JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 100, No. 2, pp. 389-395, FEBRUARY 1999. Change of Variable Method for Generalized.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 100, No. 2, pp. 389-395, FEBRUARY 1999
Change of Variable Method for Generalized Complementarity Problems M. A. NOOR1 AND E. A. AL-SAID2 Communicated by F. Giannessi
Abstract. In this paper, we establish an equivalence between the generalized complementarity problems and the Wiener-Hopf equations by using a change of variables technique. This equivalence is used to suggest and analyze a number of iterative algorithms for solving the generalized complementarity problems, Key Words. Complementarity problems, change of variables, algorithms, fixed points, convergence, Wiener-Hopf equations.
1. Introduction Complementarity theory is a branch of the mathematical sciences with a wide range of applications in industry, physical, regional, and engineering sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications. Researchers in complementarity theory have established important connections with main areas of pure and applied sciences; as a consequence, research techniques and problems are drawn from a wide range of different fields; see, for example, Refs. 1-15 and the references therein. Complementarity problems have been generalized and extended in various directions using the novel and innovative techniques. An important and useful generalization of the complementarity problems is the generalized complementarity problem; see Refs. 5-7 for numerical methods and applications. In recent years, the technique of change of variables has been used to develop some iterative methods for solving the complementarity problems. It has been shown in Refs. 8 and 9 that such type of iterative methods are 1
Professor, Department of Mathematics, King Saud University, Riyadh, Saudi Arabia. Associate Professor, Department of Mathematics, King Saud University, Riyadh, Saudi Arabia.
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quite efficient and are compatible with other types of iterative methods. This technique is mainly due to Van Bokhoven (Ref. 10). For recent applications and modifications of this technique, see Refs. 8, 9, 11, and 12. In this paper, we use the technique of change of variables as modified and extended in Refs. 8, 9, 11, and 12 to prove that generalized complementarity problems are equivalent to a system of equations, which are called Wiener-Hopf equations. This equivalence enables us to suggest and analyze a number of iterative algorithms for solving generalized complementarity problems. As special cases, we obtain iterative algorithms for various known classes of complementarity problems.
2. Formulation and Basic Results Let H be a real Hilbert space whose inner product and norm are denoted by < •, • > and || • ||, respectively. Let
be a polar cone of a close and convex cone K in H. For given nonlinear operators T, g: H->H, we consider the problem of finding ueH such that
Problem (1) is called the generalized complementarity problem. For the formulation, numerical methods, and applications of (1), see Refs. 5-7 and the references cited therein. We note that, if g=I, the identity operator, then problem (1) collapses to finding ueH such that
which is called the generalized complementarity problem; see Refs. 2 and 5-7. We remark that, if g(u) = u - m(u), where m is a point-to-point mapping, then problem (1) is equivalent to finding ueH such that
which is known as the quasi (implicit) complementarity problem; see Refs. 4-7. Let PK be the projection of H onto the close and convex cone K. We now consider the problem of finding zeH such that
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where g-1 exists and p > 0 is a constant. Equations of type (4) are called generalized Wiener-Hopf equations. It is well known that, for all zeH, From (4) and (5), it follows that which is the familiar form of the Wiener-Hopf equations. For a general treatment, formulation, and applications of the Wiener-Hopf equations, see Refs. 13-15. Problem (1) can be written in the form which is useful in developing a fixed-point formulation by using a change of variables technique. Following the ideas of Noor (Ref. 8), we consider the following change of variables:
where In view of the techniques and ideas of Noor (Ref. 8), it can be shown that (1) is equivalent to finding zeH such that that is, where p>0 is a constant and In fact, we have the following result. Lemma 2.1. Let K be a close and convex cone in H, and let g: H-»H be injective. Problem (1) has a solution ueH if, and only if, the WienerHopf equations (4) or (5) have a solution zeH, where
3. Main Results In the previous section, we have established the equivalence between (1) and the Wiener-Hopf equation (4) by using the change of variables
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technique. This alternative formulation is very useful in the approximation and numerical analysis of complementarity. One of the consequences of this formulation is that we can obtain an approximate solution by some iterative scheme. By an appropriate and suitable rearrangement of the Wiener-Hopf equations (4), we suggest new iterative algorithms for solving (1). We consider also the convergence analysis of Algorithm 3.1. The Wiener-Hopf equations (4) can be written as
implying that, using (6) and (8),
This fixed-point formulation enables us to suggest the following iterative method. Algorithm 3.1. For a given z0eH, compute z n + 1 by the iterative scheme
By an appropriate rearrangement, the Wiener-Hopf equations (4) can be written in the form
which implies that, using (6) and (8),
Using this formulation, we propose the following iterative method. Algorithm 3.2.
