Điều kiện hội tụ của lời giải cho bài toán tối ưu hóa tập hợp - Luận án tiến sĩ

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH UNIVERSITY OF SCIENCE DINH VINH HIEN CONVERGENCE CONDITIONS OF SOLUTIONS FOR SET OPTIMIZATION PROBLEMS AND RELATED PROBLEMS Doctoral Thesis Z“ VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY \ UNIVERSITY OF SCIENCE DINH VINH HIEN CONVERGENCE CONDITIONS OF SOLUTIONS FOR SET OPTIMIZATION PROBLEMS AND RELATED PROBLEMS Speciality: Applied Mathematics Code: 9460112 Reviewer 1: Prof. NGUYEN DINH Reviewer 2: Assoc. NGUYEN DINH HUY Reviewer 3: Assoc. LE THANH TUNG Independent reviewer 1: Exemption Independent reviewer 2: Exemption SUPERVISOR 1.

LAM QUOC ANH 2. NGUYEN LE HOANG ANH - Ho Chỉ Minh City - 2022 Declaration I hereby declare that this thesis is the result of my own work, done under the supervision of Professor Lam Quoc Anh and Associate Professor Nguyen Le Hoang Anh. This work is completed at the University of Science, Vietnam National University Ho Chi Minh City, and there is no part of this thesis that has been submitted for any other degree or diploma. I also obtained the consent of Dr.

Tran Quoc Duy, the co- author of the joint papers [H1—H4], and Professors Daishi Kuroiwa, Narin Petrot with the joint paper [H1], to let me include in my thesis some results in the above-mentioned papers that were not included in any thesis or dissertation before. Ho Chi Minh City, March 2022 The author Dinh Vinh Hien Acknowledgements First, I would like to express my sincere gratitude to my supervisors, Professor Lam. Quoc Anh and Associate Professor Nguyen Le Hoang Anh, who have provided me with continuous guidance throughout the course of this research. Besides my supervisors, I would also like to express my sincere thankfulness to the reviewers of this thesis and Professor Phan Quoc Khanh, who has been organized the seminar for the Group of Optimization at the University of Science, Vietnam National University.

Through this activity, I have the chance to improve my knowledge. I am also thankful to the University of Science, Vietnam National University Ho Chi Minh City, and Ho Chi Minh City University of Food Industry for their support during my work to complete the PhD program. I am also very pleased to give my thanks to Dr. Tran Quoc Duy, Professors Daishi Kuroiwa, and Narin Petrot, who give me the valuable advice during the work on our joint papers.

Finally, my appreciation also goes out to my family and friends for their encour- agement and support me during all the time of my PhD program. Ho Chi Minh City, March 2022 The author Dinh Vinh Hien ii Table of Contents Declaration i Acknowledgements ii List of Symbols and Notations vi Preface ix 1. Background and Preliminaries 1 1. The Continuity of Set-valued Mappings.

Set Less Order Relations». Painlevé-Kuratowski Convergenece. The Set Optimization Problem Model. Properties of Elements in Image Spaces.

Stability Conditions for Weak and Ideal Solutions to Set Optimization Problems 14 2. Relationships among the Kinds of Minimal Elements. External Stability Conditions for Solutions. Internal Stability Conditions for Solutions.

The Application in Welfare Economics. 0 HH HH kg va 31 iii 3. Stability Conditions for Efficient Solutions to Set Optimization Prob- lems 33 3. Properties of Elements in Image Spaces.

Stability Conditions for Efficient Solutions. Well-Posedness for the Optimistic Counterpart of Uncertain Vector Optimization Problems 50 4. Uncertain Vector Optimization Problems PU). Relationships for Well-Posedness Properties between an Optimistic Coun- terpart of P() and Corresponding Scalar Problems.

The Concepts of Well-Posedness for the Optimistic Counterpart of P(#/) and the Scalar Optimization Problems. Relationships for Well-Posedness Properties between an Opti- mistic Counterpart of P() and Corresponding Scalar Problems 56 4. Sufficient Conditions and Characterizations for Well-Posedness for Op- timistic Counterparts of P(M). Stability Conditions for Parametric Vector Quasi-Equilibrium Prob- lems with Variable Cones 70 5.

Stability of Parametric Vector Quasi-Equilibrium Problems. Sufficient Conditions for Upper Semicontinuity of Solution Map- iv 5. Sufficient Conditions for Lower Semicontinuity of Solution Map- pings. LH HH ng kh k kt 5.

Well-Posedness for Parametric Vector Quasi-Equilibrium Problems. Well-Posedness under Perturbations and Generalized Hadamard Well-Posednes. Uniquely Well-Posedness under Perturbations and Hadamard Well-Posednes. Traffic Network Problems.

