Hadamard流形上非光滑区间值多目标优化问题的理论、算法及应用

Hadamard流形上的优化问题是近年来最优化理论的重要研究方向之一，受到了国内外学者的广泛关注。本项目拟对Hadamard流形上的非光滑区间值多目标优化问题进行研究。借用多目标优化中解的概念，通过区间序关系，在Hadamard流形上引入区间值多目标优化问题的有效/弱有效解，建立这些解和对应多目标优化问题相关解的等价性。利用微分流形的基本理论和方法，结合区间分析及向量优化理论，研究解的存在性及稳定性，可行解的充分和必要最优性条件，构造逼近解的迭代算法并分析其收敛性，最后将这些研究成果应用于解决不确定环境下的一些非凸和非光滑多目标优化问题。通过本项目的研究，有望获得Hadamard流形上非光滑区间值多目标优化问题的新理论、新技巧和新算法，丰富和发展运筹学和相关学科的理论方法，对学科进步和社会经济的发展都有重要意义。

Optimization problems on Hadamard manifolds is one of the most important research directions of optimization theory in recent years, which has attracted extensive attention of many scholars. This project aims to study nonsmooth interval multi-objective optimization problems on Hadamard manifolds. By considering the notions of solution in multi-objective optimization, the (weakly) efficient solution for interval multi-objective optimization problem will be introduced on Hadamard manifolds through the interval order relation. We will establish the equivalence between such solution and the relevant solution of the associated multi-objective optimization problem. Utilizing the basic theory and method of differential manifolds, and combining with interval analysis and vector optimization theory, we will study the existence and the stability of the solution, and explore the necessary and sufficient optimality conditions for feasible point. Moreover, we will propose the approximating algorithm and analyze its convergence properties. In addition, we will apply these conclusions to solving some nonconvex and nonsmooth multi-objective optimization problems under uncertain environment. This research is not only expected to obtain new theories, new techniques and new algorithms for nonsmooth interval multi-objective optimization problems on Hadamard manifolds, but also enrich and develop theoretical approaches in Operations Research and related disciplines. It is of great significance to the progress of the disciplines and the development of the social economy.