Penalty Function Smoothing Methods For Constrained Optimization Problems

Many important problems arise in applications,such as management science,traffic assignments,network structures,national defence and finance etc.,they can be modeled as nonlinear constrained optimization problems.One of the main approach-es for solving nonlinear constrained optimization problems is to transform them into unconstrained optimization problems.Penalty function methods and Lagrangian du-ality methods are two prevailing approaches to implement the transformation.Penalty methods seek to obtain the solutions of the original constrained programming problem by solving one or more sequences of penalty problems.If each minimum of the penalty problem is a minimum of the primal constrained optimization problem,then the corre-sponding penalty function is called exact penalty function.By exact penalty function,a nonlinear constrained optimization problem can be transformed into a single uncon-strained or simple constrained optimization problem,which can avoid the appearance of the ill-conditioned case that the Hessian of the penalty function is seriously indefi-nite when the penalty parameter is too large.However,the traditional exact penalty function is non-differentiable,which causes some numerical instability problems in its implementation when the value of the penalty parameter becomes larger.So,it is very meaningful to construct smooth and exact penalty function by some means.In this thesis,we propose some new smooth penalty functions of the k-th power penalty function and smoothing of lower order penalty function.This thesis mainly consists of four chapters.In Chapter 1,we present a brief review on the fundamental theories and penalty function methods for nonlinear constrained optimization problem.In Chapter 2,a new smoothing power penalty function perturbed to the k-th is given,and the error estimates are also discussed,an algorithm is then proposed to solve the original problem.We further discuss the convergence of this algorithm and test this algorithm with some numerical examples.Numerical results show that the proposed algorithm outperforms other penalty function algorithms for nonlinear constrained optimization problems.In Chapter 3,we firstly construct a new smoothing approach for non-smooth max?x,0?P,(0<p<1).By using this smoothing approach,we present smoothing penalty function to the lower order penalty function in terms of second-order differ-entiability,which yields a second-order continuously differentiable penalty function.The error estimates between the optimal value of the smoothed penalty problem and the original problem are discussed.An algorithm based on our smoothing function is given,which is showed to be globally convergent.The numerical examples show that,the proposed algorithm is feasible and effective for solving some nonlinear constrained optimization problems.In Chapter 4,we summarized the main contribution of the paper and discussed some possible future research directions.