Vector variational inequalities and vector optimization theory and applications

Vector Optimization Qamrul Hasan Ansari Elisabeth Köbis Jen-Chih Yao Vector Variational Inequalities and Vector Optimization Theory and Applications Vector Optimization Series editor Johannes Jahn Erlangen, Germany The series in Vector Optimization contains publications in various fields of optimization with vector-valued objective functions, such as multiobjective optimization, multi criteria decision making, set optimization, vector-valued game theory and border areas to financial mathematics, biosystems, semidefinite programming and multiobjective control theory Studies of continuous, discrete, combinatorial and stochastic multiobjective models in interesting fields of operations research are also included The series covers mathematical theory, methods and applications in economics and engineering These publications being written in English are primarily monographs and multiple author works containing current advances in these fields More information about this series at http://www.springer.com/series/8175 Qamrul Hasan Ansari • Elisabeth KRobis • Jen-Chih Yao Vector Variational Inequalities and Vector Optimization Theory and Applications 123 Elisabeth KRobis Institute of Mathematics Martin Luther University Halle-Wittenberg Halle, Germany Qamrul Hasan Ansari Department of Mathematics Aligarh Muslim University Aligarh, India Jen-Chih Yao Center for General Education China Medical University Taichung, Taiwan ISSN 1867-8971 Vector Optimization ISBN 978-3-319-63048-9 DOI 10.1007/978-3-319-63049-6 ISSN 1867-898X (electronic) ISBN 978-3-319-63049-6 (eBook) Library of Congress Control Number: 2017951114 © Springer International Publishing AG 2018 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Going back to the groundbreaking works by Edgeworth (1881) and Pareto (1906), the notion of optimality in multiobjective optimization is an efficient tool for describing optimal solutions of real-world problems with conflicting criteria This branch of optimization has formally started with the pioneering work by Kuhn and Tucker (1951) The concept of multiobjective optimization is further generalized from finite-dimensional spaces to vector spaces leading to the field of vector optimization This theory has bourgeoned tremendously due to rich application fields in economics, management science, engineering design, etc A powerful tool to study vector optimization problems is the theory of vector variational inequalities which was started with the fundamental work of F Giannessi in 1980, where he extended classical scalar variational inequalities to the vector setting Later, he has shown equivalence between optimal solutions of vector optimization problems with differentiable convex objective function and solutions of vector variational inequalities of Minty type It is well known that many practical equilibrium problems with vector payoff can be formulated as vector variational inequalities In the last two decades, extensive research has been devoted to the existence theory of their solutions The objective of this book is to present a mathematical theory of vector optimization, vector variational inequalities, and vector equilibrium problems The well-posedness and sensitivity analysis of vector equilibrium problems are also studied The reader is expected to be familiar with the basic facts of linear algebra, functional analysis, optimization, and convex analysis The outline of the book is as follows Chapter collects basic notations and results from convex analysis, functional analysis, set-valued analysis, and fixed point theory for set-valued maps A brief introduction to variational inequalities and equilibrium problems is also presented Chapter gives an overview on analysis over cones, including continuity and convexity of vector-valued functions Several notions for solutions of vector optimization problems are presented in Chap Classical linear and nonlinear scalarization methods for solving vector optimization problems are studied in Chap Chapter is devoted to the vector variational inequalities and existence theory for their solutions The relationship v vi Preface between a vector variational inequality and a vector optimization problem with smooth objective function is given Chapter deals with scalarization methods for vector variational inequalities Such scalarization methods are used to study several existence results for solutions of vector variational inequalities In Chap 7, we consider nonsmooth vector variational inequalities defined by means of a bifunction and present several existence results for their solutions The relationship between nonsmooth vector variational inequalities and vector optimization problems in which the objective function is not necessarily differentiable but has some kind of generalized directional derivative is discussed Chapter presents vector variational inequalities for set-valued maps, known as generalized vector variational inequalities, and gives several existence results for their solutions It is shown that the generalized vector variational inequalities provide the optimal solutions of nonsmooth vector optimization problems Chapter is devoted to the detailed study of vector equilibrium problems, e.g., existence results, duality, and sensitivity analysis It is worth mentioning that the vector equilibrium problems include vector variational inequalities, nonsmooth vector variational inequalities, and vector optimization problems as special cases Chapter 10 deals with vector equilibrium problems defined by means of a set-valued bifunction, known as generalized vector equilibrium problems The generalized vector equilibrium problems include generalized vector variational