TY - JOUR

T1 - A class of interior proximal-like algorithms for convex second-order cone programming

AU - Pan, Shaohua

AU - Chen, Jein Shan

PY - 2008/6/1

Y1 - 2008/6/1

N2 - We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on (0, +00), and includes as a special case the entropy-like proximal algorithm [Eggermont, Linear Algebra Appl., 130 (1990), pp. 25-42], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the sequence generated is bounded, and every accumulation point is a solution of the considered problem. Preliminary numerical results are reported for two approximate entropy-like proximal algorithms, and numerical comparisons are also made with the merit function approach [Chen and Tseng, Math. Program., 104 (2005), pp. 293-327], which verify the effectiveness of the proposed method.

AB - We propose a class of interior proximal-like algorithms for the second-order cone program, which is to minimize a closed proper convex function subject to general second-order cone constraints. The class of methods uses a distance measure generated by a twice continuously differentiable strictly convex function on (0, +00), and includes as a special case the entropy-like proximal algorithm [Eggermont, Linear Algebra Appl., 130 (1990), pp. 25-42], which was originally proposed for minimizing a convex function subject to nonnegative constraints. Particularly, we consider an approximate version of these methods, allowing the inexact solution of subproblems. Like the entropy-like proximal algorithm for convex programming with nonnegative constraints, we, under some mild assumptions, establish the global convergence expressed in terms of the objective values for the proposed algorithm, and we show that the sequence generated is bounded, and every accumulation point is a solution of the considered problem. Preliminary numerical results are reported for two approximate entropy-like proximal algorithms, and numerical comparisons are also made with the merit function approach [Chen and Tseng, Math. Program., 104 (2005), pp. 293-327], which verify the effectiveness of the proposed method.

KW - Measure of distance

KW - Proximal method

KW - Second-order cone

KW - Second-order cone-convexity

UR - http://www.scopus.com/inward/record.url?scp=67649551067&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649551067&partnerID=8YFLogxK

U2 - 10.1137/070685683

DO - 10.1137/070685683

M3 - Article

AN - SCOPUS:67649551067

VL - 19

SP - 883

EP - 910

JO - SIAM Journal on Optimization

JF - SIAM Journal on Optimization

SN - 1052-6234

IS - 2

ER -