Explicit gradient information in multiobjective optimization

This work presents an algorithm that converges to points that satisfy a first-order necessary condition of weakly Pareto solutions of multiobjective optimization problems. Hints on how to include second-order information are given. Preliminary numerical results are encouraging.     

A covering theorem for set-valued mappings

The notion of covering is introduced for a set-valued mapping defined on an arbitrary set in a Banach space. A necessary and sufficient covering criterion is proved. The conditions are formulated in terms of generalized differentials and generalized normals. The covering theorem is applied to deduce formulas of generalized differential calculus and necessary optimality conditions for nonsroooth optimization problems.     

Necessary conditions for super minimizers in constrained multiobjective optimization

This paper concerns the study of the so-called super minimizers related to the concept of super efficiency in constrained problems of multiobjective optimization, where cost mappings are generally set-valued. We derive necessary conditions for super minimizers on the base of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.      

A gap between multiobjective optimization and scalar optimization

We give an example to illustrate a gap between multiobjective optimization and single-objective optimization, which solves a problem proposed in Ref. 1.     

An extension of the Hestenes necessary condition

After the pioneering approach of Hestenes, which has not been sufficiently considered, the concept of image of a constrained extremum problem has been recognized recently to be a powerful tool in optimization. Hence in this paper, we consider a necessary optimality condition for a nondifferentiable extremum problem established by Hestenes in the image space and, by again exploiting the concept of image of the problem, a necessary condition embodying that of Hestenes is given. Moreover, the study is completed by expressing the previous result in the space where the unknowns of the given problem run. In this way, we obtain a Lagrangian-type necessary condition holding for a particular class of semidifferentiable problems.     

Lagrange Multipliers for Multiobjective Programs with a General Preference

We consider a nonsmooth multiobjective optimization problems related to a new general preference between infinite dimensional Banach spaces. This preference contains preferences given by generalized Pareto as well as those given by an utility function. We use the concepts of compactly epi-Lipschitzian sets and strongly compactly Lipschitzian mappings to derive Lagrange multipliers of Karush–Kuhn–Tucker type and Fritz-John type in terms of the Ioffe-approximate subdifferentials. Dedicated to Boris Mordukhovich in honour of his 60th birthday.     

多目标优化正则条件的一个注记

给出带不等式约束的非光滑多目标优化问题正则条件的一个例子.通过该例,指出最近由Burachik和Rizvi利用线性化锥提出的可微多目标优化问题的正则条件不能利用Clarke导数推广到非光滑情形.     

A survey of recent developments in multiobjective optimization

Multiobjective Optimization (MO) has many applications in such fields as the Internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is expensive and a time consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for simple problems determining whether a point belongs to the Pareto set is -hard. In this paper, we discuss recent developments in MO. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon Pareto optimal solution, and heuristics.     

A fast Pareto genetic algorithm approach for solving expensive multiobjective optimization problems

We present a new multiobjective evolutionary algorithm (MOEA), called fast Pareto genetic algorithm (FastPGA), for the simultaneous optimization of multiple objectives where each solution evaluation is computationally- and/or financially-expensive. This is often the case when there are time or resource constraints involved in finding a solution. FastPGA utilizes a new ranking strategy that utilizes more information about Pareto dominance among solutions and niching relations. New genetic operators are employed to enhance the proposed algorithm’s performance in terms of convergence behavior and computational effort as rapid convergence is of utmost concern and highly desired when solving expensive multiobjective optimization problems (MOPs). Computational results for a number of test problems indicate that FastPGA is a promising approach. FastPGA yields similar performance to that of the improved nondominated sorting genetic algorithm (NSGA-II), a widely-accepted benchmark in the MOEA research community. However, FastPGA outperforms NSGA-II when only a small number of solution evaluations are permitted, as would be the case when solving expensive MOPs.     

Lagrange necessary conditions for Pareto minimizers in Asplund spaces and applications

In this paper, new necessary conditions for Pareto minimal points to sets and Pareto minimizers for constrained multiobjective optimization problems are established without the sequentially normal compactness property and the asymptotical compactness condition imposed on closed and convex ordering cones in Bao and Mordukhovich [10] and Durea and Dutta [5], respectively. Our approach is based on a version of the separation theorem for nonconvex sets and the subdifferentials of vector-valued and set-valued mappings. Furthermore, applications in mathematical finance and approximation theory are discussed.     

Demyanov Difference of Two Sets and Optimality Conditions of Lagrange Multiplier Type for Constrained Quasidifferentiable Optimization

In the first part of this paper, the Demyanov difference of two sets is considered. An expression for the Demyanov difference of two sets, which are the convex hulls of a finite number of points, is presented. In the second part, first-order necessary optimality conditions of the Lagrange multiplier type, for quasidifferentiable optimization with equality and inequality constraints, are given by means of the Demyanov difference of subdifferential and negative superdifferential.     

