Các phương pháp giải bất đẳng thức biến phân giả đơn điệu Chuyên ngành Toán ứng dụng

Different solution methods havebeen proposed for monotone VIs: the projection method, the Tikhonov reg-ularization method, the proximal point method, the extragradient method,etc.. In th

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

PHAM DUY KHANH

Speciality: Applied MathematicsSpeciality code: 62 46 01 12

SUMMARYDOCTORAL 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

Monotone operators have been studied since the early 1960s F Browdersystematically 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 forbrevity) with monotone operators Until now, monotone VIs continue to be asubject of the concern of many researchers Different solution methods havebeen 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 thispaper, 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 correspondingtype of generalized convexity of functions It turns out that pseudomonotoneoperator is a special case of quasimonotone operator In the last decade, solu-tion existence and solution methods for pseudomonotone and quasimonotoneVIs 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 vas, S Koml´osi, and S Schaible (2005) are among the most cited references

Hadjisav-in this field

Facchinei and Pang (2003) raised a question about the convergence of theTikhonov regularization method (TRM) for pseudomonotone VIs With theaid of a solution existence theorem based on the degree theory and someinteresting 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 solutionset which diameter tends to zero, and any selection of the solution mapping

converges to the least-norm solution of the original problem The results ofFacchinei and Pang on solution existence of pseudomonotone VIs have beenextended by B.T Kien, J.-C Yao, and N.D Yen (2008) to VIs and set-valuedVIs in reflexive Banach spaces The results of Thanh Hao on the convergence

of the TRM for pseudomonotone VIs have been developed to VIs in Hilbertspaces by N.N Tam, J.-C Yao, and N.D Yen (2008)

For monotone VIs, the convergence of the iterative sequence generated bythe 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 thisdirection 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 ofthe exact PPA and inexact PPA

The auxiliary problems of the TRM and of the PPA, applied to domonotone VIs, may not be pseudomonotone, or may remain without anysolution (if one considers the infinite-dimensional Hilbert space setting) Inaddition, if the auxiliary problems have a solution then they may have mul-tiple solutions These phenomena indicate that the auxiliary problems can

pseu-be more difficult than the original one A natural question arises: If there isany algorithm that can solve pseudomonotone VIs in an effective way? Theextragradient 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 alsopresented

Chapter 2 deals with some questions related to applying the TRM for domonotone VIs Solution uniqueness for the regularized problems is studied

pseu-in two cases: unconstrapseu-ined VIs and lpseu-inear complementarity problems Thepseudomonotonicity 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 domonotone VIs Strong convergence and error estimates for the iterativesequences are investigated in two versions of the method: the stepsizes arechosen arbitrarily from a given fixed closed interval and the stepsizes form

pseu-a non-summpseu-able decrepseu-asing sequence of positive repseu-al numbers In pseu-addition,

an interesting class of strongly pseudomonotone infinite-dimensional VIs isconsidered

Chapter 4 is devoted to a modified EGM for solving pseudomonotone VIs

in Hilbert spaces The convergence and convergence rate of the iterativesequences 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’ convergenceand 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 ofMathematics, Vietnam Academy of Science and Technology, Hanoi (June20–25, 2011);

- The 8th Vietnam-Korea Workshop “Mathematical Optimization Theoryand Applications”, University of Dalat (December 8–10, 2011);

-The VMS-SMF Joint Congress at University of Hue (August 20–24, 2012)

Chapter 1

Preliminaries

The concepts of variational inequality, complementarity problem, metric 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

Prob-lems

Let K be a nonempty subset of a real Hilbert space (H, h., i) 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

hF (u∗), u − u∗i ≥ 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∗, hF (u∗), u∗i = 0, (1.2)where

K∗ := {d ∈ H : hd, ui ≥ 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 ) Ifthe problem CP(K, F ) has a feasible vector, it is said to be feasible When

H = IRn, F is an affine mapping, i.e., F (u) = M u + q with M ∈ IRn×n,

q ∈ IRn, and K = IRn+ (in this case K∗ = IRn+), CP(K, F ) becomes the linearcomplementarity problem LCP(M, q):

u∗ ≥ 0, M u∗ + q ≥ 0, hM u∗+ q, u∗i = 0 (1.3)Here the inequalities are taken componentwise The solution set of this prob-lem is denoted by Sol(M, q)

