Smoothing Techniques for the Solution of Finite and Semi-Infinite Min-Max-Min Problems

Semi-infinite optimization problems with max-min cost or constraint functions, such as P: minx∈IR n maxy∈Y min z∈Z o (x, y, z), occur in several important areas of engineering design, as well as in economics. These problems are particularly difficult, partly because of the concatenation of max and min operators, and partly because functions of the form max y∈Y minz∈Z o (x, y, z) may fail to have directional derivatives even when o (x, y, z) is smooth. As a result, the literature dealing rigorously with their solution is very small. First we develop a first-order optimality function θ(.) for such problems. Then we show that one can use an adaptive smoothing technique to construct a sequence of finite min-max problems, of the form P N : minx∈IR nmaxy∈YN ω N (x, y), which, together with their optimality functions θ N (.) are consistent approximations to the the original pair (P, θ). This fact opens up the possibility of solving minmax-min problems using a master algorithm that calls min-max algorithms as subroutines.