The Penalty Function Method For Nonlinear Global Optimization

The appearance of optimization theory and method can be traced to the oldest extremum matter. However, being an independent subject was in the end of the 40s of last century. When Dantzing proposed the simplex algorithm for the general linear programming, and with the deepening of industrial revolution and information revolution as well as the huge development of the computer technology, for the recent decades, it has been developed rapidly. By far, the development of study for all kinds of optimization matters, such as linear programming solution, non-linear programming, non-smooth programming, multi-target programming, geometric programming 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 matter is to change it into non-constrained optimization matter or any matter with simple constraints. Two common methods are Penalty function and Lagrange Function. The former method can be realized by solving the constrained programming by solving single or several penalty matters. When the penalty parameters are sufficiently large, the minima of one single penalty matter is also the minima of the original constrained programming, thus, the penalty function in the penalty matter is called as the exact penalty function, otherwise, it will be called as sequence penalty function for which, many effective analytical methods can be applied, moreover, accuracy and smooth character are indicated under a certain circumstance.The structure of this paper is: introducing the development status of the overall optimization matter and several deterministic algorithms in chapter one. Proposing a new exact penalty function form based on the constrained non-linear programming as well as designing algorithm for the accurate penalty function whose feasibility and effectiveness have been verified by examples in chapter two. And in chapter three, a new approximation l₁ accurate penalty function for differentiable non-linear programming has been proposed, and the algorithm and incremental algorithm are also provided with example verification. Chapter four mainly focuses on the problems of penalty function by far, and outlook for the future is also presented in this paper.