Lipschitz B-Vex Functions and Nonsmooth Programming

In this paper, the equivalence between the class of B-vex functions and that of quasiconvex functions is proved. Necessary and sufficient conditions, under which a locally Lipschitz function is B-vex, are established in terms of the Clarke subdifferential. Regularity of locally Lipschitz B-vex functions is discussed. Furthermore, under appropriate conditions, a necessary optimality condition of the Slater type and a sufficient optimality condition are obtained for a nonsmooth programming problem involving B-vex functions.     

On Duality Theorems for Nonsmooth Lipschitz Optimization Problems

A nonsmooth Lipschitz vector optimization problem (VP) is considered. Using the Fritz John type necessary optimality conditions for (VP), we formulate the Mond–Weir dual problem (VD) and establish duality theorems for (VP) and (VD) under (strict) pseudoinvexity assumptions on the functions. Our duality theorems do not require a constraint qualification.     

Characterization of Quasiconvexity for Locally Lipschitz Vector-Valued Functions

The aim of the article is to characterize the locally Lipschitz vector-valued functions which are K -quasiconvex with respect to a closed convex cone K in the sense that the sublevel sets are convex. Our criteria are written in terms of a K -quasimonotonicity notion of the generalized directional derivative and of Clarke's generalized Jacobian. This work could be compared to Sach's one in which the author gives necessary and sufficient conditions for a locally Lipschitz map f between two Euclidean spaces to be scalarly K -quasiconvex in the sense that, for any continuous linear form of the nonnegative polar cone K + , the composite function f is quasiconvex.     

局部Lipschitz连续的伪线性函数的性质

本文使用Clarke次微分分析了定义在Banach空间的局部Lipschitz连续的伪线性函数的性质,并且考虑了伪线性规划解集的性质.     

Piecewise Partially Separable Functions and a Derivative-free Algorithm for Large Scale Nonsmooth Optimization

This paper introduces the notion of piecewise partially separable functions and studies their properties. We also consider some of many applications of these functions. Finally, we consider the problem of minimizing of piecewise partially separable functions and develop an algorithm for its solution. This algorithm exploits the structure of such functions. We present the results of preliminary numerical experiments.     

A Globally Convergent Algorithm for a Locally Lipschitz Function

In this paper, an algorithm of global convergence is proposed for a locally Lipschitz function, which is strictly differentiable at almost all differentiable points, and several examples are computed on an IBM PC.     

用于非光滑优化的信赖域算法的一个实施方案

本给出了处理无约束非光滑优化的信赖域算法的一个实施方案。数值实验表明：这种方案是切实可行和可靠的。     

Generalized Univex Functions in Nonsmooth Multiobjective Optimization

In this paper, we have considered a nonsmooth multiobjective optimization problem where the objective and constraint functions involved are directionally differentiable. A new class of generalized functions (d???ρ???η???θ)-type I univex is introduced which generalizes many earlier classes cited in literature. Based upon these generalized functions, we have derived weak, strong, converse and strict converse duality theorems for mixed type multiobjective dual program in order to relate the efficient and weak efficient solutions of primal and dual problem.     

一种约束非光滑优化问题的信赖域算法

提出了一种易实施的求解带线性约束的非光滑优化问题的信赖域算法，并在一定的条件下证明了该算法所产生的迭代序列的任何聚点都是原问题的稳定点．有限的数值例子表明，该方法是行之有效的．     

Second-Order Optimality Conditions in Locally Lipschitz Inequality-Constrained Multiobjective Optimization

Journal of Optimization Theory and Applications - The main goal of this paper is to give some primal and dual Karush–Kuhn–Tucker second-order necessary conditions for the existence of a...     

Dini Set-Valued Directional Derivative in Locally Lipschitz Vector Optimization

The present paper studies the following constrained vector optimization problem: min  _C f(x), g(x)∈−K, h(x)=0, where f:ℝ ⁿ →ℝ ^m , g:ℝ ⁿ →ℝ ^p and h:ℝ ⁿ →ℝ ^q are locally Lipschitz functions and C⊂ℝ ^m , K⊂ℝ ^p are closed convex cones. In terms of the Dini set-valued directional derivative, first-order necessary and first-order sufficient conditions are obtained for a point x ⁰ to be a w-minimizer (weakly efficient point) or an i-minimizer (isolated minimizer of order 1). It is shown that, under natural assumptions (given by a nonsmooth variant of the implicit function theorem for the equality constraints), the obtained conditions improve some given by Clarke and Craven. Further comparison is done with some recent results of Khanh, Tuan and of Jiiménez, Novo.     

Interval Algorithm for a Kind of Nonsmooth Global Optimization

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一类非光滑总体极值的区间算法

本文利用区间分析知识 ,构造了一类 n维非光滑函数总体极值的区间算法 ,理论分析和实例计算均表明本文算法安全可靠 ;能求出全部总体极小点 ;收敛速度也比以前方法[1] 明显加快     

Convergence of an Inexact Algorithm for Composite Nonsmooth Optimization

We describe an inexact version of Fletcher's second-order correctionalgorithm for minimizing composite nondifferentiable functions,and add a test which allows global convergence to be demonstratedwithout the assumption that a global minimum of the model functionis found at each iteration. Implementable criteria for acceptinginexact solutions of the subproblem, while retaining local convergenceproperties, are also given.     

求非光滑全局优化问题的区间算法

本文通过区间工具和目标函数的特殊导数提出了一个非光滑全局优化问题的区间算法，所提出的方法能给出问题的全部全局极小点及全局极小值，理论分析和数值结构均表明本文方法是有效的。     

非光滑多目标最优化的一个精确罚函数

本文将文献[1]中的关于单目标规划的精确罚函数的结果推广到局部Lipschitz多目标规划问题上.