Lower and Upper Bounded Equilibrium Problems. 91 General Conclusions 92 List of the author’s papers related to the thesis 94 List of the author’s conference reports related to the thesis References 97 List of Symbols and Notations Ñ Set of the natural numbers N* N* =N\ {0} R Set of the real numbers Ry Set of non-negative real numbers R = RU {-co, +00} Set of the extended real numbers R" n-dimensional Euclidean space RY Non-negative orthant of R” clA Closure of a set A intA Interior of a set A el10 e>0ande—=0 M For all 3 There exists Oo The end of a proof IỮ_¡ Aj = A x «++ x An The Cartesian product of sets 4, ---, An ) Open ball with the center a and the radius r Bla,1 ) Closed ball with the center ø and the radius r infA The infimum value of a numerical set A supA The supremum value of a numerical set A d(x, A) Distance from x to a subset A e(A, B) e(A, B) := supye, d(a, B) gphF Graph of F P(X) Family of the nonempty subsets of X L(X.Y) The space of all continuous linear mappings from X into Y dom# Domain of set-valued mapping F’ argmin(K,ƒ) Set of all minimizers of the function ƒ on vì min(A) Set of minimal points of A wmin(A) Set of weak minimal points of A max(A) Set of maximal points of A wmax(4) Set of weak maximal points of A Sr Lower type set less order relation Su Upper type set less order relation The set less order relation Strict lower type set less order relation Strict upper type set less order relation The strict set less order relation The approximating quasi-order relation A is equivalent to B (Le. A < B and B < A) Family of minimal elements of the collection of sets ⁄ WMin(¥) Family of weakly minimal elements of the collection of sets Y SMin(Z) Family of ideally minimal elements of the collection of sets G Min.() Family of e-minimal elements of the collection of sets ⁄ LiA, Set of limit points of the sequence { A„} Ls A, Set of cluster points of the sequence {A,} A, SA A,, Painlevé-lKuratowski converges to A A, SA The upper part of Painlevé-Kuratowski convergence A, <A The lower part of Painlevé-Kuratowski convergence A, 4 A A,, Hausdorff converges to A A, 2A The upper part of Hausdorff convergence A, 4A The lower part of Hausdorff convergence usc Upper semicontinuous Isc Lower semicontinuous vii Hausdorff upper semicontinuous Hausdorff lower semicontinuous C-upper semicontinuous (set-valued mapping) C-lower semicontinuous (set-valued mapping) C-upper semicontinuous (vector-valued mapping) C-lower semicontinuous (vector-valued mapping) Set optimization problem The Welfare economic problem Weak vector quasi-equilibrium problem Strong vector quasi-equilibrium problem Traffic network problem Quasi-variational inequality Lower and upper bounded equilibrium problem Uncertain vector optimization problem with uncer- tainty set U Scalar optimization problem with objective function f and feasible set Ý viii Preface Optimization is a branch of applied mathematics and has many practical appli- cations in almost all aspects of life and society. This field has been continuously developing over time and increasingly attracted the interest of many researchers.

Op- timization models appear in almost all areas, from economics, finance, engineering, technology to medicine, sports, and even social sciences, etc. Applying optimization theory to real-world problems is one of the very significant topics that mathematicians have focused on for several decades. Many research works have produced valuable re- sults in the development of theory as well as the applicability in practice. The basic problem of optimization theory is finding the minimum of a functional under certain constraints.

When considering the scalar optimization models, one of the most general ones is the equilibrium problem which was introduced by Muu and Oettli in 1992 [1]; Blum and Oettli in 1994 [2]. The mathematical formulation of this problem unifies var- ious ones such as optimization problems, variational inequality, coincidence-point and fixed-point problems, complementarity problems, Nash equilibria problems, minimax problems, and traffic network problems, etc. Moreover, there are a lot of situations in practical problems leading to generalized models of the equilibrium problem. One of the extended directions of the above problem is to consider the vector equilibrium problems instead of the scalar equilibrium ones (see [3-5]).

Vector equilibrium problems have played an important role in optimization because scalar models often do not meet the variety of situations in practice such as competition equilibrium and social welfare. In order to compare vectors, a pointed closed convex cone is used to define an ordering relation. The next developments can be listed as switching from equilibrium problems to quasi-equilibrium ones (see {6-9]), from single-valued problems to multi-valued ones [10-19], and from the problems with fixed cones (the cones that do not depend on any variables) to the ones with variable cones (see [20-22]). It is worth noting that the problems with variable cones play an important role in many practical situations.

For instance, when we want to ix choose goods or services from the market, of course, the characteristics of the chosen products have to satisfy our requirements that are considered as the required domain. It is obvious that the product with higher quality is the better choice, and hence the requirement domain is a cone which is called the requirement cone. Since our requirement cones may be different from that of the other ones, variable cones are useful in this case. However, the results related to this topic were modest and need to be studied further.

Furthermore, many problems arising from the fields such as economics, medical en- gineering, optimal control, differential inclusions, bilevel optimization, robust optimiza- tion, interval optimization, and game theory lead to new requirements for optimization problems (see [23-27]). In recent years, set-valued optimization problems where either the objective mapping and/or the constraint mappings are set-valued mappings have been intensively investigated. These problems provide an extension and unification of the scalar as well as vector optimization ones. Depending on which criterion is used to define optimal solutions, there are two main approaches for set-valued optimization problems.

The classical one is the vector approach, where the definition of a minimizer considers only an efficient point of the union of all images of the set-valued objective mapping and identifies the image set containing this minimal element [25,27]. Ob- viously, the disadvantage of this approach is that, in general, only one element does not necessarily imply that the whole image set is in a certain sense minimal with re- spect to all image sets. Along with this criterion, another approach is to use a more appropriate order relation for comparing sets, which was introduced by Kuroiwa in op- timization [28], is proposed and called the set optimization approach. With the aim of investigating preferences over sets, it seems that the set optimization approach is more natural and appropriate than the vector one.

Until now, set optimization approach has received much attention by many authors and there are a lot of works related to set optimization problems as well as their applications; see, for instance, [26,2941]. Con- cerning this approach, there are two main set order relations considered in set-valued optimization problems: upper type set less order relation and lower type set less order relation. These relations were independently employed by Young [42] in algebra, by Nishnianidze [43] in fixed-point theory, and by Chiriaev and Walster [44] in computer science and interval analysis. In 2011, Jahn and Ha [37] blended these two set order relations and obtained another one, say “set less order relation”.

Recently, Khan et al. [26] and Mordukhovich (45, Chapters 9 and 10] discussed some developments and commentaries on such problems in their monographs. Along with the topic of solution existence [46-49] and optimality conditions [36, 50,51], the stability plays an important role in optimization.