A reformulation-linearization technique for optimization over simplices

• 成果类型: Article
• 署名作者: Selvi, Aras; den Hertog, Dick; Wiesemann, Wolfram
• 署名单位: Imperial College London; University of Amsterdam
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-021-01726-y
• 发表日期: 2023
• 页码: 427-447
• 关键词: global optimization; algorithm

摘要

We study non-convex optimization problems over simplices. We show that for a large class of objective functions, the convex approximation obtained from the Reformulation-Linearization Technique (RLT) admits optimal solutions that exhibit a sparsity pattern. This characteristic of the optimal solutions allows us to conclude that (i) a linear matrix inequality constraint, which is often added to tighten the relaxation, is vacuously satisfied and can thus be omitted, and (ii) the number of decision variables in the RLT relaxation can be reduced from O(n(2)) to O(n). Taken together, both observations allow us to reduce computation times by up to several orders of magnitude. Our results can be specialized to indefinite quadratic optimization problems over simplices and extended to non-convex optimization problems over the Cartesian product of two simplices as well as specific classes of polyhedral and non-convex feasible regions. Our numerical experiments illustrate the promising performance of the proposed framework.