Mathematical methods for solving equilibrium, variational inequality and fixed point problems

VIETNAM NATIONAL UNIVERSITY-HO CHI MINH CITY UNIVERSITY OF SCIENCES PHAN TU VUONG MATHEMATICAL METHODS FOR SOLVING EQUILIBRIUM, VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS PhD Thesis in Mathematics Ho Chi Minh City - 2014 VIETNAM NATIONAL UNIVERSITY-HO CHI MINH CITY UNIVERSITY OF SCIENCES PHAN TU VUONG MATHEMATICAL METHODS FOR SOLVING EQUILIBRIUM, VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS Speciality: Mathematical Optimization Code: 62 46 20 01 Reviewer 1: Prof NGUYEN XUAN TAN Reviewer 2: Assoc Prof NGUYEN DINH PHU Reviewer 3: Assoc Prof LAM QUOC ANH Anonymous Reviewer 1: Prof NGUYEN XUAN TAN Anonymous Reviewer 2: Dr DUONG DANG XUAN THANH Supervisor 1: Assoc Prof NGUYEN DINH Supervisor 2: Prof VAN HIEN NGUYEN Ho Chi Minh City - 2014 DECLARATION I hereby declare that the work contained in this thesis has never previously been submitted for a degree, diploma or other qualifications in any University or Institution and that, to the best of my knowledge and belief, the thesis contains no material previously published or written by another person except when due reference is made in the thesis itself Ph.D Student Phan Tu Vuong Acknowledgements There have been many people who have helped me through my graduate studies In the next few lines, I would like to point out a few of these people to whom I am especially in debt First and foremost, I would like to express my deepest appreciation to my supervisors, Associate Professor Nguyen Dinh (VNU-HCM, International University) and Professor Van Hien Nguyen (Institute for Computational Science and Technology (ICST) and University of Namur, Belgium) The completion of the academic research work that led to the results published in this thesis would have not been possible without the constant encouragement, support, and advice received from Professor Van Hien Nguyen and Professor Jean Jacques Strodiot (both at ICST and University of Namur) and as such, my gratitude goes to them I would like also to thank all members of the dissertation committee and two anonymous reviewers for their useful comments and suggestions This thesis presents the results of the research carried out at ICST during the period March 2011 - November 2013 This research work was funded by the Department of Science and Technology at Ho Chi Minh City Support provided by the ICST is gratefully acknowledged I am grateful to my former advisor, Associate Professor Nguyen Bich Huy (University of Pedagogy) who introduced and helped me to start my graduate studies I would also like to thank my friends at ICST and University of Technical Education Ho Chi Minh City for their kind help Finally, to my family, I owe much more than a few words can capture I thank them for all the love and support through all the time Ho Chi Minh City, May 2014 Phan Tu Vuong Contents Introduction Preliminaries 2.1 Elements of convex analysis 2.2 The projection operator and useful lemmas 2.3 Fixed point problems 2.4 Variational inequalities 2.5 Equilibrium problems 2.5.1 Some particular equilibrium problems 2.5.2 Solution methods for solving equilibrium problems 2.6 Previous works 2.6.1 The hybrid projection method 2.6.2 The shrinking projection method 2.6.3 The viscosity approximation method 2.6.4 The extragradient method Hybrid Projection Extragradient Methods 3.1 A hybrid extragradient algorithm 3.2 Extragradient algorithms with linesearches 3.3 Shrinking projection methods 3.4 The particular case of variational inequalities 3.5 Numerical illustrations Extragradient Viscosity Methods 4.1 An extragradient viscosity algorithm 4.2 A linesearch extragradient viscosity algorithm 4.3 Applications to variational inequality problem 4.4 Numerical illustrations 9 12 14 16 17 19 21 23 24 24 25 25 27 27 34 44 50 53 57 57 64 73 74 A Mathematical Analysis of Subgradient 5.1 Previous works and motivation 5.2 A general algorithm 5.3 Two projected subgradient algorithms 5.4 Some interesting special cases Conclusions and Future Work Viscosity Methods 78 78 80 90 94 95 Basic Notation and Terminology H: a real Hilbert space ·, · : the inner product of the space · : the norm of the space domf : the domain of a function f ∂f : the subdifferential of a convex function f ∂C: the boundary of a set C F ix(S): the set of fixed points of operator S V I(F, C): the variational inequality problem whose objective operator is F and whose feasible set is C SolV I(F, C): the solution set of V I(F, C) EP (f, C): the equilibrium problem whose