A mapping F : K ⊂ H → H is said to be

(a) strongly monotone if there exists γ > 0 such that

hF (u) − F (v), u − vi ≥ γku − vk2 ∀u, v ∈ K;

(b) strongly pseudomonotone if there exists γ > 0 such that, for all u, v ∈ K,

hF (u), v − ui ≥ 0 =⇒ hF (v), v − ui ≥ γku − vk2;(c) monotone if hF (u) − F (v), u − vi ≥ 0 for all u, v ∈ K;

(d) pseudomonotone if, for all u, v ∈ K,

1.4 The Tikhonov Regularization Method

Consider the problem VI(K, F ) in a real Hilbert space H Denote the identitymapping of H by I, and put Fε = F + εI for every ε > 0 To solve VI(K, F ),one solves the sequence of problems VI(K, Fεk) where {εk} is a sequence

of positive real numbers converging to zero and Fεk = F + εkI For each

k ∈ IN , one selects a solution uk ∈ Sol(K, Fεk) and compute the limit lim

k→∞uk.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 ofsteps 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

kuk − uk−1k ≤ θ where θ > 0 is a constant

Two basic convergence theorems for the Tikhonov regularization are called in the thesis

Choose a point u0 ∈ H and a sequence {ρk} of positive real numbers satisfyingthe condition ρk ≥ ρ > 0 for all k ∈ IN If uk−1 has been defined, one canchoose uk as any solution of the auxiliary problem VI(K, F(k)) where

F(k)(u) = ρkF (u) + u − uk−1, u ∈ K, (1.5)that is uk ∈ K and

hρkF (uk) + uk − uk−1, v − uki ≥ 0, ∀v ∈ K

(Suppose that Sol(K, F(k)) is nonempty.) If the iterative scheme yields asequence {uk}, then one computes the limit lim

k→∞uk in the norm topology

or in the weak topology of H When the limit exists, one may hope thatthe obtained element belongs to the solution set of VI(K, F ) To terminatethe computation process after a finite number of steps and to obtain anapproximate solution of VI(K, F ), one introduces a stopping criterion Forexample, one can terminate the computation when kuk − uk−1k ≤ θ where

θ > 0 is a constant

Basic convergence theorems for the proximal point algorithm are recalled

in the thesis

1.6 The Extragradient Method

The extragradient method executes two projections per iteration Supposethat F is Lipschitz continuous on K with the Lipschitz constant L > 0, thatis

kF (u) − F (v)k ≤ Lku − vk, ∀u, v ∈ K (1.6)Algorithm 1.1

set k ← k + 1 and go to Step 1

Two convergence theorems for the extragradient method are recalled inthe thesis

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 aboutthe solution uniqueness of the regularized problem of a pseudomonotone VIand the preservation of the pseudomonotonicity under the regularization

Inequalities

Open questions If K ⊂ IRn is a nonempty closed convex set, F : K → IRn

is a continuous pseudomonotone mapping, and the problem VI(K, F ) has asolution, then there exists ε1 > 0 such that the mapping Fε = F + εI ispseudomonotone for each ε ∈ (0, ε1)? Is there any ε2 > 0 such that theproblem VI(K, Fε) has a unique solution for every ε ∈ (0, ε2)?

Let K be a subset of IRn A mapping F : K → IRn is said to be pseudoaffine

on K if F and −F are both pseudomonotone

Theorem 2.1 A mapping F : IRn → IRn is pseudoaffine if and only if thereexists a skew matrix M ∈ IRn×n, i.e., MT = −M , a vector q ∈ IRn, and apositive function g : IRn → IR such that

F (u) = g(u)(M u + q), ∀u ∈ IRn.Theorem 2.2 Suppose that F (u) = g(u)(M u + q) with g : IRn → IR being apositive and continuously differentiable function, M ∈ IRn×n a skew symmet-ric and nonsingular matrix, and q ∈ IRn an arbitrarily given vector Thenthere exists ¯ε > 0 such that the regularized problem VI(K, Fε) has a uniquesolution for all ε ∈ (0, ¯ε)

If detM 6= 0 and MT = −M , then n must be an even number This showsthat the assumptions of Theorem 2.2 are rather strict It is natural to findsome ways to enlarge the applicability of Theorem 2.2

Theorem 2.3 Suppose that F : IRn → IRn is a map of the form

F (u) = g(u)(M u + q),where g : IRn → IR is a positive and continuously differentiable function,

