Well posedness almost everywhere in a class of constrained convex optimization problems. In the first group the notions are based on the behaviour of a prescribed class of sequences of approximate solutions and in the second on the continuous dependence of the necessarily existing solution on the data of the problem. Specifically, three kinds of k0 well posedness and three kinds of b well posedness of a set optimization problem were discussed by zhangliteo 8 and longpeng 12, respectively. This book presents in a unified way the mathematical theory of well posedness in optimization. Recently, tikhonov type well posedness has been studied and generalized in other more complicated situations, such as scalar optimization problems, vector set optimization problems, many types of. Convexity and global well posedness in set optimization crespi, giovanni, kuroiwa, daishi, and rocca, matteo, taiwanese journal of mathematics, 2014. The corresponding concepts of well posedness in the generalized sense are also. Zolezzi, well posedness criteria in optimization with application to the calculus of variations, nonlinear anal. In this paper, three kinds of well posedness for set optimization are first introduced. We introduce generalized levitinpolyak well posedness and strongly generalized levit. No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property.
The wellposedness of optimization problems was first studied by tykhonov 1 in 1966. In this paper, we consider three kinds of pointwise wellposedness for set optimization problems. Wellposedness and scalarization in vector optimization core. Hadamard wellposedness for a setvalued optimization. Hadamard wellposedness for a setvalued optimization problem. Moreover, the hausdorff upper semicontinuity of lminimal solution mapping and uminimal solution mapping are established by assuming that the set optimization problem is lh well posed and uh well posed, respectively. Specifically, three kinds of k0wellposedness and three kinds of bwellposedness of a set optimization problem were discussed by zhangliteo 8 and longpeng 12, respectively. Wellposedness and scalarization in vector optimization. By using a notion of well posed set optimization problem due to zhang et al. Generally speaking the different notions of well posedness of a given optimization problem can be divided into two groups.
Lp well posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Wellposedness and stability of solutions for set optimization problems. Recently, this research was extended to nonconvex optimization. In this paper, a characterization of tightly properly efficient solutions of set valued optimization problem is obtained.
Studies on well posedness of optimization problems have been extended to vector optimization problems see e. In this paper, we introduce the concept of well posedness for the vector quasiequilibrium problem. The aim of this paper is to study these latter notions, and mostly the. The study of levitinpolyak well posedness for scalar convex optimization problems with functional constraints originates from 25. Wellposedness for equilibrium problems and for optimization. Bwellposedness for set optimization problems involving. Motivated and inspired by the above works, in this paper we shall investigate the well posedness of equilibrium problems and of optimization problems with equilibrium constraints. Some sufficient and necessary conditions for the levitinpolyak well posedness of.
Levitinpolyak wellposedness for generalized semiinfinite. For instance, 12 gave a survey on various aspects on well posedness of optimization problems. Pdf well posedness in vector optimization problems and. A characterization of tyhonov wellposedness for minimum. Problems in nonlinear complex systems so called chaotic systems provide well known examples of instability. Recent developments in wellposed variational problems. In this paper, levitinpolyak well posedness for set valued optimization problems with constraints is introduced. These kinds of well posedness arise from some well posedness in the vector settings. Research article wellposedness and primaldual analysis.
To download the pdf, click the download link below. Def a pde is called wellposed in the sense of hadamard, if 1 a solution exists 2 the solution is unique 3 the solution depends continuously on the data. Variational principles and wellposedness in optimization. The well posedness was generalized also to other contexts. In this paper, we give a survey on well posedness notions of tykhonovs type for vector optimization problems and the links between them with respect to the classification proposed by miglierina, molho and rocca. In this paper, three kinds of wellposedness for set optimization are first introduced. In this paper, some notions of pointwise wellposedness for set optimization problems are introduced. Wellposedness by perturbations of variational problems. Well posedness of optimization problems questions of existence of solutions and how they depend on problem s parameters are usually important for many. Universita degli studi di bergamo facolta di economia dipartimento di matematica, statistica, informatica e applicazioni dottorato di ricerca in metodi computazionali per le pre. We obtain some metric characterizations of levitinpolyak well posedness for this problem.
We establish some metric characterizations of well posedness for equilibrium problems and for optimization problems with equilibrium constraints. Scalarization and wellposedness for set optimization using. Wellposedness for generalized set equilibrium problems. Generalized levitinpolyak wellposedness in constrained. Wellposedness in vector optimization and scalarization results.
We obtain some necessary and sufficient conditions for well posedness of vector quasiequilibrium problems. By employing the notion of wellposedness for set optimization problems due to zhang et al. We give a characterization of tyhonov well posedness for the problem of minimising a convex lowersemieontinuous function f on a closed convex set k. Then, sufficient and necessary conditions of wellposedness for set optimization problems are obtained by using a generalized forcing function, respectively. We prove that under suitable conditions, the well posedness is equivalent to the existence and uniqueness of solutions. The concept of the well posedness for a special scalar problem is linked with the tightly properly efficient solutions of set valued optimization problem. In this paper we propose a new definition of global well posedness for set optimization problems. Tykhonov wellposedness of a viscoplastic contact problem. In this paper, both pointwise and global b well posedness for set op timization problems involving three kinds of set order relations are investigated. Moreover, the links between stability and lp wellposedness of such problems are established via the study on approximating solution mappings.