objective function is f and whose feasible set is C SolEP (f, C): the solution set of EP (f, C) Chapter Introduction Let H be a real Hilbert space with inner product ·, · and norm · , respectively Consider a nonempty closed and convex set C ⊂ H and a bifunction f : C × C → R such that f (x, x) = for all x ∈ C The equilibrium problem, denoted by EP (f, C), consists of finding x∗ ∈ C such that f (x∗ , y) ≥ for all y ∈ C The solution set of EP (f, C) will be denoted by SolEP (f, C) To the best of our knowledge, the term “equilibrium problem” was coined in [16] (see also [66]), but the problem itself was studied by Ky Fan in [35] (for historical comments see [36]) Equilibrium problems have been extensively studied in recent years (see, for example, [13, 15, 16, 28, 32, 45, 51, 52, 57, 60, 61, 63, 65, 67, 74, 75, 81, 82, 85, 89, 97] and the references therein) It is well known that they include, as particular cases, scalar and vector optimization problems, saddle-point problems, variational inequalities (monotone or otherwise), Nash equilibrium problems, complementarity problems, fixed point problems, and other problems of interest in many applications (see, for instance, the recent books [51, 40]) Let F : H → H be a given mapping If f (x, y) := F x, y − x , for all x, y ∈ C, then each solution x∗ ∈ C of the equilibrium problem EP (f, C) is a solution of the variational inequality F x∗ , y − x∗ ≥ for all y ∈ C, and vice versa Variational inequalities have shown to be important mathematical models in the study of many real problems, in particular in network equilibrium models ranging from spatial price equilibrium problems and imperfect competitive oligopolistic market equilibrium problems to general financial or traffic equilibrium problems (see, for example, the recent monographs [70, 34]) A point x∗ ∈ C is called a fixed point of a mapping S : H → H if Sx∗ = x∗ The set of fixed points of S is the set F ix(S) := {x ∈ H : Sx = x} The computation of fixed points is important in the study of many problems including inverse problems in science and engineering (see, for example, [11]) Construction of fixed points of nonexpansive mappings (i.e., Sx − Sy ≤ x − y for all x, y ∈ H) is an important subject in nonlinear operator theory and its applications [21]; in particular, in image recovery and signal processing [20] In 2007, S Takahashi and W Takahashi [93] introduced an iterative scheme by the viscosity approximation method for finding a common element of the solution set of EP (f, C) and the set of fixed points of a nonexpansive mapping S in a real Hilbert space and obtained, under certain appropriate conditions, a strong convergence theorem for such scheme Motivated and inspired by the ongoing results of obtaining strong convergence theorems for approximation of common elements of equilibrium problems and fixed point problems [80, 25, 94, 48, 49, 87, 31, 46, 3], we introduce, as a first contribution of our thesis, a new and different algorithm from the existing algorithms in the literature Indeed, the method used in most papers for solving the equilibrium problem EP (f, C) is the proximal point method [22, 53, 54, 81, 86, 92] This method consists in solving at each iteration a nonlinear variational inequality problem which seems not easy to solve [63, 67] In this thesis, we propose instead, to use an extragradient method with or without the incorporation of a linesearch At each iteration, one or two convex minimization problems must be solved depending on the presence or not of a linesearch Working in a Hilbert space, these methods usually generate sequences of iterates that only converge weakly to a solution of the problem while it is well known that strongly convergent algorithms are of fundamental importance for solving problems in infinite dimensional spaces [10] For obtaining the strong convergence from the weak convergence without additional assumptions on the data of the problem we propose to use the hybrid projection method [47, 68, 69, 71] In this method, the solution set of the problem is outer approximated by a sequence of polyhedral subsets and the sequence of iterates converges to the orthogonal projection of a given point onto the solution set We report some preliminary numerical tests to show the behavior of the proposed algorithms Chapter will provide a detailed description of our first contribution to the thesis Chapter and Chapter 5, which constitute the second and third contribution to our thesis work, will consider a class of ‘hierarchical optimization’: a variational inequality problem constrained