Research On Penalty Function Of Constrained Optimization Problems

Optimization theory and methods study the optimal solutions of some of the math problem.That is for practical problems,we select the best scheme from many schemes. It is the interdisciplinary of computational mathematics and operations research. It has a wide application in many areas such as national defense construction,economic plan,finance, engineering design, manufacturing, transportation and so on.And many problems of other disciplines can be attributed to the optimization problem,such as the assimilation of atmospheric science, the protein folding problem in life science, the pattern recognition problem in information science, the inversion issues of earth science and so on.The problems are often large-scale optimization problems,thus it has important theoretical and practical value which studies the optimization theory and methods.Penalty function methods are important and more practical methods for solving constrained optimization problems. Its basic idea is transforming a constrained problem into a single unconstrained problem or into a sequence of unconstrained problem and by solving these unconstrained problems to solve the constrained problem. To use unconstrained optimization problem instead of constrained optimization problem,the objective function of the unconstrained optimization problem must be a proper combination of the objective function of the constrained optimization problem and constraint functions. Usually the constraint functions who construct a penalty item are placed into the objective function via a penalty parameter in a way that penalizes any violation of the constraints.The construction principle of penalty items are: if the current iteration point is not feasible, it is necessary for its implementation of punishment, and the punishment value is increasing with the improving of the infeasibility of the point; no penalty for feasible points.The role of penalty items is to force the iterative point closer and closer and finally in the feasible domain with the progress of iteration. Constructing different penalty items corresponds different penalty function methods. Therefore, research on the different penalty items has important theoretical and practical value.1. The author constructs a new penalty function - index penalty functions for the general nonlinear constrained optimization problem. For the penalty function, the author also constructs a new penalty function algorithm and convergence theorem of the algorithm and its proof are given. Finally, numerical experiments verify the effectiveness of the algorithm.2. Geometric programming is a special nonlinear programming ,it's application is very extensive.Using the existing results and characteristics of constraints posynomial geometric programming and penalty function technique, the author designs a new algorithm for constraints posynomial geometric programming and proves the convergence of the algorithm.3. The author converts inequality constrained optimization problem into equality constrained optimization problem by using slack variables. Then we construct a new multiplier penalty function using the penalty function PE who belongs to equality constraints and was raised by Bertskas in 1982.