Incremental subgradient method for nonsmooth convex optimization with fixed point constraints

This paper proposes an incremental subgradient method for solving the problem of minimizing the sum of nondifferentiable, convex objective functions over the intersection of fixed point sets of nonexpansive mappings in a real Hilbert space. The proposed algorithm can work in nonsmooth optimization over constraint sets onto which projections cannot be always implemented, whereas the conventional incremental subgradient method can be applied only when a constraint set is simple in the sense that the projection onto it can be easily implemented. We first study its convergence for a constant step size. The analysis indicates that there is a possibility that the algorithm with a small constant step size approximates a solution to the problem. Next, we study its convergence for a diminishing step size and show that there exists a subsequence of the sequence generated by the algorithm which weakly converges to a solution to the problem. Moreover, we show the whole sequence generated by the algorithm with a diminishing step size strongly converges to the solution to the problem under certain assumptions. We also give examples of real applied problems which satisfy the assumptions in the convergence theorems and numerical examples to support the convergence analyses.     

锥扰动向量最优化问题有效解的锥次微分

目的研究锥扰动集值映射向量优化问题锥有效解的锥次可微性. 方法依据拓扑, 凸分析和泛函分析的理论, 对于锥扰动向量优化问题中特定的集值映射进行分析和研究. 结果分别给出了锥有效解和锥弱有效解的锥次可微和锥弱次可微的充分条件. 结论进一步完善了集值映射向量优化理论.     

Minimal time function with respect to a set of directions: basic properties and applications

In this paper we study a special minimal time function, given with respect to a set of directions. Several properties, concerning continuity, convexity, Lipschitz behaviour and subdifferential calculus are explored. An application to a vectorial location problem is provided.     

Saddle point criteria and Wolfe duality in nonsmooth (Φ, ρ)-invex vector optimization problems with inequality and equality constraints

In this paper, saddle point criteria and Wolfe duality theorems are established for a new class of nondifferentiable vector optimization problems with inequality and equality constraints. The results are proved under nondifferentiable (Φ, ρ)-invexity and related scalar and vector-valued Lagrangians defined for the considered nonsmooth multiobjective programming problem. It turns out that the results are established for such vector optimization problems in which not all functions constituting a vector optimization problem possess the fundamental property of invexity and the most of generalized invexity notions previously defined in the literature.     

(h，φ)-凸函数与(h，φ)-Lipschitz函数的一些广义微分性质

利用函数f与它的对应函数f(t)=φ(f(h~(-1)(t)))之间的关系,研究了(h,φ)-凸函数和(h,φ)- Lipschitz函数的广义方向导数,得到了R~n上连续(h,φ)-凸函效的广义方向导数的有限性、上半连续性以及估值不等式.在f是R~n上的(h,φ)-凸函数的假设下,给出了f为局部(h,φ)-Lipschitz的一个充分必要条件.并讨论了R~n上的(h,φ)-凸函数和(h,φ)-Lipschitz函数的关系,得到了(h,φ)-凸函数的广义次微分的几个基本性质.     

含有Lipschitz B-凸函数约束的非光滑规划问题的最优性条件

根据Clarke次微分 ,建立了局部Lipschitz函数是B_凸的一个必要条件 .并在适当条件下 ,给出了含有Lips chitzB_凸函数的非光滑规划问题的一个Slater型最优性必要条件和最优性充分条件     

Optimality condition of a nonsmooth switching control system

In this paper, a survey and refinement of recent results in discrete optimal control theory are presented. The step control problem depending on a parameter is investigated. Nonsmoothness of the cost function φ is assumed and new versions of the discrete maximum principle for the step control problem are derived. The text was submitted by the author in English.     

次可微多目标规划的最优性条件

本文主要从多目标规划相应的加权问题出发,讨论了在次可微条件下,凸多目标规划问题有效解存在的必要条件和充分条件.本文的结论是对可微情形的一种推广.     

扰动多目标规划的锥超次微分稳定性

引入了一个新的锥次微分——锥超次微分的概念，研究了当确定空间序的控制锥扰动时，它们的锥超有效点集、有效点集、弱有效点集等在锥超次微分下的超次可微性及多目标规划问题在锥超次微分意义下的稳定性．此外，进一步研究了严有效解的性质．