CLASS OF INTERIOR PROXIMAL-LIKE ALGORITHMS FORCONVEX SECOND-ORDER CONE PROGRAMMING

摘要

We propose a class of interior proximal-like algorithms for the second-order coneprogram, which is to minimize a closed proper convex function subject to general second-ordercone constraints. The class of methods uses a distance measure generated by a twice continuouslydifferentiable strictly convex function on (0, +∞), and includes as a special case the entropy-likeproximal algorithm [Eggermont, Linear Algebra Appl., 130 (1990), pp. 25-42], which was originallyproposed for minimizing a convex function subject to nonnegative constraints. Particularly, weconsider an approximate version of these methods, allowing the inexact solution of subproblems. Likethe entropy-like proximal algorithm for convex programming with nonnegative constraints, we, undersome mild assumptions, establish the global convergence expressed in terms of the objective values forthe proposed algorithm, and we show that the sequence generated is bounded, and every accumulationpoint is a solution of the considered problem. Preliminary numerical results are reported for twoapproximate entropy-like proximal algorithms, and numerical comparisons are also made with themerit function approach [Chen and Tseng, Math. Program., 104 (2005), pp. 293-327], which verifythe effectiveness of the proposed method.