Generalized Monotone Line Search SQP Algorithm For Constrained Minimax Problems

It is well known that the algorithms of sequential quadratic programming (SQP) are one of the most classical algorithms for solving nonlinear constrainted optimization problems, and they generally have good convergence properties, numerical experiments also show that SQP algorithms are very effective. However, most of the SQP algorithms are faced with an important problem, that is, at each iteration these algorithms require that the quadratic programming (QP) subproblems must be solvable. Due to the constraints of QP represent a first-ordet linearization of original problem's constraints, so the feasible set of QP may be empty. In order to overcome the shortcoming, there appear some new techniques, such as pivoting operation, introducing new variables. To obtain super linear convergence and avoid Maratos effect, one should solve one or more subprogrammings to get a second-order correction direction, this leads to large scales of computations.In this paper, the nonlinear minimax problems with general constraints are discussed. By means of pivoting operation we construct one QP subproblem to get an improved direction at each iteration. Similarly using the constructional principle of QP subproblem we only need one Systems of Linear Equations (SLE) to obtain a correction direction. In connection with a special merit function the generalized monotone line search is used to yield the step size. So a new algorithm for solving the discussed problems is presented. Under mild conditions, we can obtain the global and superlinear convergence. Remarkably, while adding some computations about approximately active constraint subset, but the constraints number of subproblems is much less than before. Finally, some numerical experiments are operated to test our algorithm, and the results demonstrate that it is satisfactory.