inequalities and vector optimization problems with nonsmooth objective function as special cases The existence of solutions, duality, and sensitivity analysis of generalized vector equilibrium problems are studied in detail We would like to take this opportunity to express our most sincere thanks to Kathrin Klamroth, Anita Schöbel, and Christiane Tammer for their support and collaboration The second author is truly grateful to her husband Markus Köbis and her parents for patience and encouragement Moreover, we are thankful to Johannes Jahn for encouraging and supporting our plan to write this monograph We are grateful to Christian Rauscher, Senior Editor, Springer, for taking a keen interest in publishing this monograph This book is dedicated to our families We are grateful to them for their support and understanding Finally, we thank our coauthors for their support, understanding, and hard work for this fruitful collaboration We are also grateful to all researchers whose work is cited in this monograph Any comment on this book will be accepted with sincere thanks Aligarh, India Halle, Germany Taichung, Taiwan Qamrul Hasan Ansari Elisabeth Köbis Jen-Chih Yao Contents Preliminaries 1.1 Convex Sets and Cones 1.2 Convex Functions and Their Properties 1.3 Generalized Derivatives 1.4 Tools from Nonlinear Analysis 1.4.1 Continuity for Set-Valued Maps 1.4.2 Fixed Point Theory for Set-Valued Maps 1.5 Variational Inequalities 1.5.1 Nonsmooth Variational Inequalities 1.5.2 Generalized Variational Inequalities 1.6 Equilibrium Problems References 1 20 33 38 38 42 48 54 58 65 72 Analysis over Cones 2.1 Orders 2.2 Some Basic Properties 2.3 Cone Topological Concepts 2.4 Cone Convexity 2.5 Cone Continuity 2.6 Nonlinear Scalarization Functions 2.7 Vector Conjugate References 79 82 92 96 102 117 127 139 141 Solution Concepts in Vector Optimization 3.1 Optimality Notions 3.2 Solution Concepts 3.2.1 Efficient Solutions 3.2.2 Weakly and Strongly Efficient Solutions 3.2.3 Properly Efficient Solutions 3.3 Existence of Solutions 3.4 Optimality Notions for Variable Ordering Structures References 143 144 154 155 159 162 171 175 179 vii viii Contents Classical Methods in Vector Optimization 4.1 Linear Scalarization 4.2 Nonlinear Scalarization Method 4.2.1 "-Constraint Method 4.2.2 Hybrid Method 4.2.3 Application: A Unified Approach to Uncertain Optimization References 181 182 192 195 199 Vector Variational Inequalities 5.1 Formulations and Preliminary Results 5.2 Existence Results for Solutions of Vector Variational Inequalities Under Monotonicity 5.3 Existence Results for Solutions of Vector Variational Inequalities Without Monotonicity 5.4 Applications to Vector Optimization 5.4.1 Relations Between Vector Variational Inequalities and Vector Optimization 5.4.2 Relations Between Vector Variational Inequalities and Vector Optimization in Finite Dimensional Spaces References 223 223 202 220 233 250 253 254 258 263 Linear Scalarization of Vector Variational Inequalities 265 References 273 Nonsmooth Vector Variational Inequalities 7.1 Formulations and Preliminary Results 7.2 Existence Results for Solutions of Nonsmooth Vector Variational Inequalities 7.3 Nonsmooth Vector Variational Inequalities and Nonsmooth Vector Optimization References 275 275 Generalized Vector Variational Inequalities 8.1 Formulations and Preliminaries 8.2 Existence Results under Monotonicity 8.3 Existence Results Without Monotonicity 8.4 Generalized Vector Variational Inequalities and Optimality Conditions for Vector Optimization Problems References 299 299 311 326 Vector Equilibrium Problems 9.1 Introduction 9.2 Existence Results 9.2.1 Existence Results for Solution of Weak Vector Equilibrium Problems 281 287 296 330 338 339 339 343 345 Contents ix 9.2.2 Existence Results for Strong Vector Equilibrium Problems 9.2.3 Existence Results for Implicit Weak Vector Variational Problems 9.3 Duality of Implicit Weak Vector Variational Problems 9.4 Gap Functions and Variational Principles 9.4.1 Gap Function for Vector Equilibrium Problems 9.4.2 Variational Principle for Weak Vector Equilibrium Problems 9.4.3 Variational Principle for Minty Weak Vector Equilibrium Problems 9.4.4 Variational Principle for WVEP f ; h/ 9.5 Vectorial Form of Ekeland’s Variational Principle 9.5.1 Vectorial Form of Ekeland-Type Variational Principle 9.5.2 Existence of Solutions for Weak Vector Equilibrium Problems Via Vectorial Form of EVP 9.5.3 Some Equivalences 9.6 Sensitivity Analysis of Vector Equilibrium Problems 9.6.1 "-Weak Vector Equilibrium Problems 9.6.2 Parametric Weak Vector Equilibrium Problems 9.6.3 Parametric Strong Vector Equilibrium Problems 9.6.4 Well-Posedness for Parametric Weak Vector Equilibrium Problems References 10 Generalized Vector Equilibrium Problems 10.1 Introduction 10.2 Generalized Abstract Vector Equilibrium Problems 10.3 Existence Results for Generalized Vector Equilibrium Problems 10.3.1 Existence Results Without Monotonicities 10.4 Duality 10.4.1 Generalized Duality 10.4.2 Additive Duality 10.4.3 Multiplicative Duality 10.5 Recession Methods for Generalized Vector Equilibrium Problems 10.6 "-Generalized Weak Vector Equilibrium Problems 10.6.1 Existence Results 10.6.2 Upper Semicontinuity of ˝ and « 10.6.3 Lower Semicontinuity of ˝ and « 10.6.4 Continuity of ˝ and « 10.7 "-Generalized Strong Vector Equilibrium Problems 10.7.1 Existence Results 10.7.2 Upper Semicontinuity of  and à 350 362 365 367 368 370 373 375 378 378 382 386 388 389 400 411 416 423 429 430 431 436 448 449 450 453 454 456 463 464 467 468 472 472 473 477 B Some Algebraic Concepts 493 Given x and y in a vector space X, we denote by Œx; y and x; yŒ the closed and open line segments joining x and y, respectively Definition B.4 Let K be a nonempty convex subset of a