A predictor-corrector algorithm for general variational inequalities

K e y w o r d s V a r i a t i o n a l inequalities, Auxiliary principle, Iterative methods, Convergence. 1. I N T R O D U C T I O N Variational inequalities theory, which was introduced by Stampacchia [1] in 1964, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, physical, regional, social, pure, and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. Variational inequalities have been generalized and extended in different directions using novel and innovative techniques. An important and useful generalization of variational inequalities is called the general variational inequality, which was introduced and studied by Noor [2] in 1988. It has been shown that a wide class of unrelated odd-order, nonsymmetric free, moving, obstacle and equilibrium problems can be studied via the general variational inequalities. There are a number of numerical methods, including projection methods, Wiener-Hopf equations, descent and decomposition for solving variational inequalities. To implement the projection method and its variant forms, one has to find the projection of the space onto the convex set, which is itself a difficult problem. On the other hand, there are some classes of variational inequalities for which one cannot use the projection method. To overcome these drawbacks, the auxiliary principle technique has been developed, the origin of which can be traced back to Lions and Stampacchia [3]. Clowinski, Lions and Tremolieres [4] used this technique to study the existence of a solution of mixed variational inequalities. In recent years, this technique has been used to suggest and analyse various iterative methods for solving various classes of variational inequalities. It has been shown that a substantial number of 0893-9659/00/$ see front mat te r (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by Aj~AS-2~X PII: S0893-9659(00) 00112-9 54 M.A. NOOR numerical methods can be obtained as special cases from this technique, see [2,4-18] and references therein. The main disadvantage of this approach is that the convergence analysis of these iterative methods requires that the operator is either strongly monotone or co-coercive. Note that co-coercive is weaker than strongly monotone, but stronger than monotone. In this paper, we use the auxiliary principle to suggest a class of predictor-corrector methods for solving general variational inequalities. The convergence of these methods requires only that the operator is partially relaxed strongly monotone, which is weaker than co-coercive. Consequently, we improve the convergence results of previously known methods, which can be obtained as special cases from our results. Our results can be considered an extension of the results of Noor [11-14] for solving variational inequalities and complementarity problems. 2. P R E L I M I N A R I E S Let H be a real Hilbert space whose inner product and norm are denoted by < .,. > and II.ll, respectively. Let K be a nonempty closed convex set in H. For given nonlinear operators T, g : H --~ H, consider the problem of finding u E H, g(u) E K such that (Tu, g(v) g(u)) >_ O, for all g(v) E K. (2.1) The inequality of type (2.1) is called the general variational inequality which was introduced and studied by Noor [2]. It has been shown [8,9,15] tha t a wide class of odd-order and nonsymmetric free, obstacle, moving, equlibrium and optimization problems arising in pure and applied sciences can be studied via the general variational inequalities (2.1). We remark that ifg -= I, the identity operator, then problem (2.1) is equivalent to finding u E K such that (Tu, v u) >_ O, for all v E K, (2.2) which are called the classical variational inequalities introduced and studied by Stampacchia [1]. For the applications, numerical methods, and formulations, see [1-21] and the references therein. If K* = {u E H : (u,v) _> 0, for all v E K} is a polar cone of a convex cone K in H, then problem (2.1) is equivalent to finding u E H such that g(u) E K , T u E K* and (Tu, g(u)} = 0, (2.3) which is known as the general complementarity problem. We note that if g(u) = u m(u), where m is a point-to-point mapping, then problem (2.4) is called the quasi(implicit)-complementarity problem, see the references for the formulation and numerical methods. It is clear tha t problems (2.2),(2.3) are special cases of the general variational inequality (2.1). In brief, for a suitable and appropriate choice of the operators T, g, and the space H, one can obtain a wide class of variational inequalities and complementarity problems. This clearly shows tha t problem (2.1) is a quite general and unifying one. Furthermore, problem (2.1) has important applications in various branches of pure and applied sciences. We also need the following concepts. LEMMA 2.1. For aB u, v E H, we have 1 (u,v) = {llu + 112 -I1 112 -Ilvll }. (2.4) PROOF. It is trivial. DEFINITION 2.1. For all u, v, z E H , an operator T : H --* H is said to be: (i) g r e l a x e d s t r o n g l y m o n o t o n e , i f there exists a constant ~/ > 0 such that > llg(u) g (v ) l l h General Variational Inequalities (ii) g p a r t i a l l y r e l a x e d s t r o n g l y m o n o t o n e , if there exists a constant a > 0 such tha t ( T ~ T v , g ( z ) g(v) ) > ~ l l g ( ~ ) g(~)ll2; (iii) g -co -coe rc ive , if there exists a constant p > 0 such that ( T u T v , g ( u ) g (v ) ) >_ , l l T u Tv l l 2. 55 We remark that if z = u, then g-partially relaxed strongly monotone is exactly g-monotone of the operator T. For g = I, the indentity operator, then Definition 2.1 reduces to the s tandard definition of relaxed strong monotonicity [21], partially relaxed strong monotonicity [17], and cocoercivity [7,18] of the operator. It has been shown [12] that g-co-coercivity implies g-partially relaxed strong monotonicity but the converse is not true. This shows that the concept of partially relaxed strong monotonicity is weaker than co-coercivity. 3. M A I N R E S U L T S In this section, we suggest and analyze a new iterative method for solving problem (2.1) by using the auxiliary principle technique of Glowinski, Lions and Tremolieres [4]. For a given u E H, g(u) E K , consider the problem of finding a unique w E H, g(w) ~ K .satisfying the auxiliary variational inequality (pTu + g(w) g(u), g(v) g(w)) > O, for all g(v) E K, (3.1) where p > 0 is a constant. We note that if w = u, then clearly w is a solution of the general variational inequality (2.1). This observation enables us to suggest the following predictor-corrector method for solving the general variational inequalities (2.1). ALGORITHM 3.1. For a given uo E H, compute the approximate solution Un-bl by the i terative ,~chemes and (pTWn -~g(Un+l) -g(Wn) ,g(v ) -g(~n+l)) )>~ 0, (gTyn + g(wn) g(yn) , g(v) g(wn)) _> 0, for all 9(v) ~ K, (3.2) for all g(v) ~ K , (3.3) ( , T u n + g (yn ) g ( u n ) , g ( v ) g(Yn)/ >0, forallg(v) e K, (3.4) where p > 0, # > 0, and/~ :> 0 are constants. Note that if g = I, the identity operator, then Algorithm 3.1 reduces to, which is a new predictor-corrector method for solving the variational inequalities (2.2). ALGORITHM 3.2. For a given uo C H, compute un+l by the iterative schemes and (pTwn + Un+l Wn,V Un+l} >_ 0, (/3Ty,~ + wn Yn, V -Wn) ~ 0, for all v E K, for all v E K. (3.5) (#Tun + Yn un, v Yn} >O, for all v E K. Using the technique of the projection, Algorithm 3.1 can be written as the following.