arXiv subjects

Bruno F. Lourenço

Publications and source records attributed to Bruno F. Lourenço.

39 records · Page 3Linked to original sources

arXiv2015

Weak Infeasibility in Second Order Cone Programming

The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a relaxation sequence of feasibility problems that 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 approach 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.

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arXiv2015

A structural geometrical analysis of weakly infeasible SDPs

In this article, we present a geometric theoretical analysis of semidefinite feasibility problems (SDFPs). This is done by decomposing a SDFP into smaller problems, in a way that preserves most feasibility properties of the original problem. With this technique, we develop a detailed analysis of weakly infeasible SDFPs to understand clearly and systematically how weak infeasibility arises in semidefinite programming. In particular, we show that for a weakly infeasible problem over $n\times n$ 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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arXiv2015

A weak constraint qualification for conic programs and a problem on duality gap

We discuss a weak constraint qualification for conic linear programs and its applications for a few classes of cones. This constraint qualification is used to give a solution to a problem proposed by Shapiro and Zǎlinescu and show that if a closed convex cone is such that all its non-trivial faces are polyhedral and all the non-trivial exposed faces of its dual are polyhedral, then the duality gap is zero as long as the primal and dual problems are feasible. Moreover, the common optimal value must be attained at least at one of the sides. We also show an example of a cone that meets the requirements our theorem but is such that previously known results cannot be used to prove its good duality properties.

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