vector space X A point x K is said to be relative algebraic interior point of K if for any u X such that x C u K, there exists " > such that x u; x C uŒ  K The set of all relative algebraic interior points of K is denoted by relint.K/ We note that if < ˛ < , then x ˛u; x C ˛uŒ  x u; x C uŒ Appendix C Topological Vector Spaces Definition C.1 (Directed Set) A set together with a reflexive and transitive ordering relation such that every finite set of has an upper bound in (that is, for ˛; ˇ , there is a such that ˛ and ˇ ) is called a directed set Definition C.2 Let X be a topological space and, and be any index sets (a) S A collection F D fO˛ g˛2 of subsets of X is said to be a cover of X if ˛2 O˛ D X; If each member of F is an open set, then F is called an open cover of X; (b) A subcollection C of a cover F of X is said to be a subcover if C is itself a cover of X; If the number of members of C is finite, then C is called a finite subcover (c) A collection C D fUˇ gˇ2 of subsets of X is said to be a refinement of the cover F D fO˛ g˛2 of X if C is an open cover of X and for each member Uˇ C , there is O˛ F such that Uˇ  O˛ Note that an open subcover is a refinement, but a refinement is not necessarily an open cover Definition C.3 An open cover F D fO˛ g˛2 of a topological space X is said to be locally finite if each point of X has a neighborhood which meets only finitely many O˛ Definition C.4 A Hausdorff space X is said to be paracompact if every open cover of X has a locally finite open refinement Definition C.5 Let X be a topological space and f W X ! R The support of f is the set Supp f / WD cl.fx X W f x/ Ô 0g/ Since a finite cover is necessarily locally finite, it follows that every open cover of a compact space has locally finite open refinement © Springer International Publishing AG 2018 Q.H Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 495 496 C Topological Vector Spaces Definition C.6 (Partition of Unity) Let X be a topological space A family fˇgi2I of continuous functions defined from X into Œ0; 1/ is called a partition of unity associated to an open cover fUi gi2I of X if (i) X for each x X, i x/ Ô ; for only finitely many ˇi ; ˇi x/ D for all x X (ii) i2I Theorem C.1 ([10, pp 68]) A Hausdorff space X is paracompact if and only if every open cover of X has a continuous locally finite partition of unity We note that every compact Hausdorff space is paracompact and every metrizable space is paracompact Definition C.7 A subset in a topological space is precompact (or relatively compact) if its closure is compact Note that every element in Rn has a precompact neighborhood Definition C.8 ([11, 12]) Let X be a Hausdorff topological vector space and L be a lattice with least one minimal element, denoted by A mapping ˚ W 2X ! L is said to be a measure of noncompactness provided that the following conditions hold for all M; N 2X : (i) ˚.M/ D if and only if M is precompact; (ii) ˚.cl.M// D ˚.M/; (iii) ˚.M [ N/ D max f˚.M/; ˚.N/g : It follows from condition (iii) that if M  N, then ˚.M/ Ä ˚.N/ Definition C.9 A net in a topological space X is a mapping ˛ 7! x˛ from a directed set into X; we often write fx˛ g˛2 , fx˛ W ˛ g, or simply fx˛ g We say that a net fx˛ g˛2 converges to x X if for any neighborhood V of x, there exists an ˛ (an index ˛) such that xˇ V for all ˇ < ˛ The point x is called a limit of the net fx˛ g When a net fx˛ g converges to a point x, we denote it by x˛ ! x We say that x is a cluster point of the net fx˛ g if for any neighborhood V of x and for any index ˛, there exists ˇ < ˛ such that xˇ V Definition C.10 A vector space X with a topology T under which the mappings x; y/ 7! x C y from X X!X and ˛; x/ 7! ˛x from R X!X are continuous, is called a topological vector space C Topological Vector Spaces 497 Theorem C.2 Let X be a vector space and B be a family of subsets of X such that the following conditions hold (i) Each U B is balanced and absorbing; (ii) For any given U1 ; U2 B, there exists U B such that U (iii) For any given U B, there exists V B such that V C V U1 \ U2 ; U Then there is a unique topology T on X such that X; T / is a topological vector space and B is a neighborhood base (or local base) at Theorem C.3 Let X be a topological vector space (a) The closure of a balanced set A  X is balanced (b) The closure of a convex set A  X is convex (c) The interior of a convex set A  X is convex Theorem C.4 Let X be a topological vector space and K be a subset of X (a) If K is balanced and int.K/, then its interior is also balanced (b) Any superset of an absorbing set is absorbing So if K is absorbing then so is its closure The interior of an absorbing set is not generally absorbing; (c) If K is open, then so is its convex hull Theorem C.5 Let X be a topological vector space and U be a neighborhood of its zero element Then U is a neighborhood of for all nonzero real Definition C.11 A topological vector space X is said to be locally bounded if there is a bounded neighborhood of Trivially, every normed space is locally bounded A well-known example of a locally bounded topological vector space is Lp for < p < which is not normable (see [13]) Theorem C.6 (a) Every neighborhood of zero in a topological vector space X is absorbing (b) Every neighborhood of zero in a topological vector space X includes a closed balanced neighborhood of zero Theorem C.7 Let X be a topological vector space with its zero element is denoted by and K be a nonempty subset of X Then cl.K/ D fK C U W U is a neighborhood of 0g: In particular, cl.K/ K C U for any neighborhood U of Proposition C.1 ([14, Proposition 15]) For each i D 1; 2; : : : ; m, let KS i be a comm pact convex subset of a Hausdorff topological vector space X Then co iD1 Ai is compact Corollary C.1 ([14, Corollary 