Smoothing Methods in Mathematical Programming

1995 i Acknowledgements My greatest thanks go to my advisor Professor Olvi L. Mangasarian for his guidance and advice. He has always been most generous with his time and encouragement. Without him, this thesis could not have been written. for the time I spent in their classes, for reading a draft of this thesis and for being members of my examination committee. I wish to thank my colleagues Steve Dirkse for providing the GAMS interface for testing our algorithm and Nick Street for many useful discussions. I also want to express my deepest gratitude to my family: my parents, Zhongkai Chen and Xioumei Jin, my husband, Yonghong Yang and my son, Michael for their love and encouragement all these years. ii Abstract A class of parametric smooth functions that approximate the fundamental plus function, (x) + = maxf0; xg, is obtained by twice integrating a probability density function. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for suuciently small positive value of the smoothing parameter . In the special case when a Slater constraint qualiication is satissed, an exact solution can be obtained for nite. Speedup over the linear/nonlinear programming package MINOS 5.4 was as high as 1142 times for linear inequalities of size 20001000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems(LCPs) were treated by converting them into a system of smooth nonlinear equations and are solved by a quadratically con-vergent Newton method. For monotone LCPs with as many as 10,000 variables, the proposed approach was as much as 63 times faster than Lemke's method. Our smooth approach can also be used to solve nonlinear and mixed comple-mentarity problems (NCPs and MCPs) by converting them to classes of smooth iii parametric nonlinear equations. For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for suuciently large value of a smoothing parameter = ?1. An eecient smooth algorithm, based on the Newton-Armijo approach with an adjusted smoothing parameter, is also given and its global and local quadratic convergence is established. For NCPs, exact solutions of our smooth nonlinear equation for various values of the parameter , generate an interior path, which is diierent from …