Tangent Projection Equations and General Variational Inequalities

Journal of Mathematical Analysis and Applications 258, 755–762 (2001) doi:10.1006/jmaa.2000.7517, available online at http://www.idealibrary.com on Tangent Projection Equations and General Variational Inequalities1 Naihua Xiu Department of Applied Mathematics, Northern Jiaotong University, Beijing 100044, People’s Republic of China E-mail:[email protected] Jianzhong Zhang Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, People’s Republic of China E-mail:[email protected] and Muhammad Aslam Noor Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, B3H 3J5 Canada E-mail:[email protected] Submitted by A. M. Fink Received December 20, 2001 In this paper we establish the equivalence between the general variational inequalities and tangent projection equations. This equivalence is used to discuss the local convergence analysis of a wide class of iterative methods for solving the general variational inequalities. We show that some existing methods can identify the optimal face after finitely many iterations under the degenerate assumption. © 2001 Academic Press Key Words: variational inequality; equation; equivalence; local convergence. 1 This research was supported by the National Natural Science Foundation of China (Grant 19971002), and a City University Strategic Research Grant. 755 0022-247X/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. 756 xiu, zhang, and noor 1. INTRODUCTION Let Rn be a real Euclidean space whose inner product and norm are denoted by
· · and  · , respectively. Let  be a nonempty closed convex set in Rn . Given nonlinear continuous operators F g  Rn → Rn , we consider the problem of finding x∗ ∈ Rn such that g x∗ ∈  and
F x∗  g y − g x∗  ≥ 0 ∀y ∈ Rn such that g y ∈  (1) which is called the general variational inequality, introduced and studied by Noor [9] in 1988. It turned out that a wide class of nonsymmetric and oddorder obstacle, free, moving, and equilibrium problems arising in various areas of pure and applied sciences can be studied via the general variational inequalities, see [10–12, 14, 15] and the references therein. If g ≡ I, the identity operator, then problem (1) is equivalent to finding x∗ ∈  such that
F x∗  w − x∗  ≥ 0 ∀w ∈  (2) which is known as the classical variational inequality, introduced and studied by Stampacchia [17] in 1964, see [3, 6–16] for the state of the art. From now on, we assume that the operator g is onto , unless otherwise specified. If  = Rn+ = x ∈ Rn  x ≥ 0, then the general variational inequality (1) reduces to finding x∗ ∈ Rn+ such that
F x∗  g x∗  = 0 F x∗ ≥ 0 g x∗ ≥ 0 (3) Problems of type (3) are called general nonlinear complementarity problems. For g x = x − m x , where m is a point-to-point mapping, problem (3) is known as a quasi-(implicit) complementarity problem. For the applications, numerical methods, and formulations of problems (1)–(3), see [3, 6–16]. Using the projection technique, Noor [9] has shown that problem (1) is equivalent to the fixed point equation of the form e x ρ = g x − P g x − ρF x  = 0 ρ > 0 (4) where P is the projection from Rn into . Noor [10] also has shown that the general variational inequality (1) is equivalent to finding z ∗ ∈ Rn satisfying the equation WH z ρ = z ∗ − P z ∗  + ρFg−1 P z ∗  = 0 ρ > 0 (5) where g x∗ = P z ∗  and z ∗ = g x∗ − ρF x∗ . Equation (5) is called the Wiener–Hopf equation and was studied and introduced by Noor [10] in 1993. The Wiener-Hopf equation technique has been used to suggest tangent equations and variational inequalities 757 and analyze various iterative methods for solving variational inequalities and related optimization problems, see [10–12, 14–16]. In this article we introduce and study a new class of equations, tangent projection equations. We show that the tangent projection