Selective Gram-Schmidt orthonormalization for conic cutting surface algorithms

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Authors:

John E. Mitchell
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, NY 12180 USA
mitchj at rpi.edu

Vasile L. Basescu
basesv at verizon.net

November 29, 2006. Revised July 16, 2007.

Mathematical Methods of Operations Research, Volume 67, Number 1, pages 91-115, 2008.

Abstract:

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number.

Keywords: Semidefinite programming, conic programming, column generation, cutting plane methods.

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