Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives

In this paper necessary and sufficient conditions for strict minimizers of a general vector optimization problem are given by means of different notions of graphical or epigraphical derivatives, extending some existing results in the literature. In first place, these conditions are established by means of contingent derivatives and secondly, by applying a nonconvex separation result, in terms of contingent epiderivatives and hypoderivatives. Moreover, through a variational approach, a scalarization method is developed in order to obtain scalar versions of these results.     

Scalarization and pointwise well-posedness in vector optimization problems

The aim of this paper is applying the scalarization technique to study some properties of the vector optimization problems under variable domination structure. We first introduce a nonlinear scalarization function of the vector-valued map and then study the relationships between the vector optimization problems under variable domination structure and its scalarized optimization problems. Moreover, we give the notions of DH-well-posedness and B-well-posedness under variable domination structure and prove that there exists a class of scalar problems whose well-posedness properties are equivalent to that of the original vector optimization problem.     

Connectedness of the Solution Sets and Scalarization for Vector Equilibrium Problems

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem. This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jinxing Province, China.     

Well-Posedness and Scalarization in Vector Optimization

In this paper, we study several existing notions of well- posedness for vector optimization problems. We separate them into two classes and we establish the hierarchical structure of their relationships. Moreover, we relate vector well-posedness and well-posedness of an appropriate scalarization. This approach allows us to show that, under some compactness assumption, quasiconvex problems are well posed.The authors thank Professor C. Zălinescu for pointing out some inaccuracies in Ref. 11. His remarks allowed the authors to improve the present work.     

Well-posedness for vector equilibrium problems

We introduce and study two notions of well-posedness for vector equilibrium problems in topological vector spaces; they arise from the well-posedness concepts previously introduced by the same authors in the scalar case, and provide an extension of similar definitions for vector optimization problems. The first notion is linked to the behaviour of suitable maximizing sequences, while the second one is defined in terms of Hausdorff convergence of the map of approximate solutions. In this paper we compare them, and, in a concave setting, we give sufficient conditions on the data in order to guarantee well-posedness. Our results extend similar results established for vector optimization problems known in the literature.      

Set-valued increasing-along-rays maps and well-posed set-valued star-shaped optimization

In this paper we introduce a class of set-valued increasing-along-rays maps and present some properties of set-valued increasing-along-rays maps. We show that the increasing-along-rays property of a set-valued map is close related to the corresponding set-valued star-shaped optimization. By means of increasing-along-rays property, we investigate stability and well-posedness of set-valued star-shaped optimization.     

Extended Well-Posedness of Quasiconvex Vector Optimization Problems

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. Research partially supported by the Cariplo Foundation, Grant 2006.1601/11.0556, Cattaneo University, Castellanza, Italy.     

Scalarization of vector optimization problems

In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.The author is grateful to the two referees for many valuable comments and suggestions which led to major imporvements of the paper.     

Image Space Analysis and Scalarization of Multivalued Optimization

Using a new method based on generalized sections of feasible sets, we obtain optimality conditions for vector optimization of objective multifunctions with multivalued constraints. The authors express their sincere gratitude to Professor F. Giannessi and the referees for comments and valuable suggestions. The second author was partially supported by the Center of Excellence for Mathematics (University of Isfahan).     

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.     

A new well-posed nonlinear nonlocal diffusion

A new nonlinear diffusion is proposed and analyzed. It is characterized by a nonlocal dependence in the diffusivity which manifests itself through the presence of a fractional power of the Laplacian. The equation is related to the well-known and ill-posed Perona-Malik equation of image processing. It shares with the latter some of its most cherished features while being well-posed. Local and global well-posedness results are presented along with numerical experiments which illustrate its interesting dynamical behavior mainly due to the presence of a class of metastable non-trivial equilibria.     

Solutions of vectorial Hamilton-Jacobi equations and minimizers of nonquasiconvex functionals

We provide a unified approach to prove existence results for the Dirichlet problem for Hamilton-Jacobi equations and for the minimum problem for nonquasiconvex functionals of the Calculus of Variations with affine boundary data. The idea relies on the definition of integro-extremal solutions introduced in the study of nonconvex scalar variational problem.     

Well-posedness for set optimization problems

In this paper, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz’s function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function, respectively. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.     

On the choice of parameters for the weighting method in vector optimization

We present a geometrical interpretation of the weighting method for constrained (finite dimensional) vector optimization. This approach is based on rigid movements which separate the image set from the negative of the ordering cone. We study conditions on the existence of such translations in terms of the boundedness of the scalar problems produced by the weighting method. Finally, using recession cones, we obtain the main result of our work: a sufficient condition under which weighting vectors yield solvable scalar problems. An erratum to this article can be found at     

Well-posedness and convexity in vector optimization

We study a notion of well-posedness in vector optimization through the behaviour of minimizing sequences of sets, defined in terms of Hausdorff set-convergence. We show that the notion of strict efficiency is related to the notion of well-posedness. Using the obtained results we identify a class of well-posed vector optimization problems: the convex problems with compact efficient frontiers.     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

Scalarization in vector optimization

In this paper some scalar optimization problems are presented whose optimal solutions are also solutions of a general vector optimization problem. This will be done for weakly minimal and minimal solutions, respectively. Finally the results will be applied to a certain class of approximation problems.     

A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures

This paper investigates some properties of approximate efficiency in variable ordering structures where the variable ordering structure is given by a special set valued map. We characterize $\epsilon$-minimal and $\epsilon$- nondominated elements as approximate solutions of a multiobjective optimization problem with a variable ordering structure and give necessary and sufficient conditions for these solutions, via scalarization.