M ∈ IRn×n is a positive semidefinite nonsingular matrix, and q ∈ IRn Thenthere exists ¯ε > 0 such that the regularized problem VI(K, Fε) has a uniquesolution for all ε ∈ (0, ¯ε)

Consider the linear complementarity problem of the form (1.3)

Theorem 2.4 Suppose that (1.3) is feasible and the mapping F (u) = M u + q

is pseudomonotone on IR+n Then the regularized problem LCP(Mε, q), where

Mε = M + εI, has a unique solution for any ε ∈ (0, +∞)

Theorem 2.5 Let K ⊂ IR be a closed convex subset and F (u) = au + b be

an affine map Then F is pseudomonotone on K if and only if one of thefollowing five cases occurs:

(a) K has a unique element;

(b) K = IR and a ≥ 0;

(c) K = [α, +∞) for some α ∈ IR, and either a ≥ 0 or a < 0 and aα + b < 0;

(d) K = (−∞, β] for some β ∈ IR, and either a ≥ 0 or a < 0 and aβ +b > 0;(e) K = [α, β], for some α, β ∈ IR with α < β, and either a ≥ 0 or a < 0and aα + b < 0, or a < 0 and aβ + b > 0.

Corollary 2.1 Let K be a closed convex set in IR and F (u) = au + b be anaffine map If F is pseudomonotone on K then there exists ¯ε > 0 such that

Fε(u) = (a + ε)u + b is pseudomonotone on K for all ε ∈ (0, ¯ε)

The pseudomonotonicity preservation of Fε for small enough ε > 0 vided that F is pseudomonotone), which was described in Corollary 2.1 forthe case K ⊂ IR, is no longer valid if K is a nonempty closed convex subset

(pro-of IRn, n ≥ 2

Theorem 2.6 Suppose that F (u) = M u + q is an affine map, where

M = diag(λ1, λ2, , λn), q = (q1, q2, , qn)T (2.1)are respectively a diagonal matrix and a vector in IRn Then F is pseu-domonotone on IRn

+ if and only if one of the following conditions holds:(i) λi ≥ 0 for every i ∈ {1, 2, , n};

(ii) There exists a unique i ∈ {1, 2, , n} such that

(

λi < 0, qi < 0,

λj = 0, qj = 0 ∀j ∈ {1, 2, , n} \ {i} (2.2)There is a class of pseudomonotone affine mappings whose regularized op-erators Fε are not pseudomonotone for all sufficiently small ε > 0

Theorem 2.7 Let F (u) = M u + q, with M = diag(λ1, λ2, , λn) being adiagonal matrix and q = (q1, q2, , qn)T being a vector in IRn If F is merelypseudomonotone on IRn

+ (i.e., F is pseudomonotone but not monotone on

IRn

+), then there exists ¯ε > 0 such that Fε(u) = F (u) + εu is not tone on IRn+ for all ε ∈ (0, ¯ε)

in the Affine Case

Observe that if F is Lipschitz continuous on K with a Lipschitz constant

L > 0, then for any uk ∈ IRn and ε > L the regularized operator

Fk,ε(u) := F (u) + ε(u − uk)

is strongly monotone.

For the PPA, we have

F(k)(u) = ρkF (u) + u − uk−1 = ρkF (u) + ε(u − uk−1), (2.3)where ε = 1 Hence, if L is a Lipschitz constant of F on K then ρkL is a Lip-schitz constant of ρkF (.) on K, so the above fact on the strong monotonicity

of the regularized operator tells us that if ε = 1 > ρkL (i.e., 0 < ρk < L−1),then F(k)(.) is strongly monotone on K with the constant αk := 1 − ρkL.The following advantage of PPA in comparison with the TRM is clear:the auxiliary problem VI(K, F(k)), for every k ∈ IN , is strongly monotone

if F is merely Lipschitzian (no monotonicity is required!), provided that thecoefficient ρk is such that 0 < ρk < L−1

Since affine operators are obviously Lipschitzian, the PPA can be effectiveapplied for pseudomonotone affine VIs Namely, if F (u) = M u + q with

M ∈ IRn×n\ {0} and q ∈ IRn then F(k)(.) is strongly monotone on K with theconstant αk := 1 − ρkkM k for any ρk satisfies 0 < ρk < kM k−1 Therefore,the iterative sequence given by the PPA, where 0 < ρ ≤ ρk < kM k−1 forall k, converges, provided that the original problem has a solution In otherwords, the PPA can solve any pseudomonotone affine VI