VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS PHAM DUY KHANH SOLUTION METHODS FOR PSEUDOMONOTONE VARIATIONAL INEQUALITIES Speciality: Applied Mathematics Speciality code: 62 46 01 12 SUMMARY DOCTORAL DISSERTATION IN MATHEMATICS HANOI - 2015 The dissertation was written on the basis of the author’s research works carried at Institute of Mathematics, Vietnam Academy of Science and Technology. Supervisors: 1. Prof. Dr. Hab. Nguyen Dong Yen 2. Dr. Trinh Cong Dieu First referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Second referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Third referee: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . To be defended at the Jury of Institute of Mathematics, Vietnam Academy of Science and Technology: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . on . . . . . . . . . . . . . . . . . . . . . 2015, at . . . . . . . . . o’clock . . . . . . . . . . . . . . . . . . . . . . . . . . . The dissertation is publicly available at: • The National Library of Vietnam • The Library of Institute of Mathematics Introduction Monotone operators have been studied since the early 1960s. F. Browder systematically employed the monotonicity of operators to study various prob- lems related to nonlinear elliptic partial differential equations. Independently, P. Hartman and G. Stampacchia studied variational inequalities (VIs for brevity) with monotone operators. Until now, monotone VIs continue to be a subject of the concern of many researchers. Different solution methods have been proposed for monotone VIs: the projection method, the Tikhonov reg- ularization method, the proximal point method, the extragradient method, etc. The concept of pseudomonotone operator introduced by S. Karamardian (1976) is an important generalization of monotone operator. Inspired by this paper, S. Karamardian and S. Schaible (1990) introduced several general- ized monotonicity concepts such as strict pseudomonotonicity, strong pseu- domonotonicity, and quasimonotonicity. For each type of generalized mono- tonicity of operators, these authors established a relation to the corresponding type of generalized convexity of functions. It turns out that pseudomonotone operator is a special case of quasimonotone operator. In the last decade, solu- tion existence and solution methods for pseudomonotone and quasimonotone VIs have been studied in many books and papers. The two-volume book of F. Facchinei and J.S. Pang (2003) and the handbook edited by N. Hadjisav- vas, S. Koml´osi, and S. Schaible (2005) are among the most cited references in this field. Facchinei and Pang (2003) raised a question about the convergence of the Tikhonov regularization method (TRM) for pseudomonotone VIs. With the aid of a solution existence theorem based on the degree theory and some interesting arguments, N. Thanh Hao (2006) solved the question in the af- firmative. Namely, she proved that if the original problem has a solution, then the Tikhonov regularized problem has a compact nonempty solution set which diameter tends to zero, and any selection of the solution mapping 1 converges to the least-norm solution of the original problem. The results of Facchinei and Pang on solution existence of pseudomonotone VIs have been extended by B.T. Kien, J C. Yao, and N.D. Yen (2008) to VIs and set-valued VIs in reflexive Banach spaces. The results of Thanh Hao on the convergence of the TRM for pseudomonotone VIs have been developed to VIs in Hilbert spaces by N.N. Tam, J C. Yao, and N.D. Yen (2008). For monotone VIs, the convergence of the iterative sequence generated by the proximal point algorithm (PPA) and the applicability of the algorithm (in the exact form as proposed by B. Martinet (1970), or in the inexact form as proposed by R.T. Rockafellar (1976)) are a novel research theme in this direction. For pseudomonotone VIs in Hilbert spaces, N.N. Tam, J C. Yao, and N.D. Yen (2008) have obtained some new results on the convergence of the exact PPA and inexact PPA. The auxiliary problems of the TRM and of the PPA, applied to pseu- domonotone VIs, may not be pseudomonotone, or may remain without any solution (if one considers the infinite-dimensional Hilbert space setting). In addition, if the auxiliary problems have a solution