概率约束优化问题的交替方向法和全局优化方法研究

Probabilistically constrained optimization problem is one of the most challenging problems. It has many applications in many important fields such as industrial and agricultural production, transportation, banking and securities. The problems with probabilistic constraints have become one of the popular research topics in operations research and management science community in the recent decades. The research of probabilistic constraints has been paid much attention and there are plentiful results of theory and algorithms for the chance constrained problems. But finding a global solution or a good approximation solution to probabilistically constrained optimization problems remains a challenge in both theory and computation, because the feasible region defined by probabilistic constraints is non-convex. This project aims at investigating the alternating direction methods of multipliers and global optimization methods for chance-constrained optimization problems with discrete distributions, and conduct numeric experiments of the algorithms. The research will be pursued in three major thrusts. By using augmented Lagrangian decomposition techniques, we will study the alternating direction method of multipliers, which is based on the 0-1 mixed integer programming reformulations, for the chance-constrained programming problems with discrete distributions, and analyze the convergence of the methods under certain conditions. We will investigate the monotone optimization methods for the chance-constrained problems and develop the corresponding global solution methods based on brand-reduce-and-cut framework. We will also discuss the global convergence of the methods. We will study the portfolio selection models with new Basel Accords risk measures，and propose the corresponding approximate and global solution methods based on the research results of the probabilistically constrained optimization problems. We will also study the performance of the new portfolio selection models and the new risk measures by empirical research. The results obtained in this project will help to enrich the studies on the applications and methodologies of chance-constrained programming.

概率约束优化问题是最具有挑战性的优化问题之一，在工农业生产、交通运输、银行证券等领域有广泛的应用，是近年运筹学和管理科学领域中的一个热点问题。概率约束优化问题的算法研究目前已有许多结果，但由概率约束导致的可行域非凸性使得求解概率约束优化问题的最优解或好的近似解在理论与算法上仍然是一个挑战。本项目旨在研究离散分布下概率约束优化问题的交替方向法和全局优化算法，并进行算法实现。我们将利用增广拉格朗日分解技术提出离散分布下概率约束优化问题的基于0-1混合整数规划变换的乘子交替方向法，并分析算法的收敛性；将研究离散分布下概率约束优化问题的单调化方法，并提出基于分枝-缩减-割框架的全局算法，研究算法的全局收敛性；将利用概率约束方法研究新巴塞尔协议风险度量下的投资模型，提出相应的近似和全局方法，并实证研究模型和新风险度量的表现。本项目的研究结果有助于丰富概率约束优化问题的求解方法和应用研究。

概率约束是处理参数不确定问题的一种重要的优化建模方法。概率约束优化问题在工农业生产、交通运输、银行证券等领域有广泛的应用，是近年运筹优化和管理科学领域中的一个研究热点问题。本项目旨在研究离散分布下概率约束优化问题的求解理论和算法。经过三年的研究，基本实现了立项时的研究目标。本项目主要集中在以下几个研究方向：带离散分布的概率约束优化问题的邻接点交替方向法，带有线性和凸二次约束的非凸二次规划问题的全局算法，带非凸二次约束的非凸二次规划问题的松弛下界问题，以及基于政府规制的航空公司收益管理博弈研究。项目取得了一些较高水平的研究成果，已发表和录用4篇SCI/SSCI/EI学术论文，包括国际上运筹优化和管理科学领域的权威期刊：INFORMS Journal on Computing，Journal of Optimization Theory and Applications， Mathematical Programming Computation，Omega-The International Journal of Management Science. 本项目的研究结果丰富了概率约束优化问题的求解理论和方法研究。