Penalty and Smoothing Methods for Convex Semi-Infinite Programming

Abstract

In this paper we consider min-max convex semi-infinite programming. In order to solve these problems we introduce a unified framework concerning Remez-type algorithms and integral methods coupled with penalty and smoothing methods. This framework subsumes well-known classical algorithms, but also provides some new methods with interesting properties. Convergence of the primal and dual sequences are proved under minimal assumptions.

Comparative study of RPSALG algorithm for convex semi-infinite programming

“…it is possible to take the regular grid T 0 = T \ r Z m for su¢ ciently large r (see [1,Remark 3.1]). When T is either a full dimensional closed convex sets or the …nite union of pairwise disjoint sets of this class (typically a box or the union of …nitely many disjoint boxes, as it happens in almost all test problems), then T = cl int T by the accessibility lemma.…”

“…A Remez penalty and smoothing algorithm (RPSALG in short) was proposed in [1] to solve min-max convex semi-in…nite programming (SIP) problems of the form (P 0 ) F := inffF (x) : x 2 Cg;…”

“…Our version of RPSALG is a particular case of the uni…ed framework described in [1], inspired by the …rst algorithm of Remez [28], which was proposed for approximating functions in the framework of linear SIP. The basic Remez's algorithm for solving (2) is described in Table 1, in which the step S1 consists of computing a minimizer x k+1 of the ordinary convex program (P k ) obtained by replacing in (2) the index set T by a grid T k ; while step S2 provides the index t k+1 of a most violated constraint at x k+1 : In practice, x k+1 is an approximate solution of (P k ) while t k+1 is the index of some constraint su¢ ciently violated by x k+1 .…”

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Abstract The Remez penalty and smoothing algorithm (RPSALG) is a uni…ed framework for penalty and smoothing methods for solving min-max convex semi-in…nite programing problems, whose convergence was analyzed in a previous paper of three of the authors [1]. RPSALG subsumes wellknown classical algorithms, but also provides some new methods with interesting computational properties.

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Subsmooth semi-infinite and infinite optimization problems

“…When X is finite dimensional, (1.2) is well studied as a semi-infinite optimization problem and has many important and interesting applications in engineering design, control of robots, mechanical stress of materials and social sciences; see the survey paper [15] and the books [3,11,28]. In the last three decades, semi-infinite optimization and its broad range of applications have been an active study area in mathematical programming (see [1,12,18,23,30] and references therein). In particular, many authors have studied first order optimality conditions of semi-infinite optimization problems with linear, convex or smooth data (cf.…”