Feasible Generalized Monotone Line Search SQP Algorithm For Nonlinear MINIMAX Problems With Inequality Constraits

Sequential quadratic programming (SQP) method is an efficient method for solving smooth constrained optimization problems because of its fast convergence rate. Thus many authors have applied the classical SQP method to solve the minimax problems and got some efficient algorithms. But many algorithms for minimax problems with constraints use penalty function, thus they are not the feasible methods, which is a drawback in some strictly feasible cases, especially in the engineering designs. Also, all the algorithms for minimax problems are either monotone or nonmonotone, and they are not unified automatically.In this paper, the nonlinear minimax problems with inequality constraints are discussed. With the help of norm-relaxed method of feasible direction and the nonmonotone line search technique, we present a new feasible SQP algorithm with a generalized monotone line search. At each iteration, a feasible direction of descent is obtained by solving a quadratic programming (QP). To avoid the Maratos effect, a high order correction direction is achieved by solving another QP. As a result, the proposed algorithm has global and superlinear convergence. Especially, the global convergence is obtained under a weak Mangasarian-Promovitz constraint qualification (MFCQ) instead of the linearly independent constraint qualification (LICQ). At last, its numerical effectiveness is demonstrated with test examples.