An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems with Many Objective Functions

A common strategy for achieving global convergence in the solution of semi-innnite programming (SIP) problems, and in particular of continuous minimax problems, is to (approximately) solve a sequence of discretized problems, with a progressively ner discretization meshes. Finely discretized minimax and SIP problems, as well as other problems with many more objec-tives/constraints than variables, call for algorithms in which successive search directions are computed based on a small but signiicant subset of the objectives/constraints, with ensuing reduced computing cost per iteration and decreased risk of numerical diiculties. In this paper, an SQP-type algorithm is proposed that incorporates this idea in the particular case of minimax problems. The general case will be considered in a separate paper. The quadratic programming subproblem that yields the search direction involves only a small subset of the objective functions. This subset is updated at each iteration in such a way that global convergence is insured. Heuristics are suggested that take advantage of a possible close relationship between \adjacent" objective functions. Numerical results demonstrate the eeciency of the proposed algorithm.