Multistage robust discrete optimization via quantified integer programming

Marc Goerigk,Michael Hartisch

doi:10.1016/j.cor.2021.105434

Marc Goerigk, Michael Hartisch

Open Access

https://doi.org/10.1016/j.cor.2021.105434

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Abstract

Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust discrete optimization has been on the analysis and solution of one- or two-stage problems, where the decision maker has limited options in reacting to additional knowledge gained after parts of the solution have been fixed. Due to its computational difficulty, multistage problems beyond two stages have received less attention.In this paper we argue that multistage robust discrete problems can be seen through the lens of quantified integer programs, where powerful tools to reduce the search tree size have been developed. By formulating both integer and quantified integer programming formulations, it is possible to compare the performance of state-of-the-art solvers from both worlds. Using selection, assignment, lot-sizing and knapsack problems as a testbed, we show that problems with up to nine stages can be solved to optimality in reasonable time.

Highlights

Uncertainty affects most aspects of decision making, and needs to be taken into account preemptively
We provide a deterministic equivalent program (DEP), i.e. an equivalent mixed-integer programs (MIPs), for which each possible scenario sequence must be listed explicitly (see model (2))
We compare the performance of Yasol on the quantified models QIP with polyhedral uncertainty (QIPPU)(Sel) and quantified integer programming (QIP)(Sel) with the performance of CPLEX on the robust counterpart DEP(Sel)

Summary

Introduction

Uncertainty affects most aspects of decision making, and needs to be taken into account preemptively. Different methodologies have been developed for this purpose, such as stochastic programming [48] or robust optimization [7], which is the focus of this paper. We consider discrete optimization problems of the form min c(ξ)x x∈X s.t. A(ξ)x ≤ b(ξ) where X ⊆ Zn is the decision space, c(ξ) is a cost vector, A(ξ) the constraint matrix, and b(ξ) the right-hand side. Vectors are always written in bold font and the transpose sign for the scalar product between vectors is dropped for ease of notation. The problem is affected by uncertainty, expressed by an uncertain parameter ξ

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