Contingent derivative of the perturbation map in multiobjective optimization

Consider a parametrized multiobjective optimization problem with parameteru. LetG(u) be the objective space image of the feasible region, and letW(u)=MinG(u) (the perturbation map) be the efficient set in the objective space. The purpose of this paper is to investigate the relationship between the contingent derivativeDW ofW with respect tou and the contingent derivativeDG ofG with respect tou. Tanino (Ref. 1) proves that MinDGDW under certain conditions. In this paper, we prove that MinDG=MinDW under weaker conditions than Tanino's and that MinDG=DW under certain conditions. The paper does this by introducing a weaker notion of set-valued derivative. Along the way, the paper improves another of Tanino's results by using weaker conditions.The author would like to thank two anonymous referees for helpful comments. The author would also like to thank Professor P. L. Yu for encouragement and useful suggestions.     

A path following method for box-constrained multiobjective optimization with applications to goal programming problems

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.     

Bounded sets of Lagrange multipliers for vector optimization problems in infinite dimension

The aim of this paper is to point out some sufficient constraint qualification conditions ensuring the boundedness of a set of Lagrange multipliers for vectorial optimization problems in infinite dimension. In some (smooth) cases these conditions turn out to be necessary for the existence of multipliers as well.     

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.     

一类非光滑优化问题解集的性质

研究了一类带不等式约束的非光滑优化问题,利用Clarke 次微分和Lagrange 乘子研究该类问题的解集的一些性质,给出了一个例子解释主要结果.主要结论是对最近一些文献中相应结果的改进与推广.     

Efficient solutions in V-KT-pseudoinvex multiobjective control problems: A characterization

In this paper, we introduce a new condition on functionals involved in a multiobjective control problem, for which we define the V-KT-pseudoinvex control problem. We prove that a V-KT-pseudoinvex control problem is characterized so that a Kuhn–Tucker point is an efficient solution. We generalize recently obtained optimality results of known mathematical programming problems and control problems. We illustrate these results with an example.     

Constraint Qualifications for Nonsmooth Mathematical Programs with Equilibrium Constraints

We study nonsmooth mathematical programs with equilibrium constraints. First we consider a general disjunctive program which embeds a large class of problems with equilibrium constraints. Then, we establish several constraint qualifications for these optimization problems. In particular, we generalize the Abadie and Guignard-type constraint qualifications. Subsequently, we specialize these results to mathematical program with equilibrium constraints. In our investigation, we show that a local minimum results in a so-called M-stationary point under a very weak constraint qualification.      

Lagrange multiplier theorem of multiobjective programming problems with set functions

In this paper, Lagrange multiplier theorems are developed for the cases of single-objective and multiobjective programming problems with set functions. Properly efficient solutions are also characterized by subdifferentials and zero-like functions.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

Generalized derivatives and nonsmooth optimization, a finite dimensional tour

During the early 1960’s there was a growing realization that a large number of optimization problems which appeared in applications involved minimization of non-differentiable functions. One of the important areas where such problems appeared was optimal control. The subject of nonsmooth analysis arose out of the need to develop a theory to deal with the minimization of nonsmooth functions. The first impetus in this direction came with the publication of Rockafellar’s seminal work titledConvex Analysis which was published by the Princeton University Press in 1970. It would be impossible to overstate the impact of this book on the development of the theory and methods of optimization. It is also important to note that a large part of convex analysis was already developed by Werner Fenchel nearly twenty years earlier and was circulated through his mimeographed lecture notes titledConvex Cones, Sets and Functions, Princeton University, 1951. In this article we trace the dramatic development of nonsmooth analysis and its applications to optimization in finite dimensions. Beginning with the fundamentals of convex optimization we quickly move over to the path breaking work of Clarke which extends the domain of nonsmooth analysis from convex to locally Lipschitz functions. Clarke was the second doctoral student of R.T. Rockafellar. We discuss the notions of Clarke directional derivative and the Clarke generalized gradient and also the relevant calculus rules and applications to optimization. While discussing locally Lipschitz optimization we also try to blend in the computational aspects of the theory wherever possible. This is followed by a discussion of the geometry of sets with nonsmooth boundaries. The approach to develop the notion of the normal cone to an arbitrary set is sequential in nature. This approach does not rely on the standard techniques of convex analysis. The move away from convexity was pioneered by Mordukhovich and later culminated in the monographVariational Analysis by Rockafellar and Wets. The approach of Mordukhovich relied on a nonconvex separation principle called theextremal principle while that of Rockafellar and Wets relied on various convergence notions developed to suit the needs of optimization. We then move on to a parallel development in nonsmooth optimization due to Demyanov and Rubinov called Quasidifferentiable optimization. They study the class of directionally differentiable functions whose directional derivatives can be represented as a difference of two sublinear functions. On other hand the directional derivative of a convex function and also the Clarke directional derivatives are sublinear functions of the directions. Thus it was thought that the most useful generalizations of directional derivatives must be a sublinear function of the directions. Thus Demyanov and Rubinov made a major conceptual change in nonsmooth optimization. In this section we define the notion of a quasidifferential which is a pair of convex compact sets. We study some calculus rules and their applications to optimality conditions. We also study the interesting notion of Demyanov difference between two sets and their applications to optimization. In the last section of this paper we study some second-order tools used in nonsmooth analysis and try to see their relevance in optimization. In fact it is important to note that unlike the classical case, the second-order theory of nonsmoothness is quite complicated in the sense that there are many approaches to it. However we have chosen to describe those approaches which can be developed from the first order nonsmooth tools discussed here. We shall present three different approaches, highlight the second order calculus rules and their applications to optimization.