This volume contains several surveys focused on the ideas of approximate solutions, well posedness and stability of problems in scalar and vector optimization, game theory and calculus of variations. Levitinpolyak wellposedness for set optimization problems. Convexity and global wellposedness in setoptimization core. By virtue of a scalarization function, we obtain some relationships between weak \\varepsilon, e\minimizers of the set valued optimization problem and \\varepsilon\approximate solutions of a scalar optimization problem. Some relationships among these notions are established. B well posedness for set optimization problems involving set order relations. These concepts are of particular interest in many fields of mathematics. Using a new nonlinear scalarization function, pointwise wellposed set optimization problems are characterized by means of a family of tykhonov wellposed scalar optimization problems. Extended and strongly extended wellposedness of set. Variational principles and wellposedness in optimization and.
By virtue of a generalized gerstewitzs 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. Wellposedness almost everywhere in a class of constrained convex optimization problems. We introduce various types of levitinpolyak lp wellposedness for set optimization problems and survey their relationships. Pdf this paper aims at studying the generalized wellposedness in the sense of bednarczuk for set optimization problems with setvalued maps. In this paper, some characterizations for the generalized lbwellposedness and the generalized ubwellposedness of set optimization problems are given. Moreover, pointwise lwellposedness and relationships be tween these kinds of pointwise wellposedness are studied. As applications, we investigate the well posedness for vector quasivariational inequality problems and vector quasi optimization problems.
Abstract optimization problems as well as applications to optimal control. Wellposedness and stability analysis in optimization. An illconditioned problem is indicated by a large condition number. Using a new nonlinear scalarization function, pointwise well posed set optimization problems are characterized by means of a family of tykhonov well posed scalar optimization problems. We derive the relations between the levitinpolyak well posedness and the upper semicontinuity of approximate solution maps for. Wellposedness and scalarization in set optimization. Finally, various criteria and characterizations of wellposedness are given for set optimization problems. Pointwise wellposedness and scalarization in set optimization. Wellposedness and primaldual analysis of some convex. Moreover, the hausdorff upper semicontinuity of lminimal. We establish some relations among the three kinds of pointwise well posedness. In the past ten years, equilibrium problem has been extensively studied and generalizedsee, e. Moreover, sufficient conditions for the generalized levitinpolyak well posedness of symmetric vector quasiequilibrium problems are established. By virtue of a generalized gerstewitzs function, the equivalent relations between the three kinds of wellposedness and the wellposedness of three kinds of scalar optimization problems are established, respectively.
Well posedness plays a crucial role in the theory and methodology of scalar optimization problems. In 1966, tykhonov first introduced the concept of well posedness for a global minimizing problem, which has become known as tykhonov well posedness. If you would like more information about how to print, save, and work with pdfs, highwire press provides a helpful frequently asked questions about pdfs. Furthermore, using the kuratowski measure of noncompactness, we. We introduce several types of the levitinpolyak well posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. An existence result for the solution set of symmetric vector quasiequilibrium problems that allows for discontinuities is obtained. Main results of wellposedness and primaldual analysis are included in section. Pdf wellposedness and stability of solutions for set.
We focus on some convex separable optimization problems, considered by the author in previous papers, for which problems, necessary and sufficient conditions or sufficient conditions have been proved, and convergent algorithms of polynomial computational complexity have been proposed for solving these problems. Most of these concepts have been generalized in the. This means that in the assumptionof weierstrasstheorem,the problem is actually wellposed in extended sense. The wellposedness for a system of generalized quasi. We obtain some criteria and characterizations of these two types of extended well posedness, further. Wellposedness and stability of solutions for set optimization problems article pdf available in optimization 661. This paper presents an approach to the stability and the hadamard well posedness of the linear semiinfinite programming problem lsip. In this paper, we introduce a notion of levitinpolyak well posedness for generalized semiinfinite multiobjective programming problems in terms of weakly efficient solutions. Wellposedness for set optimization problems request pdf. In this paper, both pointwise and global bwellposedness for set optimization problems involving three kinds of set order relations are investigated. Well posedness properties play an important role in optimization theory because of their links to several basic issues in optimization as well as the usefulness in the convergence analysis of many algorithms.
One gets in this way, as a byproduct, the hadamard well posedness of wellposedness and stability analysis in optimization 3 09 linear operator equations from a new point of view, which are natural extensions of the classical boundary value problems in mathematical physics, t o which the original hadamard well posedness definition applies. The basic concepts of well posedness and the links among them are studied, in particular hadamard and tykhonov well posedness. Further, we study the relationships between the wellposedness of a vector optimization problem and the wellposedness of an appropriate scalarized problem. We distinguish them into two classes and we establish the hierarchical structure of their relationships. Ivp initial value problem u t u x advection equation u t u xx heat equation u t u xxx airys equation u t u. Wellposedness for set optimization problems sciencedirect. Sorry, we are unable to provide the full text but you may find it at the following locations. Well posedness in vector optimization problems and vector variational inequalities. For vector optimization problems, \local properties of well posedness have been studied in 9,17, among others. Levitinpolyak wellposedness in vector quasivariational. Levitinpolyak wellposedness for equilibrium problems with.
