Smooth Exact Penalty Functions For Constrained Optimization

Many important problems arising in the fields such as engineering,national defence, economy,and finance etc,can be modeled as constrained optimization problems. One of the main approaches for solving constrained optimization problems is to transform them into unconstrained optimization problems.Penalty function methods and Lagrangian duality methods are two prevailing approaches to implement the transformation.Penalty methods seek to obtain the solutions of the original constrained optimization problem by solving one or a sequence of penalty problems.By exact penalty function,a constrained optimization problem can be transformed into a single unconstrained or simple constrained optimization problem, which can avoid the appearance of the ill-conditioning case that the Hessian of the penalty function is seriously indefinite when the penalty parameter is too large.However,the traditional exact penalty function is non-differentiable,which makes it difficult to apply fast unconstrained algorithms based on the gradients and may cause the Maratos phenomena to appear in some fast algorithms such as the Sequential Quadratic Programming methods.So it is very meaningful to construct smooth and exact penalty function by some means.In this thesis,we firstly propose new smooth exact penalty functions for several kinds of constrained optimization problems;secondly,we obtain several classes of smooth and approximately exact penalty functions by smoothing the classical exact penalty functions.The present thesis is organized as follows.In Chapter 1,we give a brief introduction to the existed research work on the penalty functions and Sequential Quadratic Programming methods,with an emphasis on exact penalty functions and smooth exact penalty functions.In Chapter 2,we construct new simple smooth exact penalty functions,where the term "simple" means that the constructed penalty function only contains the original information of the objective function and the constraint functions in the constrained optimization problem,but doesn't contain the information of their differentials. The main idea is to construct the penalty function by adding a new finite dimensional or even one dimensional decision variable.In Section 2.2,we propose a class of penalty function by adding a one-dimensional variable for the constrained optimization problem with equality constraints and simple bound constraints, and transform the original problem into a penalty problem with simple bound constraints.We discuss the smoothness of the new penalty function as well as other properties such as being bounded below.The equivalence between the optimal solutions of the original problem and those of the corresponding smoothed penalty problem is showed when the penalty parameter is sufficiently large,which implies the penalty function is exact.A simple algorithm and some numerical examples show that it is applicable to solve the smoothed penalty problem in order to solve the original problem.In Section 2.3,another smooth penalty function is proposed by adding a one-dimensional variable for the constrained optimization problem with only equality constraints,and its exactness is discussed.Numerical results show the performance of this penalty function.In Section 2.4,for inequality constrained optimization problem,a class of smooth penalty function with barrier property is proposed,and its exactness is also discussed.Compared with other barrier function methods,this smooth penalty method does not need a strict feasible interior point as the starting point.Since in the practical computation an approximate solution within the error bound is always needed,in Chapter 3 three kinds of smooth and approximately exact penalty functions are proposed.In Section 3.2 a class of smooth penalty function is proposed for inequality constrained optimization problem by adding a parameter to control its exactness.The error estimates between the optimal solutions of the original problem and of the penalty problem are discussed.An exact algorithm and an approximate algorithm are given,respectively.And some numerical results show the effect of the approximate algorithm.In Section 3.3,for general optimization problem with equality and inequality constraints,a class of smooth and approximately exact penalty function is given,and the error estimates are also discussed,with which an approximate algorithm is then proposed to solve the original problem.In Section 3.4,taking the inequality constrained optimization as an example,we give a family of smooth and approximately exact penalty functions which include the smooth penalty function given in Section 3.2.An algorithm that computes the exact solution of the original problem by solving the smoothed penalty problem is given and its convergence properties are discussed.In Chapter 4,we conclude the main contents in this thesis,and make some prospects for future works.