For a given z0 e H, compute zn+1 by the iterative scheme
If the operator T is linear and T-1 exists, then the Wiener-Hopf equations (4) can be written as
This fixed-point formulation enables us to suggest the following iterative method.
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Algorithm 3.3. For a given z0€H, compute the sequence {u n + 1 } from the iterative scheme
For g = I, the identity operator, Algorithm 3.3 reduces to the following algorithm for solving (2), which appears to be new. Algorithm 3.4. For a given z0 e H, compute zn+1 by the iterative scheme
We now study the convergence of the iterative sequence { z n + 1 } generated by Algorithm 3.1. For this purpose, we need the following concepts. Theorem 3.1. Let the operators T, g: H->H be both strongly monotone with constants a > 0, a > 0 and Lipschitz continuous with constants B>0, D>0, respectively. If
where
then there exists zeH satisfying the Wiener-Hopf equations (4), and the sequence {zn} generated by Algorithm 3.1 converges to z strongly in H. Proof. From Algorithm 3.1, we have
Since T is strongly monotone and Lipschitz continuous, so
In a similar way,
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Combining (12)-(15), we have
where
From (9), (12), and (15), we obtain
since PK is nonexpansive. Thus, From (16) and (17), we have
where From (11), it follows that t< 1; consequently, from (18), we see that the sequence {zn} is a Cauchy sequence in H; that is, there exists zeH with zn->z as n->oo. Now, by using the continuity of the operators T, g, PK and Algorithm 3.1, we have which satisfies the Wiener-Hopf equations (4) and zn->z strongly in H, the required result. D References 1. COTTLE, R. W., GIANNESSI, F., and LIONS, J. L., Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley and Sons, New York, New York, 1980. 2. COTTLE, R. W., PANG, J. S., and STONE, R., The Linear Complementarity Problem, Academic Press, New York, New York, 1992.
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3. GIANNESSI, F., and MAUGERI, A., Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, New York, 1995. 4. HARKER, P. T., and PANG, J. S., Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications, Mathematical Programming, Vol. 48, pp. 161-220, 1990. 5. HYER, D. H., ISAC, G., and RASSIAS, T. M., Topics in Nonlinear Analysis and Applications, World Scientific Publishing Company, Singapore, 1997. 6. ISAC, C., Complementarity Problems, Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 1528, 1992. 7. NOOR, M. A., General Nonlinear Complementarity Problems, Analysis, Geometry, and Groups: A Riemann Legacy Volume, Edited by H. M. Srivastava and T. M. Rassias, Hadronic Press, Palm Harbor, Florida, pp. 337-371, 1993. 8. NOOR, M. A., Fixed-Point Approach for Complementarity Problems, Journal of Mathematical Analysis and Applications, Vol. 133, pp. 437-448, 1988. 9. NOOR, M. A., and ZARAE, S., An Iterative Scheme for Complementarity Problems, Engineering Analysis, Vol. 3, pp. 240-243, 1986. 10. VAN BOKHOVEN, W. M., A Class of Linear Complementarity Problems Solvable in Polynomial Time, Technical Report, Department of Electrical Engineering, Technical University, Eindhoven, Holland, 1980. 11. AHMAD, K., KAZMI, K. R., and REHMAN, N., Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice, Journal of Optimization Theory and Applications, Vol. 93, pp. 67-72, 1997. 12. NOOR, M. A., and AL-SAID, E. A., An Iterative Technique for Generalized Strongly Nonlinear Complementarity Problems, Applied Mathematics Letters, Vol. 11, 1998. 13. NOOR, M. A., Some Recent Advances in Variational Inequalities, Part 1: Basic Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 53-80, 1997. 14. NOOR, M. A., Some Recent Advances in Variational Inequalities, Part 2: Other Concepts, New Zealand Journal of Mathematics, Vol. 26, pp. 229-255, 1997. 15. NOOR, M. A., NOOR, K. I., and RASSIAS, T. M., Some Aspects of Variational Inequalities, Journal of Computational and Applied Mathematics, Vol. 47, pp. 285-312, 1993.