1]) In a Hausdorff topological vector space, the convex hull of a finite set is compact 498 C Topological Vector Spaces Definition C.12 A topological vector space X with its topology T is said to be locally convex if there is a neighborhood base (local base) at zero consisting of convex sets The topology T is called locally convex Theorem C.8 Let T be a locally convex topology on X Then there exists a local base B whose members have the following properties: (i) Every member of B is absolutely convex and absorbing; (ii) If U B and > 0, then U B Conversely, if B is a filter base on X which satisfies conditions (i) and (ii), then there exists a unique locally convex topology T on X such that B is a local base at zero for T Definition C.13 ([15, pp 188]) Let X and Y be vector spaces A bilinear functional or bilinear form B W X Y ! R, x; y/ 7! B.x; y/ is a map which is linear in either argument when the other is held fixed A pairing or pair is an ordered pair X; Y/ of linear spaces together with a fixed bilinear functional B Usually, B.x; y/ will be denoted by hx; yi If X is a linear space and X its algebraic dual then the natural pairing of X and X is that arising from the (natural or canonical) bilinear functional on X X , which sends x; x0 / into x0 x/, that is, hx; x0 i D x0 x/ If X and Y are any two paired vector spaces, then we also have the natural pairing in the following sense: If y is any fixed element of Y, then the map y0 W X ! R, x 7! hx; yi is obviously a linear functional on X, that is, y0 X It is clear that hx; yi D 0; for all x X implies y D equivalently, y Ô implies that there is some x X such that hx; yi Ô 0: Definition C.14 Let X be a vector space A semi-norm on X is a function p W X ! R such that the following conditions hold: (i) p.x/ for all x X; (ii) p x/ D j jp.x/ for all x X and R; (iii) p.x C y/ Ä p.x/ C p y/ for all x; y X Theorem C.9 ([14, Corollary, p TVS II.24]) Let X be a topological vector space with its topology T Then T is defined by a set of semi-norms if and only if T is locally convex Definition C.15 ([15, pp 189]) Let X and Y be vector spaces The map x 7! jhx; yij D py x/ determines a semi-norm on X for all y Y The topology generated by the family of semi-norms fpy W y Yg is the weakest topology on X and it C Topological Vector Spaces 499 is called weak topology on X determined by the pair X; Y/, and it is denoted by X; Y/ Remark C.1 Clearly, X; Y/ is a locally convex topology on X and also it is a Hausdorff topology Let X and Y be Hausdorff topological vector spaces and L.X; Y/ denote the family of continuous linear functions from X to Y Let be the family of bounded S subsets of X whose union is total in X, that is, the linear hull of fU W U g is dense in X Let B be a neighborhood base of in Y, where is the zero element of Y When U runs through , V through B, the family ( M.U; V/ D L.X; Y/ W [ ) h ; xi  V x2U is a neighborhood base of in L.X; Y/ for a unique translation-invariant topology, called the topology of uniform convergence on the sets U , or, briefly, the topology (see [16, pp 79–80]) Lemma C.1 ([16]) Let X and Y be Hausdorff topological vector spaces and L.X; Y/ be the topological vector space under the -topology Then the bilinear mapping h:; :i W L.X; Y/ X ! Y is continuous on L.X; Y/ X We now present an open mapping theorem due to Brezis [17] Theorem C.10 ([17, Théorème II.5]) Let X and Y be two Banach spaces and T W X ! Y be a continuous linear surjective mapping Then there exists a constant T.B1 Œ0/, where Bc Œ0 denotes the closed ball of radius c c > such that Bc Œ0 around in Y and B1 Œ0 is the closed unit ball in X Remark C.2 From Theorem C.10, we see that T W X ! Y is an open mapping Indeed, let U be an open subset of X We show that T.U/ is open Let y T.U/, where y D T.x/ for some x U Let r > such that Br Œx U, i.e., x C Br Œ0 U Then we have y C T.Br Œ0/ T.U/ Due to Theorem C.10, we obtain Brc Œ0 T.Br Œ0/, and consequently, Brc Œy T.U/ Theorem C.11 ([18, Chapter 6, Theorem 1.1]) If X and Y are Banach spaces and T W X ! Y is a linear operator, then T is bounded if and only if it is continuous from the weak topology of X to the weak topology of Y Definition C.16 Let X; d/ be a metric space and A be a nonempty subset of X The diameter of A, denoted by ı.A/, is defined as ı.A/ D supfd.x; y/ W x; y Ag: We close this subsection by presenting the following Cantor’s intersection theorem Theorem C.12 (Cantor’s Intersection Theorem) Let X; d/ be a complete metric space and fAm g be a decreasing sequence (that is, AmC1  Am ) of nonempty closed 500 C Topological Vector Spaces subsets of X such that the diameter of Am ı.Am / ! as m ! Then the T intersection Am contains exactly one point mD1 References Q.H Ansari, Metric Spaces – Including Fixed Point Theory and Set-Valued Maps (Narosa Publishing House, New Delhi, 2010) Also Published by Alpha Science International Ltd Oxford, UK (2010) J.-P Aubin, A Cellina, Differential Inclusions (Springer, Berlin, 1994) J.