Nonlinear Optimization Problem Of A New Family Of Penalty Function Methods

Seeking fast and effective algorithms in nonlinear optimization has been a very interested research topic for the optimization researchers. Nonlinear constrained optimization are the abstract models most proximal to the practical problems. With the development of the calculative mathematics and the advances of the capability of the computers, finding dependable high effective algorithms that are easy to carry out for computers becomes focus of the era. Penalty function method is one of the effective methods to solve this kind of problems.The construction of the penalty function effects the efficience of the algorithms . In this paper we introduce the hyperbolic cosine function and construct the new hyperbolic cosine penalty function and algorithms, further more, we construct the new hyperbolic cosine multipier penalty function .In chapter 1, we first introduce the development of optimization; Some conditions to decide the optimum solution and several descent methods of unconstrained optimization; Look back to the penalty function and introduce the development and the present conditions of the augmented Lagrange function and multiplier methods.In chapter 2, using the favorable characters of the function Q(x) = ch{x) - 1, we construct the hyperbolic cosine penalty function and algorithms and proved its convergence . It can weaken the characters when the penalty genes gets too large when compared with the traditional penalty function . We propose the numerical experiments which show they are more efficient.In chapter 3, combining the traditional augmented Lagrange function and the hyperbolic cosine function, we construct a new hyperbolic penalty function multiplier method for the equality constrained optimization and deduced the iterative formulas, and proved the convergence under some conditions. We propose the numerical experiments that shows the efficience.