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{uin U} max{iin I} J_{i}(u,z), zinZ . The fact that MMP may be solved indirectly by looking for the saddle point of sum_{iin 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 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 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.

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