Study On Exact Penalty Methods And Quasi-newton Methods With Application To Nonlinear Optimization Problems

Optimization can be traced back to the extreme problem which firstly conducted research on some mathematical solution, which meant selecting the optimal scheme form a number of scheme. At the end of 1940s, optimization theory and methods developed into a independent discipline. Since the Dantzig put forward the way of solving linear programming problem of simplex method in 1947, the solving nonlinear programming, stochastic programming, multi-objective programming, semide-finite programming and other optimization problems of theoretical research have got a fast development and new methods have emerged constantly.One of the major ways on solving constrained nonlinear programming problems is to transform it into nonconstrained nonlinear programming problem. Penalty function methods and the Lagrangian methods are the two prevailing ways to implement the transformation. As that it is difficult to meet penalty function of the accuracy and differentiability at the same time. Therefore, this paper's main task is to improve the structure of the traditional penalty function and turn the optimization function into accurate and smooth type. Getting the relevant theory simultaneously, putting forward the dealing of penalty function with parameter and conducting numerical value experiment on calculating , the result shows the calculating effect improves obviously.Quasi-Newton algorithm for solving nonlinear unconstrained optimization is one of the most effective and mature algorithms. As well as ,in the quasi-Newton method, the quasi-Newton equation plays a vital role in the structure. The initial quasi-Newton equation just take charge of the objective function of the first order derivative information, without paying attention to the objective function value of the information. Therefore, many people have studied quasi-Newton equation and have made good progress. So in view of the modified quasi-Newton equation which become developing hot spot of quasi-Newton algorithm. In this paper, two kinds of nonmonotone linear search technology were submitted and three types of quasi-Newton equation were modified, and the global convergence theory were proved, and numerical experiments were conducted on the algorithm. the results show that the effect is good.This paper is organized as follows:In Chapter 1,we introduce the basics of optimization and nonmonotone linear search technology, then introduces the exact penalty function of research and development situation, and introduces the theory of quasi-Newton algorithm and the current research, finally the results of this study are given.In Chapter 2,we presents a new exact penalty function method with parameters and presents new exact penalty theorem on the base of past research : on the one hand , studying accurate penalty function's approximation accuracy, on the other hand, conducting numerical experiments to verify feasibility of ways.In Chapter 3,we presents a nonconstrained optimization of diagonal sparse quasi -Newton algorithm and adopt a new nonmonotone linear search technology, in each iteration,we exploit approximate calibration matrix diagonal matrix, so each time of the storage capacity and work-load greatly reduces. Quasi-Newton equation is corrected and a more objective function informations are exploited. Under the general assumption, the algorithm of global convergence and superlinear convergence are proved. Numerical results show the feasibility of the algorithm.In Chapter 4, we presents a nonmonotone linear search technology and modifying quasi-Newton algorithm, and proves that global convergence of nonconvex minimization problem under the linear searching, establish global convergence theorem of algorithm. Numerical results show that the using of nonmonotone quasi-Newton search methods is significantly better than the original quasi-Newton method of monotone numerical results.In Chapter 5, we poses to a limited memory quasi-Newton algorithm based on the amendments proposed by Li and Fukushima BFGS formula, which shows that the algorithm in solving large scale of minimization problem with the global convergence. The efficiency of verifying validity of the numerical algorithm is more superior than the standard limited memory BFGS algorithm.