A New Inexact Non-Interior Continuation Algorithm for Second-Order Cone Programming

Liang Fang

doi:10.24297/jam.v16i0.8152

Abstract

Second-order cone programming has received considerable attention in the past decades because of its wide range of applications. Non-interior continuation method is one of the most popular and efficient methods for solving second-order cone programming partially due to its superior numerical performances. In this paper, a new smoothing form of the well-known Fischer-Burmeister function is given. Based on the new smoothing function, an inexact non-interior continuation algorithm is proposed. Attractively, the new algorithm can start from an arbitrary point, and it solves only one system of linear equations inexactly and performs only one line search at each iteration. Moreover, under a mild assumption, the new algorithm has a globally linear and locally Q-quadratical convergence. Finally, some preliminary numerical results are reported which show the effectiveness of the presented algorithm.

Highlights

Second-order cone programming (SOCP for simplicity) is convex optimization in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of several second-order cones
Just like linear programming (LP), convex quadratic programming (CQP) and semidefinite programming (SDP) problems, SOCP problems can be solved in polynomial time by interior-point methods (IPMs) [9]
The computational effort per iteration required by these methods to solve SOCP problems is greater than that required to solve LP and CQP problems but less than that required to solve SDP with the same size and structure

Summary

Introduction

Second-order cone programming (SOCP for simplicity) is convex optimization in which a linear function is minimized over the intersection of an affine linear manifold with the Cartesian product of several second-order cones. Just like LP, CQP and SDP problems, SOCP problems can be solved in polynomial time by interior-point methods (IPMs) [9]. In order to prove their global/local-quadratic convergence, these algorithms either depend on the assumptions of uniform nonsingularity and strict complementarity or need to solve two linear systems of equations and perform at least two line searches at each iteration [13]. Motivated by this direction, the goal of the paper is to propose a new inexact non-interior continuation method for SOCP, which employs a new smoothing function to characterize the central path conditions.

Preliminaries and equivalent formulation of perturbed optimality conditions

A smoothing function and its properties

The inexact non-interior continuation algorithm

Convergence analysis

Conclusions