Power Penalty Methods For Linear Complementarity Problems

Power penalty methods are an important class of approximate methods forsolving variational inequalities and complementarity problems. The methods havebeen widely studied by researches in recent years and many important results areobtained. These results give the relationships between the solution of the originalproblem and the approximate solution. However, there are very few results to discussthe relationship between two approximate solutions. Based on some existing works,we will study the relationship between two approximate solutions. Additionally,penalized equations corresponding to lower-order penalty methods are usuallynonsmooth and nonconvex nonlinear equations, and some existing iterativealgorithms cannot be applied to solve them directly. Therefore, we shall propose somesimple iterative algorithms to solve these penalized equations.This paper is divided into four chapters. In the first chapter, we shall give thebackground, introduce some convergence results, and summarize developments initerative algorithms for solving penalized equations. In the second chapter, weintroduce several important concepts and prove that the lower-solution set of thepenalized equation is non-empty. In the third chapter, we discuss the relationshipsbetween two approximate solutions. Finally, two simple iterative algorithms areproposed to solve lower-order penalized equations, and under proper conditions, weverify that the sequence of iterates converges monotonically to the solution of thepenalized equations. Numerical results show the effectiveness of two iterativealgorithms.