equations are equivalent to the general variational inequalities. We use this equivalence to discuss its applications in analyzing local convergence behavior of a wide class of iterative methods for the solutions of the problems. Using the technique of Burke and Moré [1], we show that under the nondegenerate condition, the sequences generated by some existing methods eventually enter and remain in relative interior of the optimal face, and hence the methods have the simpler forms in a finite number of iterations. This is extremely useful in the design of algorithms for the solutions of (1). Our results can be viewed as a nice and novel application of the techniques developed in [1, 2] to general variational inequalities (1), where the optimization problems with convex constraints were considered. 2. TANGENT PROJECTION EQUATIONS In this section we introduce the tangent projection equations and show that these equations are equivalent to the general variational inequalities. First we recall some well-known concepts. Given x ∈ Rn and g x ∈ , we say that a direction v is feasible at point g x ∈  if g x + τv belongs to  for all sufficiently small τ > 0. The tangent cone T g x is the closure of the cone of all feasible directions. Since T g x is a nonempty closed convex set, −F x has a unique projection on T g x with the following form: PT g x Lemma 2.1. Then −F x  = arg minv + F x   v ∈ T g x  Let F Tg x be the tangent projection of −F x at g x ∈ . (a)
F x  F Tg x  = −F Tg x 2 (b) min
F x  v  v ∈ T g x  v ≤ 1 = −F Tg x . Proof. It is trivial; see Gao [5]. Related to the general variational inequality (1), we consider the problem of finding x ∈ Rn  g x ∈  such that F Tg x = 0 (6) the tangent projection equation. These equations play an important and significant role in studying the local convergence criteria of the iterative projection methods for solving the general variational inequality (1). 758 xiu, zhang, and noor Using Lemma 2.1, we can easily prove that the tangent projection equation (6) is equivalent to the general variational inequality (1). For the sake of completeness and to convey an idea, we give its proof. Theorem 2.1. The general variational inequality problem (1) has a solution x∗ ∈ Rn if and only if x∗ solves the tangent projection equation (6). Proof. Let x∗ ∈ Rn  g x∗ ∈ , be a solution of the general variational inequality (1). Then
F x∗  g y − g x∗  ≥ 0 ∀y ∈ R∗  g y ∈  (7) Let v be any feasible direction at point g x∗ ∈ . Then g x∗ + τv must be in  for some small τ > 0. Since g is onto , there is a point y ∈ Rn such that g y = g x∗ + τv. Substituting g y into (7), we obtain
F x∗  v ≥ 0. T g x∗ is the closure of the set of all feasible directions at g x∗ ∈ , which implies that
F x∗  w ≥ 0 ∀w ∈ T g x∗ (8) Thus, by Lemma 2.1(b), we obtain F Tg x∗ = 0 Conversely, let x∗ ∈ Rn , g x∗ ∈  be a solution of the tangent projection equation (6). Invoking Lemma 2.1(b), we obtain the inequality (8). Since g y − g x∗ ∈ T g x∗ for any g y ∈ , from (8) we obtain (7), and this shows that x∗ is a solution of the general variational inequality problem (1). Remark 2.1. If N u = w ∈ Rn 
w v − u ≤ 0 ∀v ∈  is a normal cone to the convex set  at u, then the general variational inequality (1) is equivalent to finding x∗ ∈ Rn , g x∗ ∈  such that −F x∗ ∈ N g x∗  which are known as the generalized nonlinear equations. 