Smoothing Methods For The L₁and The Lower Order Penalty Functions

Penalty function methods are a class of important methods for solving nonlinearconstrained optimization problems. They find a minimizer of a constrained optimizationproblem by solving an or a sequence of unconstrained optimization problems and thismakes the minimization process simple and efective. The study of the penalty functionmethod has been an important research topic in the field of mathematical programming.This paper gives two smoothing approximation methods of the l1exact penalty func-tion and a smoothing approximation method of the lower order exact penalty function,respectively, in order to solve inequality constrained optimization problems. Firstly, errorestimations are obtained among the optimal objective function values of the smoothedpenalty problem, of the penalty problem, and of the original problem. It is shown that,under mild assumptions, an approximate optimal solution of the original problem can beobtained by searching a sequence of optimal solutions of the smoothed penalty problems.Secondly, we develop the corresponding algorithms based on the smoothed penalty func-tions for solving the original optimization problem and prove the convergence of thesealgorithms. Finally, some numerical examples are given to illustrate the efciency of thealgorithms presented in this paper.This paper is divided into three chapters:In Chapter1, a review is given for the research status about the smoothing methodof the l1and the lower order exact penalty functions.In chapter2, two new bilateral smoothing methods about the l1exact penalty func-tion are presented. The two methods are based on a quadratic and a polynomial function,respectively, to approximate the l1exact penalty function. Firstly, it is proved that theoptimal solution of the smoothed penalty problem is an ε approximate optimal solutionof the original problem. Secondly, we develop the corresponding algorithms based onthe two bilateral smoothing methods for solving the original optimization problem andprove the convergence of the algorithms. Finally, some numerical examples are given to illustrate the efciency of the algorithms.In chapter3, a new unilateral smoothing method about the lower order exact penaltyfunction is presented. This method is based on a twice continuously diferentiable functionto approximate the lower order exact penalty function. Firstly, it is proved that theoptimal solution of the smoothed penalty problem is an ε approximate optimal solutionof the original problem. Secondly, we develop the corresponding algorithm based on theunilateral smoothing method for solving the original optimization problem and prove theconvergence of the algorithm. Finally, some numerical examples are given to illustratethe efciency of the algorithm.