Some Researches On Algorithm For Complementarity Problesm

Complementarity problem is a heated topic in the research of mathematical program-ming. Because of its wide application background and overlapping with other branch, Complementarity problem attracts more and more attentions from various fields of sci-ence and engineering in recent years. Also many achievements have been proposed in the field. Generally speaking, its research can be classified into two classes:theory and algorithms. The former is devoted to the existence, uniqueness, stability and sensitivity analysis of the solutions, while the latter is confined to solve the problems efficiently.This paper is primarily designed to solve the problems efficiently. The main contri-butions are listed as follows:1. Based on Chen-Harler-Kanzow-Smale (CHKS) smoothing function, we propose a regularized smoothing Newton method for mixed complementarity problems with a P0-function. This method is designed to handle ill-posed problems which substitutes the solution of original problem with the solution of a sequence of well-defined problems whose solutions converging to the solution of the original problem. And the regularization parameter and the smoothing parameter in our algorithm are independent variables and can be immediately obtained through iteration of linear system.2. In view of the significance of smooth functions in smoothing type algorithms for solving complementarity problems, we construct two new MCP functions by smoothing a perturbed mid function such that one can study the existence and continuity of the smooth path and boundedness of the iteration sequence. Then, three smoothing algorithms for solving the MCP with a P0-function, that is, predictor-corrector smoothing Newton algo-rithm and one-step smoothing Newton method and broyden-like quasi-Newton algorithm are proposed respectively.3. To solve general (not necessarily P0) mixed complementarity problems, a modified one-step smoothing Newton method is given. The algorithm incorporates a gradient step. Such procedure make the algorithm remove the condition that function F is required to be at least a P0-function. Under suitable assumptions, global convergence and locally superlinear convergence of the algorithm are established.4. A new C-function for symmetric cone complementarity problems is constructed. It is showed that the new C-function is coercive, strongly semismooth and its Jacobian is also strongly semismooth, which are important for the construction of the corresponding algorithm for solving symmetric cone complementarity problems with this C-function.