# Maximin without the min

December 29, 2011 Leave a comment

Experience has shown that many scholars/analysts who are not at home with the modeling aspects of the maximin paradigm are surprised to learn that true-blue maximin models such as this

can have an equivalent phrasing, without the iconic inner operation. Here represents a list of constraints on pairs.

The following result, that is used widely in game theory and robust optimization, explains how to dispose of this iconic operation:

.

assuming that the is attained on , where denotes the real line.

The sign indicates that the two optimization problems are *equivalent* in the sense that they yield the same optimal value for the objective functions and the same optimal value(s) for the decision variable . The optimal value of is equal to the optimal value of .

The following is then an immediate implication of the theorem:

.

where consists of all the constraints in , as well as the additional constraint .

Such formulations of maximin models are often called **Mathematical Programming** (MP) formulations. If you use maximin models often, you would do well to develop the skills to switch easily from the iconic formulations to the MP formulations and back.