then they may have mul- tiple solutions. These phenomena indicate that the auxiliary problems can be more difficult than the original one. A natural question arises: If there is any algorithm that can solve pseudomonotone VIs in an effective way? The extragradient method (EGM) proposed by G.M. Korpelevich (1976) is such an algorithm. The thesis has five chapters. Chapter 1 recalls some basic notions like variational inequality problem, complementarity problem, monotonicity, pseudomonotonicity, and metric pro- jection. Several fundamental solution methods for monotone VIs are also presented. Chapter 2 deals with some questions related to applying the TRM for pseu- domonotone VIs. Solution uniqueness for the regularized problems is studied in two cases: unconstrained VIs and linear complementarity problems. The pseudomonotonicity of the regularized mappings of an affine mapping defined on a polyhedral convex set is investigated. Chapter 3 presents a modified projection method for solving strongly pseu- domonotone VIs. Strong convergence and error estimates for the iterative sequences are investigated in two versions of the method: the stepsizes are chosen arbitrarily from a given fixed closed interval and the stepsizes form 2 a non-summable decreasing sequence of positive real numbers. In addition, an interesting class of strongly pseudomonotone infinite-dimensional VIs is considered. Chapter 4 is devoted to a modified EGM for solving pseudomonotone VIs in Hilbert spaces. The convergence and convergence rate of the iterative sequences generated by this method are studied. Chapter 5 proposes a new EGM for solving strongly pseudomonotone VIs in Hilbert spaces. A detailed analysis of the iterative sequences’ convergence and of the range of applicability of the method is provided. The results of the thesis were reported by the author at - Seminar of Department of Numerical Analysis and Scientific Computing of Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi; - Summer School “Variational Analysis and Applications”, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi (June 20–25, 2011); - The 8 th Vietnam-Korea Workshop “Mathematical Optimization Theory and Applications”, University of Dalat (December 8–10, 2011); -The VMS-SMF Joint Congress at University of Hue (August 20–24, 2012). 3 Chapter 1 Preliminaries The concepts of variational inequality, complementarity problem, metric pro- jection, together with three basic solution methods for variational inequalities (the Tikhonov regularization method, the proximal point algorithm, the ex- tragradient method) are described in this chapter. 1.1 Variational Inequalities and Complementarity Prob- lems Let K be a nonempty subset of a real Hilbert space (H, ., .) and let F : K → H be a single-valued mapping. The variational inequality defined by K and F which is denoted by VI(K, F ) is the problem of finding a vector u ∗ ∈ K such that F (u ∗ ), u − u ∗  ≥ 0, ∀u ∈ K. (1.1) The set of solutions to this problem is denoted by Sol(K, F ). The complementarity problem given by a convex cone K and a mapping F : K → H is the problem of finding a vector u ∗ ∈ H with u ∗ ∈ K, F(u ∗ ) ∈ K ∗ , F(u ∗ ), u ∗  = 0, (1.2) where K ∗ := {d ∈ H : d, u ≥ 0 ∀u ∈ K} is the dual cone of K. Problem (1.2) is abbreviated to CP(K, F ). If u ∈ K and F (u) ∈ K ∗ then u is called a feasible vector of CP(K, F ). If the problem CP(K, F ) has a feasible vector, it is said to be feasible. When H = IR n , F is an affine mapping, i.e., F(u) = Mu + q with M ∈ IR n×n , 4 q ∈ IR n , and K = IR n + (in this case K ∗ = IR n + ), CP(K, F ) becomes the linear complementarity problem LCP(M, q): u ∗ ≥ 0, Mu ∗ + q ≥ 0, Mu ∗ + q, u ∗  = 0. (1.3) Here the inequalities are taken componentwise. The solution set of this prob- lem is denoted by Sol(M, q). 1.2 Monotonicity and Generalized Monotonicity A mapping F : K ⊂ H → H is said to be (a) strongly monotone if there exists γ > 0 such that F (u) − F (v), u − v ≥ γu −v 2 ∀u, v ∈ K; (b) strongly pseudomonotone if there exists γ > 0 such that, for all u, v ∈ K, F (u), v − u ≥ 0 =⇒ F (v), v − u ≥ γu − v 2 ; (c) monotone if F (u) − F (v), u −v ≥ 0 for all u, v ∈ K; (d) pseudomonotone if, for all u, v ∈ K, F (u), v − u ≥ 0 =⇒ F (v), v − u ≥ 0; (e) quasimonotone if, for all u, v ∈ K, F (u), v − u > 0 =⇒ F (v), v − u ≥ 0. 