by a fixed point problem and/or an equilibrium problem This class of problems (also known as ‘bilevel problems’) has been studied extensively in the literature (see, for example, [2, 32] and the references cited therein) Such hierarchical and equilibrium models are of interest in energy markets, and particularly in electricity and natural gas markets [40] More precisely, our aim in the second contribution is to study new numerical algorithms for finding a solution of a variational inequality problem whose constraint set is the set of common elements of the set of fixed points of a mapping and the set of solutions of an equilibrium problem in a real Hilbert space The strategy is to use the extragradient methods with or without linesearch instead of the proximal methods to solve equilibrium problems To obtain the strong convergence of the iterates generated by these algorithms, a regularization procedure is added (the so-called viscosity approximation method; see, for example, [1, 55, 57, 64, 93]) after an extragradient method Preliminary numerical tests are presented to show the behavior of the extragradient methods when a viscosity step is added For more details, please see Chapter The third contribution of this thesis, Chapter 5, contains some numerical methods for finding a solution of a variational inequality problem over the solution set of an equilibrium problem defined on a subset C of a real Hilbert space The strategy used in this chapter is to combine viscosity-type approximations with projected subgradient techniques to obtain the strong convergence of the iterates to a solution of the problem First a general scheme is considered, and afterwards two practical realizations are studied depending on the characteristics of the feasible set C When this set is simple, the projections onto C can be easily computed and all the iterates remain in C On the other hand, when C is described by convex inequalities, the projections onto C are replaced by projections onto half-spaces containing C with the consequence that most iterates are outside the feasible set C This strategy has been recently used in [56], and partly in [12, 13, 85], for finding a solution of a variational inequality problem over the solution set of another variational inequality problem defined on C Here we develop a similar approach but for equilibrium constraints instead of variational inequality constraints For more details, please see Chapter The results presented in this dissertation have been published in • Journal of Optimization Theory and Applications (SCI) and Vietnam Journal of Mathematics for Chapter ([98, 90]); • Optimization (SCIE) for Chapter ([99]); • Journal of Global Optimization (SCI) for Chapter ([100]) Step Choose the sequences {αn } ⊂ [0, 1), {ρn } ⊂ (0, ∞), {βn } ⊂ (0, 1) and { n } ⊂ (0, 1) Step Let x0 ∈ C Set n = Step Select gn ∈ ∂2n f (xn , xn ) Compute un = gn + αn F xn and λn = Step Compute xn+1 = PC (xn − λn un ) Step Set n := n + 1, and go to Step βn max{ρn , un } We note that some existing algorithms can be recovered from Algorithm 5.1.2 when the function f is defined by (5.41): (i) With the parameter ρn = ρ for all n, Algorithm 5.1.2 coincides with Algorithm proposed by Maing´e in [56]; see also Algorithm 3.1 in Xia et al [104] where T is a θ-pseudomonotone+ ∗ operator; (ii) When F (·) = · − u with (u ∈ H), Algorithm 5.1.2 is nothing but the method proposed by Tang and Huang in [96]; (iii) When F (·) = · − u with (u ∈ H) and T = 0, Algorithm 5.1.2 collapses to Algorithm proposed by Maing´e in [55]; (iv) When F = 0, ρn = and αn = for all n, Algorithm 5.1.2 reduces to the method proposed by Xia et al in [103]; (v) When F = 0, T = 0, and ρn = ρ for all n, Algorithm 5.1.2 coincides with the method proposed in [6] Now we can state our strong convergence theorem for Algorithm 5.1.2 as follows Theorem 5.3.1 Let C be a nonempty, closed and convex subset of a Hilbert space H Let f be a bifunction from H × H into R satisfying conditions (A1bis),(A2), (A7)(A9) such that Ω = ∅ Suppose that the sequences of parameters {ρn }, {βn }, { n } and {αn } satisfy conditions (CP) Then the sequence {xn }, generated by Algorithm 5.1.2, converges strongly to x∗ ∈ Ω, the unique solution of V I(F, Ω) 93 Proof We only have to prove that the assumption of monotonicity of f used in Theorem 5.2.1 can be replaced by the assumption that f is pseudomonotone on C Since this assumption is used in the second part of Claim of Theorem 5.2.1 when Lemma 2.5.1 (ii) is applied, we replace Claims and of this theorem by the following claim: For