On Saddle Point Of Augmented Lagrangian For Constrained Optimization And The Convergence Of The Optimal Path

Optimization method is an important part of operations research. The Lagrangian dual problem and the optimization conditions of the saddle point play important parts in optimization problem. In this paper, we present a new augmented Lagrangian function for inequality-constrained optimization. Then we propose the sufficient and necessary conditions for the existence of the local and global saddle point of the new and classical Lagrangian functions. This paper is composed of two chapters.Chapter 1 is the introduction of this paper, which introduces the augmented Lagrangian function and the main results obtained in this paper.In Chapter 2, we generalize and extend the four classes of augmented Lagrangian functions in [1].We first show that the local saddle point condition of the new augmented Lagrangian function is a sufficient condition for the local optimality. We then prove that under mild conditions, the new augmented Lagrangian function possesses local and global saddle point. And then, the augmented second order sufficient condition which is weaker than the second order sufficient condition is given. In [2], for a given iterate generated by the special augmented Lagrangian method, the author only derived computable estimates for the distance to the primal solution of the underlying optimization problem, and the saddle point problem was not considered. Here for the mix-constrained optimization, we prove that the local saddle point condition of thespecial augmented Lagrangian function in [2] is a sufficient condition for the local optimality. And we show that local saddle point of the special Lagrangian function exists under augmented second order sufficiency condition.In the end, the convergence of the optimal paths is studied.