Theory of globally convergent probability-one homotopies for nonlinear programming

作     者：Watson, LT 

作者机构：Virginia Polytech Inst & State Univ Dept Comp Sci & Math Blacksburg VA 24061 USA 

出 版 物：《SIAM JOURNAL ON OPTIMIZATION》 (工业与应用数学会最优化杂志)

年 卷 期：2001年第11卷第3期

页      面：761-780页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

主　　题：constrained optimization globally convergent homotopy algorithm nonlinear equations probability-one homotopy 

摘      要：For many years globally convergent probability-one homotopy methods have been remarkably successful on di cult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed point theory exists, but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking. This paper derives some probability-one homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity. Some insight is provided into why the homotopies used in engineering practice are so successful, and why this success is more than dumb luck. By presenting the theory as variations on a prototype probability-one homotopy convergence theorem, the essence of such convergence theory is elucidated.