Research On The Method Of Simple Smoothly Exact Penalty Function

Constrained optimization problems abound in many department such as econ-my, engineering, national defence, energy, traffic and so on, and exist in manyfields such as information science, environmental science, and military etc. One ofhe important approaches for solving constrained optimization problems is penaltyunction method. The main idea of penalty function methods is to transform theonstrained optimization problems into solving a sequence of unconstrained opti-mization problems or constrained programming only with box constraints. Exactpenalty function methods is the method that by exact penalty function the con-trained optimization problems can be transformed into a single unconstrained op-imization problem or a constrained optimization problem with simple constraints,which can avoid the appearance of the ill-conditioning case that the Hessian of thepenalty function is seriously indefinite when the penalty parameter is too large.However, for the traditional penalty functions, if it is simple and exact, then it isnot smooth; if it is simple and smooth, then it is not exact; if it is exact and smooth,hen it is not simple. Here the word \simple" means that the constructed penaltyunction only contains the original information of the objective function and theonstraint functions in the constrained optimization problems, but does not con-ain the information of their differentials. Therefore in the practical algorithms, its difficult to apply fast unconstrained algorithms based on the gradients such asuasi-Newton method to the traditional exact penalty function methods(smoothbut not simple or simple but not smooth). So it is very meaningful to construct theimple smooth and exact penalty functions for constrained optimization problems.In this thesis, several simple smooth and exact penalty functions are devel-ped. In chapter 1, a brief introduction is given to local optimality conditions foronstrained minimization and the existing research work on the penalty functionsmethod and sequential quadratic programming method. In chapter 2, by addingvariable, we propose a new simple exact penalty for the constrained optimiza-ion problems without box constraints. In the section 2.2, we present a simplexact penalty function by adding a variable for the constrained optimization prob- lems with only equality constraints and transform the original problem into anunconstrained penalty problem. We discuss the smoothness and exactness of theproposed penalty function. In the section 2.3, for the constrained optimizationproblems with only inequality constraints, another simple exact penalty functionis given and its exactness is discussed. In section 2.4, similar to the discussion insection 2.2 and 2.3, for the constrained optimization problems with equality andinequality constraints, we construct a new simple exact penalty function by addinga variable, which have the same properties with the proposed penalty functions insection 2.3 and 2.4.Continuous differentiability of the penalty functions proposed in chapter 2can't be guaranteed whereε*= 0, and it is difficult to adopt the algorithms forunconstrained optimization problems to solve the corresponding penalty problembecause these algorithms usually requires the problems with continuous differen-tiability for some problems. On this basis, in chapter 3, we construct three simplesmooth exact penalty functions and discuss the smoothness and exactness undersome moderate conditions. In section 3.5, based on the penalty function proposedin section 3.3, we give some numerical experiments for constrained minimizationproblems.In chapter 4, Combined with the filled function method which is applied tosolve the global minimum solution for unconstrained optimization problems, wepresent a global algorithm for constrained optimization problems which is based ona simple smooth and exact penalty function. And some analysis of the algorithmis proposed.