Bruno F. Lourenço

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 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.

An extension of Chubanov’s algorithm to symmetric cones

In this work we present an extension of Chubanov’s algorithm to the case of homogeneous feasibility problems over a symmetric cone

The p-cones in dimension n ≥ 3 are not homogeneous when p ≠ 2

{\mathcal {K}}

Optimality conditions for nonlinear semidefinite programming via squared slack variables

K. As in Chubanov’s method for linear feasibility problems, the algorithm consists of a basic procedure and a step where the solutions are confined to the intersection of a half-space and

Facial Reduction and Partial Polyhedrality

{\mathcal {K}}

The automorphism group and the non-self-duality of

K. Following an earlier work by Kitahara and Tsuchiya on second order cone feasibility problems, progress is measured through the volumes of those intersections: when they become sufficiently small, we know it is time to stop. We never have to explicitly compute the volumes, it is only necessary to keep track of the reductions between iterations. We show this is enough to obtain concrete upper bounds to the minimum eigenvalues of a scaled version of the original feasibility problem. Another distinguishing feature of our approach is the usage of a spectral norm that takes into account the way that