Min-Max Problem

Abstract

The min-max problem is a model for decision making under uncertainty. The aim is to minimize the function f (x, y) but the decision maker only has control of the vector x ∈ R ⁿ. After he selects a value for x, an “opponent” chooses a value for y ∈ R ^m which alternatively can be viewed as a vector of disturbances. When the decision maker is risk averse and has no information about how y will be chosen, it is natural for him to assume the worst. In other words, the second decision maker is completely antagonistic and will try to maximize f (x,y) once x is fixed. The corresponding solution is called the min-max solution and is one of several conservative approaches to decision making under uncertainty. When stochastic information is available for y other approaches might be more appropriate (e.g., see [S4, E3]).