Set valued optimization problem, tightly proper efficiency, well posedness. The concepts of well posedness of optimization problems in the sense of tychonov. We establish some metric characterizations of well posedness. In this paper, we introduce a kind of hadamard well posedness for a set valued optimization problem. Using an embedding technique proposed by kuroiwa and nuriya 2006, we prove well posedness property of a class of generalized convex set valued maps. Phan quoc khanh, somyot plubtieng and kamonrat sombut.
The k0 well posedness at a minimizer introduced in 8 was clari. Request pdf wellposedness for set optimization problems in this paper, three kinds of wellposedness for set optimization are first introduced. Wellposedness of lset optimization problem under variable. Wellposedness of generalized vector quasivariational. We study the well posedness for generalized set equilibrium problems gsep and propose two types of the well posed concepts for these problems in topological vector space settings. Both the pure and applied side of these topics are presented. Wellposedness for set optimization problems semantic. These two concepts are generalizations of the extended well posedness of realvalued optimization probems defined by zolezzi. Wellposedness, lset optimization, gap function, variable order structure 1 introduction in the last several decades, two types of criteria of optimization in terms of a setvalued mapping f. When we call a solution unique, we sometimes mean unique within a certain class of functions.
Wellposedness of optimization problems questions of existence of solutions and how they depend on problem s parameters are usually important for many. Wellposedness for vector quasiequilibrium problems with. Moreover, we relate vector well posedness and well posedness of an appropriate scalarization. We give a characterization of tyhonov wellposedness for the problem of minimising a convex lowersemieontinuous function f on a closed convex set k.
Scalarization and pointwise wellposedness for set optimization. According to hadamard, a problem is well posed or correctly set if a. Existence and uniqueness involves boundary conditions ex u. The purpose of this paper is introduce several types of levitinpolyak well posedness for bilevel vector equilibrium and optimization problems with equilibrium. The k0wellposedness at a minimizer introduced in 8 was clari.
It is well known that the wellposedness is very important for both optimization the. Existence and wellposedness for symmetric vector quasi. Well posedness for optimization problems with constraints defined by. In this paper, we study a setvalued optimization problem. Wellposedness for tightly proper efficiency in setvalued. More recently, some attempts to extend thee results to set valued optimization have been proposed, mainly applying some scalarization. Research article wellposedness and primaldual analysis of. In this paper, we consider levitinpolyaktype well posedness for a general constrained optimization problem. Naturally, it is meaning to study scalarization and wellposedness for these set optimization problems. If it is not well posed, it needs to be reformulated. Criteria and characterizations of these types of the levitinpolyak well posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. Pdf generalized bwellposedness for set optimization problems. Main results of well posedness and primaldual analysis are included in section.
Elliptic pdes are coupled with boundary conditions, while hyperbolic and parabolic equations get initialboundary and pure initial conditions. After that, sufficient and necessary conditions for the reference problems to be lp wellposed are given. Jul 28, 2006 2019 levitinpolyak wellposedness for set optimization problems involving set order relations. One important problem in vector optimization is to find the efficient points. Then, sufficient and necessary conditions of well posedness for set. Furthermore, using the kuratowski measure of noncompactness, we study characterizations of wellposedness for set optimization problems. In this paper, the concept of extended wellposedness of scalar optimization problems introduced by zolezzi is generalized to vector optimization problems in three ways. To get this result, we use ekelands theorem on theapproximate variational principle21.
In this paper, we deal with the extended well posedness and strongly extended well posedness of set valued optimization problems. Wellposedness in unconstrained polynomial optimization. Abstract optimization problems as well as applications to optimal control, calculus of variations and mathematical programming are considered. In this paper, some characterizations for the generalized lb well posedness and the generalized ub well posedness of set optimization problems are given. We also study the relationship between these well posedness concepts and present several criteria for the well posedness of gsep. If the problem is well posed, then it stands a good chance of solution on a computer using a stable algorithm. Lp well posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints khanh, phan quoc, plubtieng, somyot, and sombut, kamonrat, abstract and applied analysis, 2014. By virtue of a generalized nonlinear scalarization function, we obtain the equivalence relations between the three kinds of pointwise well posedness for set optimization problems and the well posedness of three kinds of scalar optimization problems, respectively. In this paper, we introduce a kind of hadamard wellposedness for a setvalued optimization problem. Tools and techniques used in this study and our results are different from existing ones in the literature. Abstract in this paper, three kinds of well posedness for set optimization are first introduced. Characterizations and sufficient andor necessary conditions of these types of well posedness are given.
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