every x∗ ∈ Ω and all n, we have xn+1 − x∗ ≤ xn − x∗ + 2λn f (xn , x∗ ) − 2λn αn F xn , xn − x∗ + 3βn2 (5.42) Since xn ∈ C and x¯n = PC xn = xn , this inequality can be proven immediately using (5.7) and (5.13) Finally the proof of the corresponding Claim is the same as in Theorem 5.2.1 if we observe that θ can be taken equal to zero and x¯n = xn 5.4 Some interesting special cases In this section, we consider the special case when F x = x − u with u a fixed element in H This mapping F is 1-Lipschitz and 1-strongly monotone, and in this situation, the variational inequality problem V I(F, Ω) becomes the problem of finding the projection of u onto the solution set of the equilibrium problem Concretely, Step of Algorithm 5.1.2 becomes Step 2a Select gn ∈ ∂2n f (xn , xn ) and compute un = gn + αn (xn − u) and λn = βn max{ρn , un } Under the same assumptions as the ones used in Theorem 5.3.1, we can conclude that the sequence {xn } generated by the modified algorithm converges strongly to x∗ ∈ Ω, the projection of u onto Ω Remark 5.4.1 When we choose u = x0 in Step 2a, the sequence {xn } converges strongly to x∗ = PΩ (x0 ) So the choice of u is crucial to obtain a desired type of solution For example, when u = 0, the sequence {xn } converges strongly to x∗ = PΩ (0), the minimum-norm solution in the solution set Ω Finding such a solution is important in many applications A typical example is the least-squares solution to the constrained linear inverse problem (see, for instance, [84]) 94 Chapter Conclusions and Future Work In conclusion, we have proposed some new algorithms to solve the following three problems: • (P1 ): Finding the projection of a given vector onto the common elements of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space • (P2 ): Finding a solution of a variational inequality problem whose feasible set is the set of all common elements of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in a Hilbert space • (P3 ): Finding a solution of a variational inequality problem whose feasible set is the set of solutions of an equilibrium problem in a Hilbert space All of them generate a strongly convergent sequence of iterates These proposed methods are either based on • a hybrid projection method or a shrinking projection method combined with an extragradient method with and without a linesearch for solving Problem (P1 ); • or a viscosity approximation method combined with an extragradient method with and without a linesearch for solving Problem (P2 ); • or a projected subgradient method combined with a viscosity approximation method for solving Problem (P3 ) A possible future study would be to combine in a clever way the hybrid projection method or the shrinking projection method with the extragradient method in order to obtain a strongly convergent algorithm for solving not only Problem (P1 ) but also the more general Problem (P2 ) So doing, we would obtain a unified algorithmic framework for studying this kind of problems 95 Publications from this thesis P T Vuong, J J Strodiot, V H Nguyen (2012), “Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems”, J Optim Theory Appl., 155, 605-627 J J Strodiot, V H Nguyen, P T Vuong (2012), “Strong convergence of two hybrid extragradient methods for solving equilibrium and fixed point problems”, Vietnam J Math., 40, 371-389 P T Vuong, J J Strodiot, V H Nguyen (2013), “On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space”, Optimization, DOI:10.1080/02331934.2012.759327 P T Vuong, J J Strodiot, V H Nguyen (2014), “Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces”, J Glob Optim., 59, 173-190 96 Author’s Publications and Conferences Publications P T Vuong, J J Strodiot, V H Nguyen (2012), “Extragradient methods and linesearch algorithms for solving Ky Fan inequalities and fixed point problems”, J Optim Theory Appl., 155, 605-627 J J Strodiot, V H Nguyen, P T Vuong (2012), “Strong convergence of two hybrid extragradient methods for solving equilibrium and fixed point problems”, Vietnam J Math., 40, 371-389 P T Vuong, J J Strodiot, V H Nguyen (2013), “On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space”, Optimization, DOI:10.1080/02331934.2012.759327 P T Vuong, J J Strodiot, V H Nguyen (2014), “Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces”, J Glob Optim., 59, 173-190 P D Khanh, P T Vuong (2014), “Modified projection method for strongly pseudomonotone variational inequalities ”, J Glob Optim., 58, 341-350 D S Kim, P T Vuong, P D Khanh (2013), “On the qualitative properties of strongly pseudomonotone variational inequalities”, Submitted to J Glob Optim P T Vuong, J J Strodiot, V H Nguyen (2013), “A gradient projection method for solving split equality and split feasibility problems in Hilbert spaces”, Submitted to Optimization Conferences The 8th Vietnamese Mathematical Conference, August 2013, Nha Trang, Vietnam Talk: See Ref above The Vietnam-France 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Math Comput., 186, 1551-1558 [109] C Zalinescu (2002), Convex Analysis in General Vector Spaces, World Scientific, Singapore 107 [...]