3. APPLICATIONS In this section we apply the tangent projection equation (6) to derive a characterization of a wide class of iterative methods which identify the optimal face in a finite number of iterations. The analysis is in the spirit of Burke and Morè [1]. We show that some existing methods have such a characterization. For this purpose, we recall the following concepts. For a set S ⊆ Rn , the affine hull aff S is the smallest affine set which contains S, and the relative interior ri S is the interior of S relative to aff S . For a cone K ⊆ Rn , the linearity linK of the cone is the largest tangent equations and variational inequalities 759 subspace contained in K. For a nonempty closed convex set  ⊆ Rn , a convex set face ⊆  is said to be a face of  if the endpoints of any closed line segment in  whose relative interior intersects face are contained in face . Thus, if x and y are in  and θx + 1 − θ y lies in face for some 0 < θ < 1, then x and y must also belong to face . Lemma 3.1. [1, Theorem 2.3] If face is face of the convex set , then N u is independent of u for any u ∈ ri face . We denote T u for u ∈ ri face by T face , and N u by N face . Definition 3.1. A solution x∗ ∈ Rn of problem (1) is said to be nondegenerate if −F x∗ ∈ ri N g x∗ For the special case, g x = x, this definition is due to Dunn; see, for example, [1]. This implies that Definition 3.1 is a further generalization of the strict complementarity condition. For example, consider the general variational inequality problem (1) with linear constraints, which is to find x∗ ∈ Rn such that g x∗ ∈ X and
F x∗  g y − g x∗  ≥ 0 ∀y ∈ Rn  y ∈ X where X = u ∈ Rn  AT u ≥ b, A = a1  a2  b1  b2   bm T ∈ Rm . Its KKT system is F x = Aλ
λ AT g x − b = 0 (9)  am ∈ Rn×m and b = λ ≥ 0 AT g x − b ≥ 0 (10) If x∗ is a solution of (9), then there is a vector λ∗ ∈ Rm of Lagrange multipliers such that x∗  λ∗ is a solution (10). According to Definition 3.1, x∗ is nondegenerate if λ∗ and AT g x∗ − b are strictly complementary. Now we consider those algorithms which generate the sequence xk  with uk = g xk  ⊆ . By using only the knowledge of face geometry (which is independent of any algorithm), Burke and Moré [1] proved the following result. Lemma 3.2 [1, Lemma 3.3] Assume that face is a quasi-polyhedral face of the convex set  (i.e., aff face = u + linT u  for any u ∈ ri face ) with u∗ ∈ ri face and d ∗ ∈ ri N face . If uk ∈  and d k ∈ N uk for all k, and the sequences uk  and d k  converge to u∗ and d ∗ , respectively, then uk ∈ ri face for all k sufficiently large. In view of Lemmas 3.1 and 3.2, we state and prove the following result. 760 xiu, zhang, and noor Theorem 3.1. Assume that xk  with g xk ∈  is a sequence which converges to a nondegenerate solution x∗ of problem (1). If face is a quasipolyhedral face of  with g x∗ ∈ ri face , then g xk ∈ ri face for all sufficiently large k if and only if F Tg xk  converges to zero. Proof. Its proof parallels line for line the proof in [1, Theorem 3.4]. When  is the polyhedral set X, (1) reduces to (9), and Theorem 3.1 can be simplified as follows. Theorem 3.2. Assume that xk  with g xk ∈ X is a sequence which converges to a nondegenerate solution x∗ of problem (10). Then
g xk =
g x∗ for all sufficiently large k if and only if FT g xk  converges to 0, where
u = j  aTj u = bj  j = 1 2  m is the active set at u. We now consider some applications of Theorems 3.1 and 3.2 to three iterative methods for solving the general variational inequality (1), due to Noor [9, 10, 13], Noor and Al-Said [16] and He [8]. Algorithm 3.1 [10, 16]. For a given z 0 ∈ Rn , compute z k+1 by the iterative scheme g xk = P z k  z k+1 = g xk − ρF xk  k = 0 1 2 0 Algorithm 3.2 [9, 10]. For a given x ∈ Rn , compute xk+1 by the iterative scheme g y k = P g xk − ρF xk  xk+1 = xk − g xk + g y k  k = 0 1 2 Algorithm 3.3 [8, 13]. For a given x0 ∈ Rn , compute xk+1 by the iterative scheme g y k = P g xk − ρF xk  g xk+1 + ρF xk+1 = g xk + ρF xk − γ g xk − g y k  k = 0 1 2  where γ is a constant in 0 2 . For Algorithm 3.1, we know that the sequence xk  satisfies the relation g xk+1 = P g xk − ρF xk  k = 0 1 2 Noor [9, 10] proved that under certain assumptions, the sequence xk  converges to a solution x∗ of problem (1). One can easily prove that Algorithm 3.1 has the following global and local convergence properties under the assumption of nondegeneracy. However, to give the flavour of the main ideas involved, we give its proof. tangent equations and variational inequalities 761 Theorem 3.3. If Algorithm 3.1 produces a sequence xk  which converges to a solution x∗ of problem (1), then F Tg xk  converges to 0. In addition, if x∗ is nondegenerate, then (a) g xk ∈ ri face g x∗ k (b) For  = X,
