Finite termination of a Newton-type algorithm for a class of affine variational inequality problems

• Published in: Applied mathematics and computation Vol. 217; no. 7; pp. 3368 - 3378
• Main Authors: Zhao, Na, Huang, Zheng-Hai
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.12.2010; Elsevier
• Subjects: 65K10; 90C33; Affine variational inequality problem; Algorithms; Calculus of variations and optimal control; Exact sciences and technology; Exact solutions; Finite termination; Generalized Newton method; Global analysis, analysis on manifolds; Inequalities; Mathematical analysis; Mathematical models; Mathematics; Newton methods; Nonlinear algebraic and transcendental equations; Numerical analysis; Numerical analysis. Scientific computation; Optimization; Sciences and techniques of general use; Smoothing; Smoothing Newton method; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; 65K10; Affine variational inequality problem; Generalized Newton method; Smoothing Newton method; Finite termination; 90C33; Transcendental equation; Smoothing methods; Optimization method; Iteration; Algorithm; Exact solution; Equation system; Non linear equation; Variational problem; Numerical analysis; Variational inequality; Applied mathematics; Algebraic equation; Newton method
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.08.069

Many optimization problems can be reformulated as a system of equations. One may use the generalized Newton method or the smoothing Newton method to solve the reformulated equations so that a solution of the original problem can be found. Such methods have been powerful tools to solve many optimization problems in the literature. In this paper, we propose a Newton-type algorithm for solving a class of monotone affine variational inequality problems (AVIPs for short). In the proposed algorithm, the techniques based on both the generalized Newton method and the smoothing Newton method are used. In particular, we show that the algorithm can find an exact solution of the AVIP in a finite number of iterations under an assumption that the solution set of the AVIP is nonempty. Preliminary numerical results are reported.