Solution Point Characterizations and Convergence Analysis of a Descent Algorithm for Nonsmooth Continuous Complementarity Problems

• Published in: Journal of optimization theory and applications Vol. 110; no. 3; pp. 493 - 513
• Main Authors: Fischer, A., Jeyakumar, V., Luc, D. T.
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 01.09.2001; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of variations and optimal control; Computer science; control theory; systems; Control theory; Control theory. Systems; Exact sciences and technology; Function theory, analysis; Mathematical analysis; Mathematical methods in physics; Mathematics; Optimal control; Optimization; Physics; Sciences and techniques of general use; Global convergence; Approximate method; Continuous; Complementarity problem; Nonlinear problems; Stationary condition; Convergence; Necessary condition; Algorithms; Nonsmooth analysis; Solutions; Descent method; Optimality condition; Nonlinearity; Optimality criterion; Jacobian function
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1023/A:1017580126509

We consider a nonlinear complementarity problem defined by a continuous but not necessarily locally Lipschitzian map. In particular, we examine the connection between solutions of the problem and stationary points of the associated Fischer-Burmeister merit function. This is done by deriving a new necessary optimality condition and a chain rule formula for composite functions involving continuous maps. These results are given in terms of approximate Jacobians which provide the foundation for analyzing continuous nonsmooth maps. We also prove a result on the global convergence of a derivative-free descent algorithm for solving the complementarity problem. To this end, a concept of directional monotonicity for continuous maps is introduced.