1.3 Metric Projection Let K ⊂ H be a closed convex set. Then for each u ∈ H, there is a unique v ∈ K such that u − v = inf w∈K u − w. (1.4) The unique vector v satisfying (1.4) is called the metric projection of u on K. It is denoted by P K (u). Several basic properties of the metric projection are recalled in the thesis. 5 1.4 The Tikhonov Regularization Method Consider the problem VI(K, F) in a real Hilbert space H. Denote the identity mapping of H by I, and put F ε = F + εI for every ε > 0. To solve VI(K, F ), one solves the sequence of problems VI(K, F ε k ) where {ε k } is a sequence of positive real numbers converging to zero and F ε k = F + ε k I. For each k ∈ IN, one selects a solution u k ∈ Sol(K, F ε k ) and compute the limit lim k→∞ u k . When such limit exists, we may hope that the obtained vector is a solution of VI(K, F ). To terminate the computation process after a finite number of steps and to obtain an approximate solution of VI(K, F ), one has to introduce a stopping criterion. For example, we can terminate the computation when u k − u k−1  ≤ θ where θ > 0 is a constant. Two basic convergence theorems for the Tikhonov regularization are re- called in the thesis. 1.5 The Proximal Point Algorithm Choose a point u 0 ∈ H and a sequence {ρ k } of positive real numbers satisfying the condition ρ k ≥ ρ > 0 for all k ∈ IN. If u k−1 has been defined, one can choose u k as any solution of the auxiliary problem VI(K, F (k) ) where F (k) (u) = ρ k F (u) + u − u k−1 , u ∈ K, (1.5) that is u k ∈ K and ρ k F (u k ) + u k − u k−1 , v − u k  ≥ 0, ∀v ∈ K. (Suppose that Sol(K, F (k) ) is nonempty.) If the iterative scheme yields a sequence {u k }, then one computes the limit lim k→∞ u k in the norm topology or in the weak topology of H. When the limit exists, one may hope that the obtained element belongs to the solution set of VI(K, F ). To terminate the computation process after a finite number of steps and to obtain an approximate solution of VI(K, F), one introduces a stopping criterion. For example, one can terminate the computation when u k − u k−1  ≤ θ where θ > 0 is a constant. Basic convergence theorems for the proximal point algorithm are recalled in the thesis. 6 1.6 The Extragradient Method The extragradient method executes two projections per iteration. Suppose that F is Lipschitz continuous on K with the Lipschitz constant L > 0, that is F (u) − F (v) ≤ Lu − v, ∀u, v ∈ K. (1.6) Algorithm 1.1 Data: u 0 ∈ K and α ∈ (0, 1/L). Step 0: Set k = 0. Step 1: If u k = P K (u k − αF (u k )), stop. Step 2: Compute ¯u k = P K (u k − αF (u k )), ¯u k+1 = P K (u k − αF (¯u k )); set k ← k + 1 and go to Step 1. Two convergence theorems for the extragradient method are recalled in the thesis. 7 Chapter 2 On the Tikhonov Regularization Method and the Proximal Point Algorithm for Pseudomonotone Problems This chapter presents our partial solutions for the some open questions about the solution uniqueness of the regularized problem of a pseudomonotone VI and the preservation of the pseudomonotonicity under the regularization. 2.1 Open Questions on Pseudomonotone Variational Inequalities Open questions. If K ⊂ IR n is a nonempty closed convex set, F : K → IR n is a continuous pseudomonotone mapping, and the problem VI(K, F ) has a solution, then there exists ε 1 > 0 such that the mapping F ε = F + εI is pseudomonotone for each ε ∈ (0, ε 1 )? Is there any ε 2 > 0 such that the problem VI(K, F ε ) has a unique solution for every ε ∈ (0, ε 2 )? 2.2 Solution Uniqueness of the Regularized Problems Let K be a subset of IR n . A mapping F : K → IR n is said to be pseudoaffine on K if F and −F are both pseudomonotone. 8 . existence and solution methods for pseudomonotone and quasimonotone VIs have been studied in many books and papers. The two-volume book of F. Facchinei and J.S. Pang (2003) and the handbook edited. bound for the constant a and an upper bound for the constant b in Algorithm 4.1 and in Theorems 4.1-4.3. Example 4.1 Consider the problem VI(K, F ) with K = IR and F(u) = u. We can easily check. Vietnam Academy of Science and Technology, Hanoi; - Summer School “Variational Analysis and Applications”, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi (June 20–25, 2011); -