-P Aubin, H Frankowska, Set-Valued Analysis (Birkhäuser, Boston, 1990) C Berge, Topological Space (Oliver & Boyd, Edinburgh, 1963) R.S Burachik, A.N Iusem, Set-Valued Mappings and Enlargements of Monotone Operators (Springer, New York, 2008) L Górniewicz, Topological Fixed Point Theory of Multivalued Mappings (Kluwer Academic Publishers, Dordrecht, 1999) V.I Istrˇatescu, Fixed Point Theory: An Introduction (D Reidel Publishing Company, Dordrecht, 1981) E Klein, A.C Thompson, Theory of Correspondences, Including Applications to Mathematical Economics (Wiley, New York, 1984) J Jahn, Vector Optimization: Theory, Applications, and Extensions (Springer, Berlin, 2004) 10 C.D Aliprantis, K.C Border, Infinite Dimensional Analysis (Springer, New York, 1994) 11 W.V Petryshyn, P.M Fitzpatrick, Fixed point theorems of multivalued noncompact inward maps J Math Anal Appl 46, 756–767 (1974) 12 W.V Petryshyn, P.M Fitzpatrick, Fixed point theorems of multivalued noncompact acyclic mappings Pac J Math 54, 17–23 (1974) 13 G Köthe, Topological Vector Spaces (Springer, Berlin, 1983) 14 N Bourbaki, Topological Vector Spaces (Springer, Berlin, 1987) 15 L Narice, E Beckenstein, Topological Vector Spaces (Marcel Dekker, Inc., New York, 1985) 16 X.P Ding, E Tarafdar, Generalized vector variational-like inequalities without monotonicity, in [19] (2000), pp 113–124 17 H Brézis, Analyse Fonctionnelle (Masson, Paris, 1983) 18 J.B Conway, A Course in Functional Analysis, 2nd edn (Springer, New York, 1990) 19 F Giannessi (ed.), Vector Variational Inequality and Vector Equilibria: Mathematical Theories (Kluwer Academic, Dordrecht, 2000) Bibliography L Altangerel, Scalarized gap functions for vector variational inequalities via conjugate duality Mongolian Math J 11, 46–54 (2007) L Altangerel, R.I Bot, G Wanka, Conjugate duality in vector optimization and some applications to the vector variational inequality J Math Anal Appl 329, 1010–1035 (2007) L Altangerel, G Wanka, O Wilfer, An oriented distance function application to gap functions for vector variational inequalities, in Optimization, Simulation, and Control, ed by A Chinchuluun et al (Springer, New York, 2013), pp 17–34 Q.H Ansari, J.C Yao, On strong solutions of the generalized implicit vector variational problem Adv Nonlinear Var Inequal 2, 1–10 (1999) C Topological Vector Spaces 501 A.S Antipin, On convergence of proximal methods to fixed point of extremal mappings and estimates of their rate of convergence Comput Math Math Phys 35, 539–551 (1995) M Arana, R Rufián, R Osuna, Necessary and sufficient conditions for weak minima, in Minutes of the XXIV National Congress of Statistics and Operations Research, Almería, Spain (1998) K Bergstresser and P.L Yu, Domination structures and multicriteria problem in N-person games Theor Decis 8, 5–47 (1977) J.M Borwein, On the existence of Pareto efficient points Math Oper Res 8(1), 64–73 (1983) C Charitha, J Dutta, Regularized gap functions and error bounds for vector variational inequalities Pac J Optim 6, 497–510 (2010) 10 G.Y Chen, B.D Craven, A vector variational inequality and optimization over an efficient set ZOR-Math Meth Oper Res 34, 1–12 (1990) 11 G.Y Chen, C.J Goh, X.Q Yang, On gap functions for vector variational inequalities, in [19], pp 55–72 (2000) 12 G.Y Chen, C.J Goh, X.Q Yang, Existence of a solution for generalized vector variational inequalities Optimization, 50(1), 1–15 (2001) 13 G.Y Chen, X.X Huang, Stability results for Ekeland’s " variational principle for vector valued functions Math Meth Oper Res 48, 97–103 (1998) 14 G Cohen, F Chaplais, Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms J Optim Theory Appl 59, 360–390 (1988) 15 H.W Corley, Existence and Lagrange duality for maximization of set-valued functions J Optim Theory Appl 54, 489–501 (1987) 16 H.W Corley, Optimality conditions for maximization of set-valued functions J Optim Theory Appl 58, 1–10 (1988) 17 B.D Craven, Nondifferentiable optimization by smooth approximation Optimization, 17, 3– 17 (1986) 18 G Debreu, A social equilibrium existence theorem Proc Natl Acad Sci U S A 38, 886–893 (1952) 19 F Ferro, A minimax theorem for vector-valued functions, J Optim Theory Appl 60(1), 19–31 (1989) 20 F Ferro, Optimization and stability results through cone lower semicontinuity Set-valued Anal 5, 365–375 (1997) 21 J Fu, Simultaneous vector variational inequalities and vector implicit complementarity problem J Optim Theory Appl 93, 141–151 (1997) 22 C.J Goh, X.Q Yang, On the solutions of vector variational inequalities, in Proceedings of Optimization Techniques and Applications, ed by L Caccetta et al (Curtin University of Technology, Perth, 1998), pp 1158–1164 23 X.H Gong, Efficiency and Henig efficiency for vector equilibrium problems J Optim Theory Appl 108, 139–154 (2001) 24 A Göpfert, Chr Tammer, C Z˘alinescu, On the vectorial Ekeland’s variational principle and minimal points in product spaces Nonlinear Anal 39, 909–922 (2000) 25 R Hartley, On cone-efficiency, cone-convexity and cone-compactness SIAM J Appl Math 34(2), 211–222 (1978) 26 D.W Hearn, The gap function of a convex program Oper Res Lett 1, 67–71 (1982) 27 S Helbig, On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces J Optim Theory Appl 65(2), 257–270 (1990) 28 M.I Henig, Proper efficiency with respect to cones J Optim Theory Appl 36, 387–407 (1982) 29 C Hillermeier, Nonlinear Multiobjective Optimization: A Generalized Homotopy Approach (Birkhäuser Verlag, Basel, 2001) 30 N.-J Huang, J Li, J.C Yao, Gap functions and existence of solutions for system of vector equilibrium problems J Optim Theory Appl 133, 201–212 (2006) 31 T Husain, E Tarafdar, Simultaneous variational inequalities, minimization problems and related results Math Japon 39, 221–231 (1994) 502 C Topological Vector Spaces 32 V Jeyakumar, A generalization of minimax theorem of Fan via a theorem of the alternative J Optim Theory Appl 48, 525–533 (1986) 33 K Kimura, X.H Gong, J.C Yao, Sensitivity analysis of strong vector equilibrium problems J Nonlinear Convex Anal 9(1), 83–94 (2008) 34 K Kimura, Y.C Liou, J.C Yao, Parametric equilibrium problems with applications to optimization problems under equilibrium constraints J Nonlinear Convex Anal 7(2), 237– 243 (2006) 35 S Komlòsi, On the Stampacchia and Minty variational inequalities, in Generalized Convexity and Optimization for Economic and Financial Decisions, ed by G Giorgi, F Rossi (Pitagora Editrice, Bologna, 1998), pp 231–260 36 I.V Konnov, Combined relaxation method for solving vector equilibrium problems Russ Math (Izvestiya Vuzov Math.) 39, 51–59 (1995) 37 I.V Konnov, A general approach to finding stationary point and the solution of related problems Comput Math Math Phys 36, 585–593 (1996) 38 I.V Konnov, A class of combined relaxation methods for decomposable variational inequalities Optimization 51(1), 109–125 (2002) 39 S Kum, G.M Lee, Remarks on implicit vector variational inequalities Taiwan J Math 6(3), 369–382 (2002) 40 G.M Lee, D.S Kim, B.S Lee, Generalized vector variational inequality Appl Math Lett 9(1), 39–42 (1996) 41 G.M Lee, S Kum, On implicit vector variational inequalities J Optim Theory Appl 104(2), 409–425 (2000) 42 J Li, G Mastroeni, Vector variational inequalities involving set-valued mappings via scalarization with applications to error bounds for gap functions J Optim Theory Appl 145, 355–372 (2010) 43 S.J Li, G.Y Chen, Properties of gap function for vector variational inequalities, in Variational Analysis and Applications, ed by F Giannessi, A Maugeri (Springer, Berlin, 2005), pp 605– 631 44 S.J Li, H Yan, G.Y Chen, Differential and sensitivity properties of gap functions for vector variational inequalities Math Meth Oper Res 57, 377–391 (2003) 45 Z.F Li, G.Y Chen, Lagrangian multipliers, saddle points, and duality in vector optimization of set-valued maps J Math Anal Appl 215, 297–316 (1997) 46 Z.F Li, S.Y Wang, A type of minimax inequality for vector-valued mappings, J Math Anal Appl 227, 68–80 (1998) 47 M.B Lignola, J Morgan, Well-posedness for optimization problems with constrains defined by variational inequalities having a unique solution J Global Optim 16(1), 57–67 (2000) 48 D.T Luc, Connectedness of the efficient point sets in quasiconcave vector maximization J Math Anal Appl 122, 346–354 (1987) 49 D.T Luc, N.X Tan, P.N Tinh, Convex vector functions and their subdifferential Acta Math Vietnam 23, 107–127 (1998) 50 D.T Luc and C.A Vargas, A saddle point theorem for set-valued maps, Nonlinear Anal Theory Methods Appl 18, 1–7 (1992) 51 H.-Q Ma, N.-J Huang, M Wu, D O’Regan, A new gap function for vector variational inequalities with an application J Appl Math 2013, Article ID 423040, pages (2013) 52 G Mastroeni, Separation methods for vector variational inequalities Saddle point and gap function, in Nonlinear Optimization and Applications, ed by G Di Pillo, F Giannessi (Kluwer Academic, Dordrecht, 2000), pp 207–217 53 G Mastroeni, On Minty vector variational inequality, in [19] (2000), pp 351–361 54 K.W Meng, S.J Li, Differential and sensitivity properties of gap functions for Minty vector variational inequalities J Math Anal Appl 337, 386–398 (2008) 55 J Morgan, Constrained well-posed two-level optimization problem, in Nonsmooth optimization and Related Topics, ed F.H Clarke, V.F Demyanov, F Giannessi (Plenum Press, New York, 1989), pp 307–325 56 U Mosco, Dual variational inequalities J Math Anal Appl 40, 202–206 (1972) C Topological Vector Spaces 503 57 E Muselli, Upper and lower semicontinuity for set-valued mappings involving constraints J Optim Theory Appl 106, 527–550 (2000) 58 J.F Nash, Equilibrium point in n-person games Proc Natl Acad Sci U S A 36, 48–49 (1950) 59 W Oettli, A remark on vector-valued equilibria and generalized monotonicity Acta Math Vietnam 22(1), 213–221 (1997) 60 R Reemtsen, J.J Rückmann (eds.), Semi-Infinite Progamming (Kluwer Academic, Dordrecht, 1998) 61 W.D Rong, Y.N Wu, Epsilon-weak minimal solution of vector optimization problems with set-valued maps J Optim Theory Appl 103(3), 569–579 (2000) 62 H.H Schaefer, Topological Vector Spaces (Springer, Berlin, 1971) 63 W Song, Vector equilibrium problems with set-valued mappings, in [19] (2000), pp 403–432 64 F Szidarovszky, M.E Gershon, L Duckstein, Techniques for Multiobjective Decision Making in System Management (Elsevier, Amsterdam, 1986) 65 T Tanaka, Generalized quasiconvexities, cone saddle points, and minimax theorem for vectorvalued functions J Optim Theory Appl 81, 355–377 (1994) 66 T Tanaka, Cone-quasiconvexity of vector-valued functions Sci Rep Hirosaki Univ 42, 157– 163 (1995) 67 T Tanaka, Generalized semicontinuity and existence theorems for cone saddle points Appl Math Optim 36, 313–322 (1997) 68 T Tanaka, D Kuroiwa, The convexity of A and B assures intA C B D int.A C B/ Appl Math Lett 6, 83–86 (1993) 69 X.Q Yang, C.J Goh, On vector variational inequalities: applications to vector equilibria J Optim Theory Appl 95, 431–443 (1997) 70 G.X.-Z Yuan, G Isac, K.-K Tan, J Yu, The study of minimax inequalities, abstract economies and applications to variational inequalities and Nash equilibria Acta Appl Math 54, 135–166 (1998) 71 E Zeidler, Nonlinear Functional Analysis and Its Applications, I: Fixed Point Theorems (Springer, New York, 1986) 72 M Zeleny, Games with multiple payoffs Int J Game Theory 4, 179–191 (1976) 73 L.C Zeng, J.C Yao, An existence result for generalized vector equilibrium problems without pseudomonotonicity Appl Math Lett 19, 1320–1326 (2006) Index B-pseudomonotone map weighted, 272 C-h-convex function, 287 strictly, 288 C-h-pseudoconvex function, 288 strongly, 288 weakly, 288 C-bi-pseudomonotone function, 373 C-bounded set, 96 C-closed set, 96 C-compact set, 98 C-complete set, 101 C-continuous function, 117 C-convex function, 102 C-convex-like function, 104 C-diagonally concave function, 362 C-diagonally convex function, 362 C-hemicontinuous function, 124 C-lower hemicontinuous function, 124 C-lower semicontinuous function, 117 C-monotone bifunction, 343 C-properly subodd function, 278 C-pseudocontinuous function, 117 C-pseudoconvex function, 110 C-pseudomonotone bifunction, 343, 348 maximal, 344 strictly, 343 C-pseudomonotone function, 277 strongly, 277 weakly, 278 C-pseudomonotoneC function, 284 C-quasi-strictly convex set, 160 C-quasiconcave-like function, 