g x for all k sufficiently large =
g x∗ for all k sufficiently large. Proof. Let ǫ > 0 be given, and let vk be a feasible direction at g xk ∈  with vk  ≤ 1 such that F Tg xk  ≤
−F xk  vk  + ǫ (11) Since g xk+1 = P g xk − ρF xk  can be written in the form of variational inequality as
g xk+1 − g xk + ρF xk  g w − g xk+1  ≥ 0 ∀g w ∈  from which it follows that −ρ
F xk g w − g xk+1  ≤ g xk+1 − g xk  · g w − g xk+1  Take some τ > 0 such that g w = g xk+1 + τvk+1 ∈ . Then, from the above inequality, vk+1  ≤ 1, and the continuity of g, we derive lim sup −
F xk  vk+1  ≤ lim g xk − g xk+1  = 0 k→∞ k→∞ This, together with the continuity of F, implies that lim sup −
F xk+1  vk+1  k→∞ ≤ lim sup −
F xk  vk+1  + lim sup F xk − F xk+1  ≤ 0 k→∞ (12) k→∞ From (11) and (12), we have limk→∞ sup F Tg xk+1  ≤ ǫ Since ǫ > 0 is arbitrary, the sequence F Tg xk  converges to 0. If xk  converges to a solution x∗ of (1) and x∗ is nondegenerate, then, invoking Theorems 3.1 and 3.2, we obtain the required results (a) and (b). In view of the conclusions of Theorem 3.3, the computation of the projection P z k  eventually reduces to solving an equality-constrained subproblem min u − z k 2 s.t. aTj u = bj  j ∈
g x∗  the solution of which can easily be computed. Similarly, we can prove that the sequence y k  generated by Algorithms 3.2 and 3.3 have similar local convergence property if both xk  and y k  converge to a nondegenerate solution x∗ of (1). 762 xiu, zhang, and noor ACKNOWLEDGMENT The authors thank the referee for useful comments and suggestions. REFERENCES 1. J. V. Burke and J. J. Moré, On the identification of active constraints, SIAM J. Numer. Anal., 25 (1988), 1197–211. 2. P. H. Calamai and J. J. Moré, Projected gradient methods for linearly constrained problems, Math. Program., 39 (1987), 93–116. 3. R. W. Cottle, J. S. Pang, and R. E. Stone, “The Linear Complementarity Problem,” Academic Press, New York, 1992. 4. M. C. Ferris and J. S. Pang, Nondegenerate solutions and related concepts in affine variational inequalities, SIAM J. Control Optim. 34 (1996), 244–263. 5. D. Y. Gao, “Duality Principles in Nonconvex Systems: Theory, Methods and Applications,” Kluwer Academic, Amsterdam, 1999. 6. F. Giannessi and A. Maugeri, “Variational Inequalities and Network Equilibrium Problems,” Plenum Press, New York, 1995. 7. P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161–220. 8. B. He, Inexact implicit methods for monotone general variational inequalities, Math. Program., 86 (1999), 199–217. 9. M. A. Noor, General variational inequalities, Appl. Math. Letters 1 (1988), 119–121. 10. M. A. Noor, Wiener–Hopf equations and variational inequalities, J. Optim. Theory Appl., 79 (1993), 197–206. 11. M. A. Noor, Some recent advances in variational inequalities, Part I: Basic concepts, N. Z. J. Math., 26 (1997), 53–80. 12. M. A. Noor, Some recent advances in variational inequalities, Part II: Other concepts, N. Z. J. Math., 26 (1997), 229–255. 13. M. A. Noor, Algorithms for general mixed variational inequalities, J. Math. Anal. Appl. 229 (1999), 330–343. 14. M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217–229. 15. M. A. Noor, K. I. Noor, and T. M. Rassias, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), 285–312. 16. M. A. Noor and E. A. Al-Said, Change of variable method for generalized complementarity problems, J. Optim. Theory Appl. 100 (1999), 389–395. 17. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413–4416.