An extension of Chubanov’s algorithm to symmetric cones

Bruno F. Lourenço, Tomonari Kitahara, Masakazu Muramatsu, Takashi Tsuchiya

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)117-149
Number of pages33
JournalMathematical Programming
Volume173
Issue number1-2
DOIs
Publication statusPublished - Jan 23 2019

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Symmetric Cone
Cones
Semidefinite Programming
Intersection
Second-order Cone Programming
Spectral Norm
Second-order Cone
Half-space
Concretes
Cone
Upper bound
Eigenvalue
Iteration
Norm
Necessary

All Science Journal Classification (ASJC) codes

  • Software
  • Mathematics(all)

Cite this

An extension of Chubanov’s algorithm to symmetric cones. / Lourenço, Bruno F.; Kitahara, Tomonari; Muramatsu, Masakazu; Tsuchiya, Takashi.

In: Mathematical Programming, Vol. 173, No. 1-2, 23.01.2019, p. 117-149.

Research output: Contribution to journalArticle

Lourenço, Bruno F. ; Kitahara, Tomonari ; Muramatsu, Masakazu ; Tsuchiya, Takashi. / An extension of Chubanov’s algorithm to symmetric cones. In: Mathematical Programming. 2019 ; Vol. 173, No. 1-2. pp. 117-149.
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