112 C-quasiconvex function, 103 C-quasimonotone bifunction, 344 C-semicompact set, 98 C-strictly convex set, 160 C-subconvex-like function, 104 C-upper hemicontinuous function, 124 C-upper semicontinuous function, 117 C-upper sign continuous function, 278, 352 Cx -convex function, 178, 254 strictly, 178, 254 Cx -monotone operator, 233 Cx -pseudoconvex function, 178, 254 strongly, 178, 254 weakly, 178, 254 Cx -pseudomonotone operator, 233 strongly, 233 weakly, 233 Cx -pseudomonotoneC operator, 236 Cx -upper sign continuous map, 228 strongly, 227 weakly, 228 ˚-condensing map, 47 -transfer lower semicontinuous function, 71 -transfer upper semicontinuous function, 71 H -continuous set-valued map, 42 H -hemicontinuous map, 305 "-constraint method, 195 "-equilibrium point, 382 "-generalized strong vector equilibrium problem, 473 "-weak vector equilibrium problem, 390 h-vector variational inequality problem, 276 Minty, 276 Minty strong, 276 Minty weak, 276 strong, 276 weak, 276 © Springer International Publishing AG 2018 Q.H Ansari et al., Vector Variational Inequalities and Vector Optimization, Vector Optimization, DOI 10.1007/978-3-319-63049-6 505 506 v-coercive condition, 241 v-coercive condition C1 , 241 v-coercive operator, 245 weakly, 245 v-coercive set-valued map, 442 v-hemicontinuous operator, 228 absolutely convex set, 491 absorbing set, 491 acute cone, additive dual vector equilibrium problem, 453 additive duality, 453 affine combination, affine function, 20 affine hull, affine set, algebraic boundary, 492 algebraic bounded set, 492 algebraic closed set, 492 algebraic closure, 492 algebraic interior, 492 algebraic open set, 492 approximating net, 417 arc-concave-like function, 114 asymptotic cone, 14 balanced set, 491 base, 10 bilinear functional, 498 binary relation, 82 Bouligand tangent cone, 17 boundedly order complete space, 100 Cantor’s intersection theorem, 499 Clarke directional derivative, 37 Clarke generalized subdifferential, 37 closed cone, coercive operator, 245 weakly, 245 completely continuous map, 239, 326 concave function, 21 cone, cone combination, conic hull, contingent cone, 17 continuous set-valued map, 38 convex combination, convex cone, convex function, 20 strictly, 21 convex hull, Index convex set, core of a set, 492 correct cone, Daniell cone, 100 decision (variable) vector, 79 decision variable space, 79 diagonally quasiconvex function, 70 -generalized, 70 Dini directional derivative, 34 lower, 34 upper, 34 directed set, 495 direction of recession, 13 directional derivative, 33 domination factor, 86 domination structure, 86 downward directed function, 113 downward directed set, 113 dual cone, 12 dual generalized weak vector equilibrium problem, 449 dual implicit weak vector variational problem, 365 duality operator, 450 efficient element, 144 efficient solution, 155 dominated, 175 dominated properly in the sense of Henig, 175 dominated strong, 175 dominated weakly, 175 global properly in the sense of Henig, 166 local properly in the sense of Henig, 166 properly in the sense of Benson, 163 properly in the sense of Borwein, 162 properly in the sense of Geoffrion, 167 efficient solution set, 155 Ekeland variational principle, 378 vectorial form, 378 epigraph, 21, 109 equilibrium problem, 65 dual, 67 Minty, 67 equivalence relation, 83 explicitly C-quasiconvex function, 104 externally stable function, 140 feasible objective region, 79 feasible region, 79 Index finite concave-like function, 113 fixed point, 43 fixed point problem, 51 Fréchet derivative, 34 Gâteaux derivative, 34 gap function, 367, 368 for vector equilibrium problems, 369 generalized Cx -monotone set-valued map, 312 strongly, 312 generalized Cx -pseudomonotone set-valued map, 313 strongly, 312 generalized Cx -quasimonotone set-valued map, 325 generalized Cx -upper sign continuous map, 303 strongly, 302 weakly, 303 generalized d-coercive condition, 318 weakly, 318 generalized v-coercive condition, 317, 323 weakly, 317, 323 generalized v-hemicontinuous map, 304 generalized complementarity problem, 60, 248 vector, 248, 324 generalized duality, 450 generalized KKM map, 44 generalized Minty variational inequality problem, 61 weak, 61 generalized variational inequality problem, 58 generalized vector equilibrium problem, 429, 431 abstract, 431 strong, 431 weak, 430 generalized vector variational inequality problem Minty, 300 Stampacchia, 300 507 ideal minimal element, 144 implicit variational problem, 68 implicit weak vector variational problem, 342 improperly efficient solution in the sense of Geoffrion, 167 Jameson lemma, 12 Jensen’s inequality, 22 KKM-map, 43 Kneser minimax theorem, 58 L-condition, 239 line segment, linear combination, linear function, 20 linear hull, linear order, 83 linear scalarization, 182 linearly accessible element, 492 Lipschitz continuous function, 26 locally bounded set, 497 Locally Lipschitz function, 25 lower bound, 87 lower semicontinuous function, 28 lower semicontinuous set-valued map, 38 Hausdorff metric, 42 hemicontinuous function, 53 lower, 52 upper, 52 weighted, 269 hybrid method, 199 maximal element, 87, 144 measure of noncompactness, 496 minimal element, 87, 144 minimal solution, 155 properly in the sense of Geoffrion, 167 minimization problem, 66 Minty lemma, 53 Minty variational inequality, 51 Minty vector equilibrium problem, 340 strong, 340 weak, 340 monotone operator, 30 strictly, 30 strongly, 30 weighted, 269 multicriteria optimization problem, 79 multiobjective optimization problem, 79 multiplicative dual problem, 454 multiplicative duality, 454 multivalued map, 487 ideal efficient element, 144 ideal efficient solution, 160 ideal maximal element, 144 Nadler theorem, 42 Nash equilibrium point, 66 Nash equilibrium problem, 66 508 natural C-quasiconvex