非线性标量化及其在向量优化问题中的应用

项目名称： 非线性标量化及其在向量优化问题中的应用

项目编号： No.11301567

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 陈纯荣

作者单位： 重庆大学

项目金额： 23万元

中文摘要： 本项目以向量（多目标）优化领域中著名的非线性标量化工具- - Gerstewitz(简称G)函数为主要研究对象。随着向量优化研究的深入发展，G函数的现有成果已不能完全支撑研究需要，对G函数的更多有用性质开发和应用及其改进或推广也变得非常必要和迫切。围绕G函数的性质及其应用，我们的研究内容包括：（一）研究G函数当前存在和需要解决的关键问题：Lipschitz性和强单调性，以及变动控制结构下的函数构造及其性质；（二）借助G标量化技巧，在非凸框架下研究参数向量拟平衡问题的解映射的Holder连续性；（三）利用G函数辅助设计向量值优化问题的Newton型迭代方法，在目标映射弱凸性设置下对算法的可行性和收敛性进行分析。本课题研究将进一步完善和充实G函数的性质及其应用，从而丰富和发展向量优化研究的有关理论、方法与技巧，同时也为现实世界中广泛存在于各个领域的多目标决策提供必要的理论支持。

中文关键词： 向量优化；非线性标量化；Gerstewitz函数；稳定性分析；向量平衡问题

英文摘要： This project mainly focuses on the Gerstewitz function, which as a powerful tool of nonlinear scalarization is well-known in vector (multiobjective) optimization. With the development of vector optimization, current researches on the Gerstewitz function could not fully support the requirements in vector optimization. So it is very necessary and important to further exploit many useful properties of the Gerstewitz function with applications, also including its various modifications and generalizations. To study more properties and applications of the Gerstewitz function, it will concentrate on the following issues: (i) the key problems about the Gerstewitz function itself, such as the Lipschitz continuity, the strong monotonicity, and the Gerstewitz function with a variable domination structure; (ii) the application of Gerstewitz nonlinear scalarization on Holder continuity of solutions to parametric vector quasiequilibrium problems without convexity; (iii) the design of Newton-type methods for vector-valued optimization with the help of the Gerstewitz function, together with the feasibility and convergence analysis for the proposed algorithm under weak convexity. These researches will enrich and complete the properties and applications of the Gerstewitz function, thus develop related theory, methods and techniqu

英文关键词： Vector optimization；Nonlinear scalarization；Gerstewitz function；Stability analysis；Vector equilibrium problem