Weak infeasibility in second order cone programming

Lourenço, Bruno F.; Muramatsu, Masakazu; Tsuchiya, Takashi

doi:10.1007/s11590-015-0982-4

Cited by 7 publications

(6 citation statements)

References 16 publications

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“…We also developed an analogous bound for second order cone programs. If K is a product of m Lorentz cones, then we have the bound m for the dimension of the affine space [11]. We note that this is tighter than the bound predicted by [8].…”

Section: Recent Developments and Conclusionmentioning

confidence: 70%

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: Recent Developments and Conclusionmentioning

confidence: 70%

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Lourenço

Muramatsu

Tsuchiya

2016

JORSJ

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“…We will use the following pair of results to certify infeasibility of (1) in cases where it is primal and/or dual strongly infeasible; we refer the reader to [LMT16] for more details on strong and weak infeasibility.…”

Section: Infeasibility Certificatesmentioning

confidence: 99%

Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

Banjac

Goulart

Stellato

et al. 2019

J Optim Theory Appl

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The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well-known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive information regarding problem infeasibility for optimization problems with linear or quadratic objective functions and conic constraints, which includes quadratic, second-order cone, and semidefinite programs. In particular, we show that in the limit the iterates either satisfy a set of first-order optimality conditions or produce a certificate of either primal or dual infeasibility. Based on these results, we propose termination criteria for detecting primal and dual infeasibility.

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“…From ( 22), ( 23), (24) and recalling that θ ≤ 0, we obtain u, z = 0. This implies that C ⊆ H and θ = 0.…”

Section: Proof: Letmentioning

confidence: 97%

“…In [50], Waki showed that weakly infeasible problems sometimes arise from polynomial optimization. There is also a discussion on weak infeasibility semidefinite programming and second-order cone programming in [22,24], respectively. Some of the results in [22] were generalized to arbitrary closed convex cones by Liu and Pataki, see [18] for more details.…”

Section: Background and Previous Workmentioning

confidence: 99%

“…One of the features of this work is the interior point oracle O int and its application to the complete solvability of semidefinite programs. Related to this task, the usage of reducing directions to the construction of almost optimal solutions and almost feasible solutions for general conic linear programs seems to be new, although some of those ideas were present in our previous works on semidefinite programming [22] and second-order cone programming [24]. Technical results on facial reduction and double facial reduction such as Proposition 4.2, Theorems 4.1 and 4.2 seem to be novel as well.…”

Section: Comparison With Other Approachesmentioning

confidence: 99%

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Solving SDP completely with an interior point oracle

Lourenço

Muramatsu

Tsuchiya

2021

Optimization Methods and Software

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We suppose the existence of an oracle which solves any semidefinite programming (SDP) problem satisfying strong feasibility (i.e. Slater's condition) simultaneously at its primal and dual sides. We note that such an oracle might not be able to directly solve general SDPs even after certain regularization schemes are applied. In this work we fill this gap and show how to use such an oracle to 'completely solve' an arbitrary SDP. Completely solving entails, for example, distinguishing between weak/strong feasibility/infeasibility and detecting when the optimal value is attained or not. We will employ several tools, including a variant of facial reduction where all auxiliary problems are ensured to satisfy strong feasibility at all sides. Our main technical innovation, however, is an analysis of double facial reduction, which is the process of applying facial reduction twice: first to the original problem and then once more to the dual of the regularized problem obtained during the first run. Although our discussion is focused on semidefinite programming, the majority of the results are proved for general convex cones.

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Weak infeasibility in second order cone programming

Cited by 7 publications

References 16 publications

A Structural Geometrical Analysis of Weakly Infeasible SDPS

A Structural Geometrical Analysis of Weakly Infeasible SDPS

Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

Solving SDP completely with an interior point oracle

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