Applications of Smoothing Functions for Solving Optimization Problems Involving Second-Order Cone

Abstract

無中文摘要In this thesis, we apply smoothing methods for solving two optimization problems over a second-order cone, namely the absolute value equation associated with second-order cone (abbreviated as SOCAVE) and convex second-order cone programming (abbreviated as CSOCP). For SOCAVE, numerical comparisons are presented to illustrate the kind of smoothing functions which work well along with the smoothing Newton algorithm. In particular, the numerical experiments show that the well-known loss function widely used in engineering community is the worst one among the constructed smoothing functions. It indicates that other proposed smoothing functions can be considered for solving engineering problems. For CSOCP, we use the penalty and barrier functions as smoothing functions. These methods are motivated by the work presented in [2]. Under the usual hypothesis that the CSOCP has a nonempty and compact optimal set, we show that the penalty and barrier problems also have a nonempty and compact optimal set. Moreover, any sequence of approximate solutions of these penalty and barrier problems is shown to be bounded whose accumulation points are solutions of the CSOCP. Finally, we provide numerical simulations to illustrate the theoretical results. More specifically, we use various penalty and barrier functions in solving the CSOCP and compare their efficiency by means of performance profiles.