What is Robust Optimal Solution

We define this here for a two-stage minimization problem with decision variables x, recourse variables y, and depending on u, a p-component vector the components of which represent the various parameters subject to uncertainty; assuming that u is known to belong to some given (finite or compact) uncertainty set U in p-dimensional space, the value h(x,v) of any solution x in case u=v (with v an element of U) is the minimum objective function value achievable by adjusting the recourse variables y. The worst-case value w(x) of any given solution x is then defined as the maximum taken over v in U of h(x,v), and an optimal robust solution x* is a solution minimizing w(x). It is thus realized that such a x* is obtained as the result of a Min-Max optimization process. The problem of minimizing w(x) is called the deterministic equivalent to the robust optimization problem under consideration.

Robust Two-Stage and Multistage Optimization: Complexity Issues and Applications

Michel Andre Minoux (University P. and M. Curie, France)