On Direct Methods for Lexicographic Min-Max Optimization

The approach called the Lexicographic Min-Max (LMM) optimization depends on searching for solutions minimal according to the lex-max order on a multidimensional outcome space. LMM is a refinement of the standard Min-Max optimization, but in the former, in addition to the largest outcome, we minimize also the second largest outcome (provided that the largest one remains as small as possible), minimize the third largest (provided that the two largest remain as small as possible), and so on. The necessity of point-wise ordering of outcomes within the lexicographic optimization scheme causes that the LMM problem is hard to implement. For convex problems it is possible to use iterative algorithms solving a sequence of properly defined Min-Max problems by eliminating some blocked outcomes. In general, it may not exist any blocked outcome thus disabling possibility of iterative Min-Max processing. In this paper we analyze two alternative optimization models allowing to form lexicographic sequential procedures for various nonconvex (possibly discrete) LMM problems. Both the approaches are based on sequential optimization of directly defined artificial criteria. The criteria can be introduced into the original model with some auxiliary variables and linear inequalities thus the methods are easily implementable.