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

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.