Optimality conditions for directionally differentiable multi-objective programming problems

This paper is concerned with the optimality for multi-objective programming problems with nonsmooth and nonconvex (but directionally differentiable) objective and constraint functions. The main results are Kuhn-Tucker type necessary conditions for properly efficient solutions and weakly efficient solutions. Our proper efficiency is a natural extension of the Kuhn-Tucker one to the nonsmooth case. Some sufficient conditions for an efficient solution to be proper are also given. As an application, we derive optimality conditions for multi-objective programming problems including extremal-value functions.This work was done while the author was visiting George Washington University, Washington, DC.     

广义凸函数及非光滑多目标规划真有效解的充分条件

本文给出了一类广义凸函数的统一定义，在锥意义下，得出了非光滑多目标规划真有效解的充分条件，推广了以往的结论．     

On gap functions for nonsmooth multiobjective optimization problems

A set-valued gap function, \(\phi \), existing in the literature for smooth and nonsmooth multiobjective optimization problems is dealt with. It is known that \(0\in \phi (x^*)\) is a sufficient condition for efficiency of a feasible solution \(x^*\), while the converse does not hold. In the current work, the converse of this assertion is proved for properly efficient solutions. Afterwards, to avoid the complexities of set-valued maps some new single-valued gap functions, for nonsmooth multiobjective optimization problems with locally Lipschitz data are introduced. Important properties of the new gap functions are established.     

Convex composite multi-objective nonsmooth programming

This paper examines nonsmooth constrained multi-objective optimization problems where the objective function and the constraints are compositions of convex functions, and locally Lipschitz and Gâteaux differentiable functions. Lagrangian necessary conditions, and new sufficient optimality conditions for efficient and properly efficient solutions are presented. Multi-objective duality results are given for convex composite problems which are not necessarily convex programming problems. Applications of the results to new and some special classes of nonlinear programming problems are discussed. A scalarization result and a characterization of the set of all properly efficient solutions for convex composite problems are also discussed under appropriate conditions.This research was partially supported by the Australian Research Council grant A68930162.This author wishes to acknowledge the financial support of the Australian Research Council.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Isolated and Proper Efficiencies in Semi-Infinite Vector Optimization Problems

This paper deals with a nonsmooth semi-infinite multiobjective/vector optimization problem (SIMOP, for short). We first establish necessary and sufficient conditions for (local) strongly isolated solutions and (local) positively properly efficient solutions of an SIMOP. Then, we propose a dual problem to the SIMOP under consideration and examine weak and strong duality relations between them.     

Lagrange Multiplier Characterizations of Solution Sets of Constrained Nonsmooth Pseudolinear Optimization Problems

This paper deals with the minimization of a class of nonsmooth pseudolinear functions over a closed and convex set subject to linear inequality constraints. We establish several Lagrange multiplier characterizations of the solution set of the minimization problem by using the properties of locally Lipschitz pseudolinear functions. We also consider a constrained nonsmooth vector pseudolinear optimization problem and derive certain conditions, under which an efficient solution becomes a properly efficient solution. The results presented in this paper are more general than those existing in the literature.     

Concepts of proper efficiency

The centrol concept of proper efficiency has been largely that of Geoffrion. There are, however, other concepts, and this paper considers two of them, viz. those of Klinger and Kuhn and Tucker, in relationship to each other and to Geoffrion. This is done in terms of various properties which characterise the efficient sets. Geoffrion's concept is a global one, whereas the other two concepts are local ones, and their significance is somewhat different. This is examined specifically in the context of optimal solutions, where, for example, it is shown that Geoffrion and Kuhn and Tucker proper efficiency fails to meet an optimality condition, which is satisfied by Klinger proper efficiency.     

Characterization of (weakly/properly/robust) efficient solutions in nonsmooth semi-infinite multiobjective optimization using convexificators

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.     

