非线性优化问题的二阶和高阶对偶性

This project aims to study second and higher order duality in nonlinear optimization problems, producing new theory and improvement of existing results on second and higher order dual models and duality theory in nonlinear multiobjective optimization, nonlinear minimax optimization and fractional optimization. Special emphases are: second and higher order converse duality in nonlinear optimization problems; unified second and higher order dual models and duality theory in nonlinear optimization problems; second and higher order duality with cone constrained optimization problems; generalized second derivative and its applications to second order duality in nonlinear optimization problems; and second and higher order duality in nonlinear optimization problems under approximation solutions. Implementation and completion of this project will result in novel new theory enriching nonlinear optimization theory, in particular, duality theory. This new theory will provide fundamental theoretical basis for the design of nonlinear optimization algorithms.

本项目研究非线性优化问题的二阶和高阶对偶性，主要内容是丰富和完善非线性多目标优化、非线性极大极小优化和分式优化的二阶和高阶对偶模型与对偶理论，特别重点研究非线性优化问题二阶和高阶逆对偶性；多类非线性优化问题二阶或高阶模型的统一性及相应对偶性；锥约束优化问题的二阶和高阶对偶性；广义二阶导数在非线性优化问题二阶对偶理论中的应用以及非线性优化问题近似解下的二阶和高阶对偶性等。本项目的实施和完成不仅能够丰富非线性优化理论，特别是对偶理论，而且能够为非线性优化新算法的设计提供理论依据。

本项目研究非线性优化问题的二阶和高阶对偶性，我们按照预期计划主要研究了非线性多目标优化问题二阶和高阶对偶模型中的逆对偶性；研究了可微和不可微锥约束多目标优化问题二阶和高阶对偶性；研究了分式多目标优化问题的二阶和高阶对偶性等。从2013年1月至2016年12月四年间，本项目取得了丰富的研究成果，发表了20多篇SCI论文，包括在中国科学（中，英文），Mathematics of Operations Research和Optimization等刊物,同时项目所取得成果得到国内外同行的大量引用和好评，且项目负责人获得2014年度重庆市自然科学一等奖。