Explicit and Implicit Continuation Algorithms for Strongly Monotone Variational Inequalities with Box Constraints

  • Authors:

  • Jin-Bao Jian;Xing-De Mo;Jian-Ling Li

  • Affiliations:

  • Department of Mathematics and Information, Guangxi University 530004, Nanning, P.R. China (e-mail: jianjb@gxu.edu.cn);Department of Mathematics and Information, Guangxi University 530004, Nanning, P.R. China;Department of Mathematics and Information, Guangxi University 530004, Nanning, P.R. China

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2004

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Abstract

In this paper we discuss the variational inequality problems VIP(X, F), where F is assumed to be a strongly monotone mapping from {\frac {R}}^{n} to {\frac {R}}^{n}, and the feasible set X = [l, u] has the form of box constraints. Based on the Chen-Harker-Kanzow smoothing functions, first we present an explicit continuation algorithm (ECA) for solving VIP(X, F). The ECA possesses main features as follows: (a) at each iteration, it yields a new iterative point by solving a system of equations in {\frac {R}}^{(n + s)} with a parameter and nonsingular Jacobian matrix, where s = |{j: -∞ lj uj X. Secondly we give an implicit continuation algorithm (ICA) for solving VIP(X,F), the prime character of the ICA is that it solves only one, rather than a series of, system of nonlinear equations to obtain a solution of VIP(X,F). The two proposed algorithms are shown to possess strongly global convergence. Finally, some preliminary numerical results of the two algorithms are reported.