Proper efficiency and duality for a class of multiobjective fractional variational problems containing arbitrary norms

Necessary and sufficient conditions are established for properly efficient solutions of a class of nonsmooth nonconvex variational problems with multiple fractional objective functions and nonlinear inequality constraints. Based on these proper efficiency criteria. two multiobjective dual problems are constructed and appropriate duality theorems are proved. These proper efficiency and duality results also contain as special cases similar rcsults fer constrained variational problems with multiplei fractional. and conventional objective functions, which are particular cases of the main variational problem considered in this paper     

两类锥广义伪不变凸性的刻画

研究了一类非光滑带约束的向量优化问题. 首先引入锥意义下的 FJ-伪不变凸I(II)型的概念; 然后将经典的Gordan择一定理推广到了带锥的情形,并在此基础上利用FJ向量驻点与(弱)有效解间的关系, 研究了锥FJ-伪不变凸I(II)型的等价刻画.     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

非光滑非凸向量极值问题的真有效解

本文考虑非光滑非凸向量极值问题的真有效解,其主要结果如下:(1)Borwein真有效解与Benson真有效解的等价性;(2)向量极值问题的真有效解与标量极值问题的最优解的等价性;(3)广义鞍点定理;(4)真有效解的必要和充分条件。     

A graphical characterization of the efficient set for?convex multiobjective problems

In this paper, a graphical characterization, in the decision space, of the properly efficient solutions of a convex multiobjective problem is derived. This characterization takes into account the relative position of the gradients of the objective functions and the active constraints at the given feasible solution. The unconstrained case with two objective functions and with any number of functions and the general constrained case are studied separately. In some cases, these results can provide a visualization of the efficient set, for problems with two or three variables. Besides, a proper efficiency test for general convex multiobjective problems is derived, which consists of solving a single linear optimization problem.     

Value functions,domination cones and proper efficiency in multicriteria optimization

In the absence of a clear objective value function, it is still possible in many cases to construct a domination cone according to which efficient (nondominated) solutions can be found. The relations between value functions and domination cones and between efficiency and optimality are analyzed here. We show that such cones must be convex, strictly supported and, frequently, closed as well. Furthermore, in most applications potential optimal solutions are equivalent to properly efficient points. These solutions can often be produced by maximizing with respect to a class of concave functions or, under convexity conditions, a class of affine functions.     

一类（h，Ψ）─—意义下的半无限多目标规划有效解的充分条件

本文利用Ｂｅｎ－Ｔａｌ广义代数运算对一类可微及不可微半无限多目标规划进行了讨论。在目标及约束为（ｈ，）─—凸的情况下得出了有效解的几个充分条件。     

Infine Functions, Nonsmooth Alternative Theorems and Vector Optimization Problems

In this paper we introduce a new notion of infine nonsmooth functions and give several characterizations of infineness property. We prove alternative theorems with mixed constraints (i.e., inequality and equality constraints) being described by invex-infine nonsmooth functions. We establish a necessary and sufficient condition for a solution of a vector optimization problem involving mixed constraints to be a properly efficient solution.     

单调Minkowski泛函与Henig真有效性的标量化

没有凸锥的闭性和点性假设,该文考虑由一般凸锥生成的单调Minkowski泛函并研究其性质.由此,在偏序局部凸空间的框架下,通过利用单调连续Minkowski泛函和单调连续半范,该文分别获得了一般集合及锥有界集合的弱有效点的标量化.利用此弱有效性的标量化,该文分别推导出一般集合及锥有界集合的Henig真有效点的标量化.进而,当序锥具备有界基时,该文获得局部凸空间中超有效性的一些标量化结果.最后,该文给出Henig真有效性和超有效性的稠密性结果.这些结果推广并改进了有关的已知结果.     

Optimality and duality for proper and isolated efficiencies in multiobjective optimization

We use some advanced tools of variational analysis and generalized differentiation such as the nonsmooth version of Fermat’s rule, the limiting/Mordukhovich subdifferential of maximum functions, and the sum rules for the Fréchet subdifferential and for the limiting one to establish necessary conditions for (local) properly efficient solutions and (local) isolated minimizers of a multiobjective optimization problem involving inequality and equality constraints. Sufficient conditions for the existence of such solutions are also provided under assumptions of (local) convex/affine functions or $L$-invex-infine functions defined in terms of the limiting subdifferential of locally Lipschitz functions. In addition, we propose a type of Wolfe dual problems and examine weak/strong duality relations under $L$-invexity-infineness hypotheses.