A Structural Geometrical Analysis of Weakly Infeasible SDPS

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

doi:10.15807/jorsj.59.241

Cited by 13 publications

(22 citation statements)

References 21 publications

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“…A chain of faces of K is a finite sequence of faces satisfying F 1 · · · F and the length is defined to be . For the SDP case, they showed that the bound can be refined to S n + − 2, which matches the bound discussed in [7], since S n + = n + 1. If K = K n 1 × · · · × K n m , then the largest chain of faces of K * has length 2m + 1.…”

Section: The Number Of Directions Needed To Approach Ksupporting

confidence: 72%

“…Recently, Liu and Pataki generalized the results in [7] and showed that the dimension of L +c can be taken to be less or equal than K * −1, see items (ii) and (iii) of Theorem 9 in [5]. The quantity K * corresponds to the length of the longest chain of faces of K * .…”

Section: The Number Of Directions Needed To Approach Kmentioning

confidence: 96%

“…In addition, it is closely related to the issue of closedness of image of closed convex cones, e.g., [1,12]. See [7] for a more detailed review about this issue. We review the literature in more detail in the next section mainly in view of finite/infinite certificates of weak infeasibility.…”

Section: Introductionmentioning

confidence: 99%

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Weak infeasibility in second order cone programming

2015

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The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a sequence of feasibility problems which mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasible problem at most m directions are needed to get arbitrarily close to the cone, where m is the number of Lorentz cones. We also tackle a closely related question and show that given a bounded optimization problem satisfying Slater's condition, we may transform it into another problem that has the same optimal value but it is ensured to attain it. From solutions to the new problem, we discuss how to obtain solution to the original problem which are arbitrarily close to optimality. Finally, we discuss how to obtain finite certificate of weak infeasibility by combining our own techniques with facial reduction. The analysis is similar in spirit to previous work by the authors on SDPs, but a different approach is required to obtain tighter bounds on the number of directions needed to approach the cone.

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Section: The Number Of Directions Needed To Approach Ksupporting

confidence: 72%

Section: The Number Of Directions Needed To Approach Kmentioning

confidence: 96%

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Weak infeasibility in second order cone programming

2015

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“…Also, detection of infeasibility is a complicated task. Some interior point methods, such as the one discussed in [30] by Nesterov, Todd and Ye, are able to obtain a certificate of infeasibility if the problem is dual or primal strongly infeasible, but the situation is less clear in the presence of the so-called weak infeasibility [22]. These issues are also discussed by Karimi and Tunçel in [14], in the context of their software DDS (Domain-Driven Solver) [13,15].…”

Section: Background and Previous Workmentioning

confidence: 99%

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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“…This term is referring to inconsistent problems whose decision problem is ill-posed. For more details, see [31][32][33] and the references therein. For reasons of simplicity, in the following we assume the problems to be either consistent or strictly inconsistent.…”

Section: Cone-lps With Non-degenerate Unique Solutionsmentioning

confidence: 99%

Verification methods for conic linear programming problems

Lange

2020

NOLTA

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This paper gives an overview of verification methods for finite dimensional conic linear programming problems. Besides the computation of verified tight enclosures for unique non-degenerate solutions to well-posed conic linear programming instances, we discuss a rigorous treatment of problems with multiple or degenerate solutions. It will be further shown how a priori knowledge about certain boundedness qualifications can be exploited to efficiently compute verified bounds for the optimal objective value. The corresponding approach is applicable even to ill-posed programming problems. Examples from linear and semidefinite programming are used to illustrate the respective approaches and give further explanations. Another topic is the treatment of programming problems whose parameters are subject to uncertainties. Rough but inclusive estimates for the variability of the corresponding programming problems are given, and also a best and worst case analysis is taken into account. At the end of this paper, special consideration is given to typical issues when applying interval arithmetic in the context of conic linear programming. Different ways are shown on how to resolve these issues.

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A Structural Geometrical Analysis of Weakly Infeasible SDPS

Cited by 13 publications

References 21 publications

Weak infeasibility in second order cone programming

Weak infeasibility in second order cone programming

Solving SDP completely with an interior point oracle

Verification methods for conic linear programming problems

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