The Algorithms Of Constrained Nonlinear Programming

The appearance of optimization theory and method can be traced to the oldestextremum matter; however, being an independent subject was in the end of the40s oflast century. When Dantzing proposed the simplex algorithm for the general linearprogramming, and with the deepening of industrial revolution and informationrevolution as well as the huge development of the computer technology, for the recentdecades, it has been developed rapidly. By far, the development of study for all kindsof optimization matters, such as linear programming solution, non-linear programmingas well as non-smooth programming, multi-target programming, geometricprogramming and integer programming have been undergoing fast with new methods,moreover, it has been widely used in economic, military and scientific aspect,becoming an active subject.One of the main methods to solve the constrained optimization problem is tochange it into non-constrained optimization problem. Lagrange multiplier method iswidely used in practice because it can transfer the constrained optimization probleminto a non-constrained optimization problem effectively. A corresponding relationshipexists between the solution of the multiplier method and that of original constrainedproblem. For nonlinear programming problems with constraints, some documents haveput forward a number of Augmented Lagrange multiplier functions, which can useplenty of effective analytical methods and also have relationships with the solutions ofthe original problem in given condition.This paper is structured as follows: Chapter One introduces the basic definitionand some methods of Optimization Theory. In terms of non-linear programming withinequality constraints, Chapter Two raises one kind of new Augmented Lagrangemultiplier functions, proves the corresponding relationship between its stable point&global minimum point and the original constrained problem’s KKT point&global minimum point, and testifies that the local minimum points of Augmented Lagrangemultiplier functions is that of the original problem. Based on the numerical calculationon the new Augmented Lagrange multiplier functions, its feasibility and effectivenessare confirmed. Chapter Three proposes a new function concerning non-linearprogramming of general constrains, and demonstrates some of its properties. ChapterFour briefs the present research and results home and abroad of trust region methodand filter method, and presents the prospect for future work.