对称锥互补问题的非连续内部算法研究

Symmetric cone complementarity problems (SCCP) provide an unified frame-work for the nonlinear complementarity problem, the second-order cone complementarity problem, and the semi-definite complementarity problem. It has wide applications in combination optimization, robust optimization, stochastic optimization, and other fields. In this project, the Euclidean Jordan algebra and the differential properties of the Löwner’s operator and spectral are used as tools to study the non-interior continuous algorithm and its theory for SCCP. Firstly, the property of FB complementarity function and its smoothing function will be analysised. And then, the non-interior continuous algorithm for SCCP with the monotone property and the Cartesian P0-property will be proposed. Finally, the quickly convergence of the proposed algorithm will be analysised by the property of FB complementarity function and its smoothing function. Moreover, the new non-monotone property will be found by the structure character of FB smoothing function and the non-interior continuous algorithmto will be used to slove the new class of non-monotone SCCP. The key problems include how to divide the Rn space properly and to find out the structure character of FB smoothing function by the decomposition theorem of Euclidean Jordan algebra.

对称锥互补问题不仅为标准非线性互补问题、二阶锥互补问题、半定互补问题提供了统一框架，还与组合优化、鲁棒优化、不确定优化等分支有密切联系。本项目拟利用欧几里得若当代数和该空间中Löwner算子和谱函数的微分性质, 研究求解对称锥互补问题的非连续内部算法及其相关理论。包括分析对称锥互补问题的FB互补函数及其光滑逼近函数的性质，建立求解单调对称锥互补问题和满足Cartesian P0性质的对称锥互补问题的非内部连续算法，然后利用FB互补函数及其光滑逼近函数的性质分析所提算法的快速收敛性。进一步，根据FB函数的结构特征，寻找新的非单调性质，将非连续内部算法和完全光滑牛顿法的使用对象扩大至更大的一类非单调对称锥互补问题。拟解决的关键问题包括如何将Rn空间进行合理划分和根据欧几里得若当代数的分解定理来寻找光滑FB函数的结构特征。

对称锥互补问题不仅为标准非线性互补问题、二阶锥互补问题、半定互补问题提供了统一框架，还与组合优化、鲁棒优化、不确定优化等分支有密切联系。本项目利用欧几里得若当代数和该空间中Löwner算子和谱函数的微分性质, 研究了求解对称锥互补问题的光滑化牛顿算法及其相关理论。分析了对称锥互补问题的一类新的互补函数及其光滑逼近函数的性质，建立了求解满足Cartesian P0性质的对称锥互补问题的光滑牛顿算法，然后在非奇异假设之下，利用新互补函数及其光滑逼近函数的性质，得到了所提算法的适定性和快速收敛性。录用论文两篇，参加国际会议一次，国内会议一次。