function, 103 natural pairing, 498 net, 496 nondominated set, 155, 159 nonlinear complementarity problem, 50 nonlinear scalarization, 192 nonlinear scalarization function, 128, 133 nonsmooth variational inequalities, 54 nontrivial cone, objective function, 79 objective space, 79 open lower section of a set, 46 ordered structure, 83 pairing, 498 paracompact space, 495 parametric strong vector equilibrium problem, 411 parametric weak vector equilibrium problem, 401 parametrically well-posed problems, 417 unique, 418 Pareto efficient solution, 155 Pareto optimal solution, 155 properly in the sense of Geoffrion, 167 Pareto solution dominated, 175 dominated properly in the sense of Henig, 175 dominated strong, 175 dominated weakly, 175 partial order, 83 partition of unity, 496 pointed cone, polar cone strict, 12 positively homogeneous function, 20 precompact set, 496 preorder, 83 proper C-quasimonotone mapping, 286 strongly, 286 weakly, 286 proper cone, properly C-quasiconvex function, 103 properly efficient element, 147 properly efficient solution in the sense of Kuhn-Tucker, 169 properly maximal element, 147 properly minimal element, 147 pseudoconvex function, 29 strictly, 29 pseudolinear function, 29 Index pseudomonotone bifunction, 36 pseudomonotone map, 31 strictly, 31 quasi-order, 83 quasiconvex function, 26 semistrictly, 26 strictly, 26 quasimonotone map, 31 semistrictly, 32 strictly, 32 radially semicontinuous function, 53 lower, 34, 52 upper, 34, 52 regular saddle point problem, 341 relative algebraic interior point, 493 relative boundary, relative interior, relatively compact set, 496 relatively open set, reproducing cone, saddle point problem, 66 selection, 325 continuous, 325 set-valued bifunction H/-C-pseudomonotone, 454 C-pseudomonotone, 440 C-quasimonotone, 445 G-C-pseudomonotone, 451 m.H/-C-pseudomonotone, 455 maximal H/-C-pseudomonotone, 454 maximal C-pseudomonotone, 440 maximal m.H/-C-pseudomonotone, 455 set-valued fixed point problem, 60 set-valued inclusion problem, 59 set-valued Lipschitz map, 43 set-valued map, 487 C-continuous, 125 C-convex, 115 C-lower semicontinuous, 125 C-proper quasiconcave, 115 C-quasiconcave, 114 C-quasiconvex, 115 C-quasiconvex-like, 115 C-upper semicontinuous, 125 H -hemicontinuous, 42 u-hemicontinuous, 42 completely semicontinuous, 327 explicitly ı-C-quasiconvex, 115 explicitly C-quasiconvex-like, 116 Index generalized hemicontinuous, 62 generalized pseudomonotone, 63 generalized weakly pseudomonotone, 63 inverse, 489 properly C-quasiconvex, 115 strictly C-proper quasiconcave, 115 strictly C-quasiconcave, 114 strongly semicontinuous, 327 weakly lower semicontinuous, 436 solid cone, starshaped set, 18 strict order, 83 strict partial order, 83 strict properly C-quasiconvex function, 103 strictly C-convex function, 102 strictly C-quasiconcave-like function, 112 strictly C-quasiconvex function, 103 strictly convex body, 225 strong C-pseudomonotone bifunction, 352 strong C-quasimonotone bifunction, 352 strong efficient element, 144 strong maximal element, 144 strong minimal element, 144 strong order, 83 strong properly C-quasimonotone bifunction, 353 strongly C-complete set, 101 strongly C-diagonally concave function, 363 strongly C-diagonally convex function, 362 strongly v-coercive condition C1 , 240 strongly v-coercive condition C2 , 241 strongly efficient set, 160 strongly efficient solution, 155, 160 strongly nondominated set, 160 strongly nonlinear variational inequality problem, 68 strongly nonlinear weak vector variational inequality problem, 343 subodd function, 20 subspace, support of a function, 495 tangent, 17 topological vector space, 496 locally convex, 498 total preorder, 84 transfer closed-valued map, 45 transfer open-valued map, 45 upper bound, 87 upper semicontinuous function, 28 upper semicontinuous set-valued map, 38 upper sign continuous bifunction, 54 509 variational inequality generalized, 58 weak generalized, 58 variational inequality problem, 48 variational principle, 367, 370, 371 for Minty weak vector equilibrium problems, 373 for weak vector equilibrium problems, 370 for WVEP f ; h/, 375 vector biconjugate function, 139, 365 vector conjugate function, 139, 365 vector equilibrium problem, 339, 340 dual, 340 Minty, 340 strong, 340 weak, 340 vector optimization problem, 79 vector saddle point problem, 341 vector variational inequality problem, 223 Minty, 226 Minty strong, 226 Minty weak, 226 perturbed, 229 perturbed strong, 229 perurbed weak, 229 strong, 223 weak, 224 weak order, 83 weak subdifferential of a vector-valued function, 139 weak subgradient of a vector-valued function, 139 weakly v-coercive condition C1 , 241 weakly v-coercive condition C2 , 241 weakly efficient element, 147, 148 weakly efficient set, 159 weakly efficient solution, 155, 159 weakly maximal element, 148 weakly minimal element, 147, 148 weakly minimal solution, 159 weakly Pareto efficient solution, 155 weakly Pareto optimal solution, 159 Weierstrass theorem, 96 vectorial form, 384 weighted pseudomonotone operator, 269 maximal, 269 weighted variational inequality problem, 266 Minty, 268 well-ordered, 83 Zermelo’s theorem, 83 Zorn’s lemma, 87 ... Equilibrium Problems Vector Optimization Vector Optimization Problem Vector Variational Inequality Vector Variational Inequalities Vector Variational Inequality Problem Vector Variational Inequality... duality, and sensitivity analysis It is worth mentioning that the vector equilibrium problems include vector variational inequalities, nonsmooth vector variational inequalities, and vector optimization. .. to present a mathematical theory of vector optimization, vector variational inequalities, and vector equilibrium problems The well-posedness and sensitivity analysis of vector equilibrium problems