An extension of Chubanov’s algorithm to symmetric cones

Abstract

In this work we present an extension of Chubanov's algorithm to the case of homogeneous feasibility problems over a symmetric cone 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 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 K is decomposed as simple cones. In several key cases, including semidefinite programming and second order cone programming, these norms make it possible to obtain better complexity bounds for the basic procedure when compared to a recent approach by Peña and Soheili. Finally, in the appendix, we present a translation of the algorithm to the homogeneous feasibility problem in semidefinite programming.

An improved version of Chubanov's method for solving a homogeneous feasibility problem

“…Moreover, some authors successfully generalized Chubanov's method from LO to conic optimization (see, e.g. [7,10,11]). …”

A New Extension of Chubanov's Method to Symmetric Cones

“…If result (3) is returned, the problem is scaled appropriately and the basic procedure is called again. It should be noted that the purpose of rescaling differs between [2] and [11].…”

“…In [2], the authors assumed that the symmetric cone K is given by the Cartesian product of ps simple symmetric cones K 1 , K 2 , . .…”

“…Chubanov's method has also been extended to the feasibility problem over the second-order cone [8] and the symmetric cone [11,2]. The feasibility problem over the symmetric cone is of the form,…”

“…As proposed in [11,2], for problem P(A), the structure of Chubanov's method remains the same; i.e., the main algorithm calls the basic procedure, and the basic procedure returns one of the following in a finite number of iterations:…”