Robust Multistage Optimization with Decision-Dependent Uncertainty

Abstract

Quantified integer (linear) programs (QIP) are integer linear programs with variables being either existentially or universally quantified. They can be interpreted as two-person zero-sum games between an existential and a universal player on the one side, or multistage optimization problems under uncertainty on the other side. Solutions are so called winning strategies for the existential player that specify how to react on moves—certain fixations of universally quantified variables—of the universal player to certainly win the game. In this setting the existential player must ensure the fulfillment of a system of linear constraints, while the universal variables can range within given intervals, trying to make the fulfillment impossible. Recently, this approach was extended by adding a linear constraint system the universal player must obey. Consequently, existential and universal variable assignments in early decision stages now can restrain possible universal variable assignments later on and vice versa resulting in a multistage optimization problem with decision-dependent uncertainty. We present an attenuated variant, which instead of an NP-complete decision problem allows a polynomial-time decision on the legality of a move. Its usability is motivated by several examples.