不确定性互补问题及其鲁棒解的若干研究

项目名称： 不确定性互补问题及其鲁棒解的若干研究

项目编号： No.11201074

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 唐嘉

作者单位： 福建师范大学

项目金额： 22万元

中文摘要： 不确定性互补问题是数学规划中一个新的分支，它与互补理论、鲁棒优化技术以及随机规划等数学分支紧密相联，在工程和经济方面有着广泛应用。目前国内外研究的互补问题均基于确定性的数据，而对数据的不确定性影响有免疫力的稳健算法尚不多见。本项目拟以鲁棒优化为工具，探讨当输入数据不精确或不确定（属于某一不确定集）的互补问题的算法及理论分析。基于新的鲁棒优化模型，讨论不确定性互补问题的可行性、鲁棒解的存在性、唯一性。在此基础上，将求解经典互补问题的有效算法进行推广，充分利用模型和算法的各自优势，使设计的算法对大规模的不确定性互补问题保持高效性和稳健性，以求其解对于"集合"内的任意不确定数据都满足"硬性"约束条件，并且确保最坏情况下的目标函数值达到最优。对这一问题的研究既有理论上的深刻性又有应用前景的广泛性，同时将充实鲁棒优化和互补问题的理论与方法，促进相关学科的发展。

中文关键词： 不确定性互补问题；变分不等式；非线性极小化问题；迭代算法；收敛性

英文摘要： Different from general and classical complementarity problems, uncertain complementarity problems are new branch in mathematical programming which have close correlations with complementarity theory, robust optimization technology, stochastic programming problems and other mathematics branches and have widespread applications in the engineering and economical domains. Till now, the researches on complementary problems are usually based as deterministic data, so steady algorithm which has immunity to the effect of uncertain data still rare to see. This project mainly study theory and algorithm on uncertain complementarity problems for which the data is not specified exactly or uncertain, and it is only known to belong to a given uncertain set. By means of new robust optimization models, we will explore the feasibility of uncertain complementarity problems , the existence and uniqueness of its robust solutions.Furthermore, taking good advantages of the new models and effective algorithm for classical complementarity problems, we will pay much attention to make the extended corresponding algorithm keep high efficient and stable for large-scale uncertain complementarity problems so that the robust solutions will satisfy all hard constrains for each uncertain data in the set and the optimal value of the objective

英文关键词： uncertain complementarity problems；variational inequalities；nonlinear minimization problems；iterative methods；convergnece