Convex Cone

Cited by 53 publications

References 22 publications

A Structural Geometrical Analysis of Weakly Infeasible SDPS

A Structural Geometrical Analysis of Weakly Infeasible SDPS

A Conic Duality Frank–Wolfe-Type Theorem via Exact Penalization in Quadratic Optimization

Set Intersection Theorems and Existence of Optimal Solutions

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