The Use of "Continuous Method" in Complementarity Problems

Abstract

Let Rn be the n-dimensional real euclidean space and let x =(x1, x2,...xn)T eRn, where T means transpose for x , y eRn. x , y >= ∑ i = 1 n X i Y i is the scalar product between x and y and || x ||2= x , x >½ is the euclidean norm of x . We will consider the following problems: Problem 1. (Linear complementarity problem). Let M be an nxn matrix and q eRn. Find w , z e Rn such that (1) w _ = q _ + M z _ w _ ≥ 0 _ z _ ≥ 0 _ (2) w _ , z _ > = 0 where w ≥0, z ≥0 means that each component of w and z is greater than or equal to zero.and Problem 2. (Nonlinear complementarity problem) Let f :Fn→ Rn be a given map. Find x eRn such that: (3) f _ ( x ) ≥ 0 _ x _ ≥ 0 (4) f _ ( x _ ) , x _ > = 0 The linear complementary problem contains as particular cases the problems of linear programming and quadratic programming [1], and has been used for example in the search of equilibrium points of bimatrix games [2]. The nonlinear complementarity problem is also of great generality; for example the first order Kuhn Tucker conditions for a constrained optimum can be reformulated as a nonlinear complementarity problem [3]. The linear and the nonlinear complementarity problems are usually attacked with pivotal methods. In this paper using the results of Hangasarian [4] we first transform the complementarity problems into a system of nonlinear equations then we apply to them two continuous of optimization theory developed by the authors. The first method [5], [6], [7] associates a system of ordinary differential equations, with the equations whose roots we are interested in, and integrates the former numerically. The system of differential equations is inspired by classical mechanics and is of second order. The second method [8], [9], [10] associates a system of stochastic differential equations with the equations, whose roots we are interested in, and integrates the former numerically. The system of stochastic differential equations is inspired by quantum physics. Both methods are implemented in high quality mathematical software [6], [7], [9], [10]. The use of continuous in the study of complementarity problems is shown to be useful when dealing with: (i) nonconvex linear and nonlinear complementarity problems (ii) very ill-conditioned problems (iii) problems involving a large number of variables (image reconstruction. tomography) especially if only an approximate solution is sought. Finally. some numerical experience on test problems motivated by image reconst ruction applications [11] is shown.