... obtain xn+1 satisfying (2.9) 2.6 Previous works For finding a common solution of an equilibrium problem and a fixed point problem, the strategy is to combine a method for solving equilibrium problems with a method for solving fixed point problems Most of the methods for solving equilibrium problems in the literature are based on the proximal point method These methods require that the function f is assumed... bk+1 } Then {τ (n)}n≥n0 is a nondecreasing sequence verifying lim τ (n) = ∞, n→∞ and, for all n ≥ n0 , the following two estimates hold bτ (n) ≤ bτ (n)+1 and 13 bn ≤ bτ (n)+1 2.3 Fixed point problems Let S : H → H be a mapping, the fixed point problem associated with S is to find a point x∗ ∈ H such that x∗ = Sx∗ The fixed point set of S is denoted F ix(S) Definition 2.3.1 Let C be a subset of H The... Solving the complementarity problem amounts to solving EP (f, C) with f (x, y) = F x, y − x (e) Variational inequality problems Given a nonempty closed convex set C ⊂ Rn and a mapping F : Rn → Rn , the Stampacchia variational inequality problem asks to determine a point x∗ ∈ C such that F x∗ , y − x∗ ≥ 0 for any y ∈ C Solving this problem amounts to solving EP (f, C) with f (x, y) = F x, y − x If... solution methods for equilibrium problems in this subsection Other interesting solution methods for equilibrium problems can be found in [15] To begin with, let us recall the two basic assumptions for the function f associated with the equilibrium problem EP (f, C): (i) f (x, x) = 0 for all x ∈ C; (ii) f (x, ·) is convex, subdifferentiable and lower semicontinuous for all x ∈ C (a) Fixed point method This... recall some definitions and fundamental results related to the theory of convex analysis and nonlinear mappings in Hilbert spaces We focus mainly on the background material needed to approach our work, specially the existing results for fixed point problems, variational inequalities and equilibrium problems The interested reader can find more comprehensive informations in these fields, for example, in [33,... in combining the proximal point method for solving equilibrium problems with the CQ projection method for solving fixed point problem Here the sequence {xn } is generated as follows: Given x0 ∈ C, compute for all n ∈ IN ,  1   zn ∈ C such that f (zn , y) + rn y − zn , zn − xn ≥ 0 for every y ∈ C       wn = (1 − αn )zn + αn Szn    Cn = {z ∈ C : tn − z ≤ xn − z } and      Dn = {z ∈ C... particular fixed point of S For example, if g(x) = x0 for every x ∈ C, then the sequence {xn } converges strongly to the projection of x0 onto the fixed point set F ix(S) Related papers on viscosity approximation method can be found in [1, 18, 55, 57, 64, 78, 105] 2.6.4 The extragradient method When the problem is to find a common solution to a variational inequality problem and a fixed point problem,... finding x∗ ∈ C and u∗ ∈ F x∗ such that u∗ , y − x∗ ≥ 0 for any y ∈ C amounts to solving EP (f, C) with f (x, y) = max u, y − x u∈F x n n Given two mappings F, g : R → R and a function h : Rn → (−∞, +∞], another kind of generalized variational inequality problem asks to find a point x∗ ∈ Rn such that F x∗ , y − g(x∗ ) + h(y) − h(g(x∗ )) ≥ 0, for every y ∈ Rn Solving this problem amounts to solving EP... EP (f, C) with C = Rn and f (x, y) = F x, y − g(x) + h(y) − h(g(x)) (f ) Fixed point problems Given a closed set C ⊂ H, a fixed point of a mapping S : C → C is any x∗ such that Sx∗ = x∗ Finding a fixed point amounts to solving EP (f, C) with f (x, y) = x − Sx, y − x 20 If S : C ⇒ C is a set-valued mapping with compact values, then finding x∗ ∈ C such that x∗ ∈ F x∗ amounts to solving EP (f, C) with... − x ≥ 0 for all x, y ∈ C; (d) L-Lipschitz continuous on C (for some L > 0) if Fx − Fy ≤ L x − y for all x, y ∈ C The implications (a) ⇒ (b) and (b) ⇒ (c) are obvious Proposition 2.4.1 [42] Let F : H → H be a L-Lipschitz continuous and γ-strongly monotone mapping and C is a nonempty closed convex set of H, then the variational inequality problem V I(F, C) has a unique solution 2.5 Equilibrium problems