A piecewise closed form algorithm for a family of minmax and vector criteria problems

Abstract

This short paper describes the theory and a new algorithm for computing the parameterized solution to a family of minmax problems (MMP):\min{u\in U} \max{i\in I} J_{i}(u,z), z\inZ . The fact that MMP may be solved indirectly by looking for the saddle point of \sum_{i\in I}c_{i}J_{i} (u,z) enables an important special class of MMP to be reduced by analytic manipulation into a family of inequality constrained programming problems. Over partitioning subsets of Z , the solution ω to this latter family of problems may be found by solving appropriate equality constrained problems. Two important new results are established: one concerns the continuity of the solution ω in Z and the other concerns linearity of the interset boundaries separating the partitioning subsets of Z . These results are incorporated into the new algorithm which proves to be excellent for obtaining the parameterized solution of certain types of families of minmax problems.