A tractable approach for designing piecewise affine policies in two-stage adjustable robust optimization

• 成果类型: Article
• 署名作者: Ben-Tal, Aharon; El Housni, Omar; Goyal, Vineet
• 署名单位: Technion Israel Institute of Technology; Tilburg University; Columbia University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01385-0
• 发表日期: 2020
• 页码: 57-102
• 关键词: finite adaptability

摘要

We consider the problem of designing piecewise affine policies for two-stage adjustable robust linear optimization problems under right-hand side uncertainty. It is well known that a piecewise affine policy is optimal although the number of pieces can be exponentially large. Asignificant challenge in designing a practical piecewise affine policy is constructing good pieces of the uncertainty set. Here we address this challenge by introducing a new framework in which the uncertainty set is approximated by a dominating simplex. The corresponding policy is then based on amapping from the uncertainty set to the simplex. Although our piecewise affine policy has exponentially many pieces, it can be computed efficiently by solving a compact linear program given the dominating simplex. Furthermore, we can find the dominating simplex in a closed form if the uncertainty set satisfies some symmetries and can be computed using a MIP in general. We would like to remark that our policy is an approximate piecewise-affine policy and is not necessarily a generalization of the class of affine policies. Nevertheless, the performance of our policy is significantly better than the affine policy for many important uncertainty sets, such as ellipsoids and norm-balls, both theoretically and numerically. For instance, for hypersphere uncertainty set, our piecewise affine policy can be computed by an LP and gives a O(m(1/4))-approximation whereas the affine policy requires us to solve a second order cone program and has a worst-case performance bound of O(root m).