Relaxations and approximations of chance constraints under finite distributions

• 成果类型: Article; Proceedings Paper
• 署名作者: Ahmed, Shabbir; Xie, Weijun
• 署名单位: University System of Georgia; Georgia Institute of Technology; Virginia Polytechnic Institute & State University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-018-1295-z
• 发表日期: 2018
• 页码: 43-65
• 关键词: optimal power-flow; decomposition algorithms; PROGRAMS; formulations; RISK

摘要

Optimization problems with constraints involving stochastic parameters that are required to be satisfied with a prespecified probability threshold arise in numerous applications. Such chance constrained optimization problems involve the dual challenges of stochasticity and nonconvexity. In the setting of a finite distribution of the stochastic parameters, an optimization problem with linear chance constraints can be formulated as a mixed integer linear program (MILP). The natural MILP formulation has a weak relaxation bound and is quite difficult to solve. In this paper, we review some recent results on improving the relaxation bounds and constructing approximate solutions for MILP formulations of chance constraints. We also discuss a recently introduced bicriteria approximation algorithm for covering type chance constrained problems. This algorithm uses a relaxation to construct a solution whose (constraint violation) risk level may be larger than the pre-specified threshold, but is within a constant factor of it, and whose objective value is also within a constant factor of the true optimal value. Finally, we present some new results that improve on the bicriteria approximation factors in the finite scenario setting and shed light on the effect of strong relaxations on the approximation ratios.