Smooth Penalty Functions And Penalty Algorithms For Solving Nonlinear Programming Problems

This thesis mainly considers smooth penalty functions and penalty algo-rithms for solving nonlinear programming problems. The paper consists of three chapters.Chapter 1 introduces the current developments of nonlinear programming problems and the smooth penalty methods. The main contribution of this paper is also listed in this chapter.In Chapter 2, we propose a new class of penalty functions for solving non-linear programming problems with inequality constraints, a subclass of which smoothly approximates the l1 penalty function. This approximation can be gen-eralized to lower order penalty functions. Based on the new class of penalty functions, we consider a penalty algorithm and, at each iteration, an exact global optimal solution or an inexact global optimal solution is obtained. Under very weak conditions, the algorithm is always applicable. We further present the global convergence property without any constraint qualification. Finally, numerical ex-periments are given.In Chapter 3, we give another penalty algorithm and the global convergence property. A perturbation theorem is established, that is, the limit of the objec-tive function value generated by our algorithm is equal to that of perturbation function at zero. From the perturbation theorem, we obtain the global conver-gence property, which is that any accumulation point of the sequence generated by the algorithm is the optimal solution of nonlinear programming problem. In addition, some useful corollaries are followed. With generalized M-F constraint qualification, we show further convergence property. Finally, we establish dual programming with the new penalty functions, and obtain zero duality gap prop- erty.