A New Superlinearly Convergent SQP Algorithm for Nonlinear Minimax Problems

Abstract

In this paper, the nonlinear minimax problems are discussed. By means of the Sequential Quadratic Programming (SQP), a new descent algorithm for solving the problems is presented. At each iteration of the proposed algorithm, a main search direction is obtained by solving a Quadratic Programming (QP) which always has a solution. In order to avoid the Maratos effect, a correction direction is obtained by updating the main direction with a simple explicit formula. Under mild conditions without the strict complementarity, the global and superlinear convergence of the algorithm can be obtained. Finally, some numerical experiments are reported.

A sequential quadratically constrained quadratic programming method for unconstrained minimax problems

“…So, many authors have applied the idea of SQP method to present effective algorithms for solving the minimax problems, such as in Refs. [4][5][6][7][8][9][10][11][12]. It is a key problem of various SQP methods to overcome the so-called Maratos effect [13] under suitable conditions, for example, to solve one or more additional quadratic programs or systems of linear equations, or compute explicit correction directions.…”

A hybrid algorithm for nonlinear minimax problems

“…This class of problems occur, for instance, in curve fitting, 1 and ∞ approximation problems, systems of nonlinear equations, problems of finding feasible points of systems of inequalities, nonlinear programming problems, multiobjective problems, engineering design, optimal control and many other situations. The finite minimax problem is a very important class of nonsmooth optimization problems, which has attracted attention from more and more researchers (see Overton 1982;Murray and Overton 1980;Du 1995;Polak et al 1991;Ploak 1989;Fletcher 1982;Yuan 1985;Zhou and Tits 1993;Gaudioso and Monaco 1982;Charalambous and Conn 1978;Han 1981;Vardi 1992;Luksan 1986;Shen and Wang 2005;Di Pillo et al 1993;Dem'yanov and Malozemov 1974;Xu 2001;Gigola and Gomez 1990;Li and Fang 1997;Yi 2002Yi , 2004Jian et al 2007;Watson 1979;Conn et al 2000). Many algorithms have been developed for past decades, they can reduce to two classes.…”

“…For this reason, one is always trying to overcome it, which causes that most of existing algorithms on minimax problem are line search rather than trust region based (see Overton 1982;Polak et al 1991;Charalambous and Conn 1978;Han 1981;Dem'yanov and Malozemov 1974). Recently, Zhou and Tits (1993) proposed a nonmonotone SQP line search method with second-order correction for finite minimax problem (Yi 2002(Yi , 2004) developed a class of line search algorithms based on a new approximation model to problem (1.1), and Jian et al (2007) proposed another SQP line search algorithm for problem (1.1). Typically, under mild assumptions, these algorithms exhibit a locally superlinear (or two-step superlinear) rate of convergence.…”

A New Trust-Region Algorithm for Finite Minimax Problem

“…where x 6 R" and 2; S R. Many line search algorithms were proposed by using this features (see [8][9][10][11][12][13][14][15][16]), under mild assumptions, these methods have good properties of both global convergence and locally superlinear convergence.…”