几类特殊优化问题的数值方法研究

Mathematical programming with complementarity constraints, semi-infinite programming and minimax optimization are three special kinds of optimization problems which have wide real world applications such as engineering design, optimal control and financial management, etc. However the numerical methods of traditional optimization problems either cannot be applied directly to these problems or have undesirable numerical results. This project aims to put forward a group of new methods with good theoretical convergence and efficient numerical performance regarding the three special kinds of optimization problems mentioned above, together with their derivitives. Our major research content and innovation are the following: (1) to present new smoothing functions, new conversion techniques and new approximation techniques, in order to effectively transform and approximate the objectives; (2) to combine subproblems of new quadratic programming, quadratically constrained quadratic programming and linear equations with the new active set identification, bundle technique and norm-relaxed technique to produce the main search direction and the higher-order correction direction of numerical methods for these problems; (3) to study search technique for effective steps and designing new hybrid line searches which can integrate these three methods: P-strongly sub-feasible direction method, penalty function method and filter method; (4) to weaken the traditional strong assumptions and to obtain the global and superlinear convergence for the proposed numerical methods by using new analyzing methods; (5) to realize the breakthrough of large-scale numerical experiments and developing software packages for practical applications.

互补约束数学规划、半无限规划和极大极小优化等三类特殊优化问题在工程设计、最优控制和金融管理等实际领域有着广泛应用，而传统优化问题的数值方法要么不能直接应用于求解，要么数值效果不理想。本项目以以上三类及其组合衍生的特殊优化问题为研究对象，旨在提出一批具有良好理论收敛性和高效数值表现的新方法。主要研究内容与创新之处有：(1)提出新的光滑化函数、转化技术和逼近技术，对研究对象进行有效转化或近似；(2)使用新型二次规划、二次约束二次规划及线性方程组等子问题，并结合积极集识别新技术、bundle技术和模松弛技术等，构建数值方法的主搜索方向和高阶修正方向；(3)研究有效步长搜索技术，设计P-强次可行方向法、罚函数法及滤子法有机集成的新型杂交线搜索；(4)在减弱传统较强假设条件的基础上，利用新的分析论证技术，获得新数值方法的全局和超线性收敛性；(5)实现大规模数值试验的突破，并制成软件包，以供实际应用。

几类特殊优化问题的数值方法项目始终按原计划展开研究工作，已取得一批有特色、有影响的成果，正式发表学术论文37篇，其中SCI收录20篇，中文核心15篇。成果的主要贡献和创新有：1、非线性极大极小问题的可行下降束方法、序列二次约束二次规划（SQCQP）方法、模松弛序列二次规划（SQP）算法；2、不等式约束优化问题的强次可行原始对偶内点算法、QP-free算法、SQP算法；3、均衡约束优化问题的QP-free类算法和广义梯度投影类算法；4、半无限优化问题的SQCQP模松弛算法和强次可行模松弛SQP算法；5、非光滑约束优化的可行SQP算法、无约束优化的共轭梯度法、乘子交替方向法及非线性半定规划综述。. 在项目经费的资助下，四年期间课题组共主办了4次学术会议、参加了5次学术会议并作学术邀请报告；课题组成员中有1人晋升为教授、3人晋升为副教授、2人晋升为讲师和培养了博士6人，硕士7人。