Existence of local saddle points for a new augmented Lagrangian function.(Research Article)(Report)

1. Introduction

Consider the nonlinear optimization problem

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (P)

where f, [g.sub.i] : [R.sup.n] [right arrow] R for i = 1, ... , m and [h.sub.j] : [R.sup.n] [right arrow] R for j = 1,2, ... ,l are twice continuously differentiable functions and X [[subset].bar] [R.sup.n] is a nonempty closed subset.

The classical Lagrangian function associated with (P) is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where [lambda] = [([[lambda].sub.1], [[lambda].sub.2], ... , [[lambda].sub.m]).sup.T] [member of] [R.sup.m.sub.+] and [mu] = [([[mu].aub.1], [[mu].aub.2], ... , [[mu].aub.l]).sup.T] [member of] [R.sup.l].

The Lagrangian dual problem (D) is presented:

max [theta]([lambda],[mu]), (D)

s.t. [lambda] [greater than or equal to] 0,

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Lagrange multiplier theory not only plays a key role in many issues of mathematical programming such as sensitivity analysis, optimality conditions, and numerical algorithms, but also has important applications, for example, in scheduling, resource allocation, engineering design, and matching problems. According to both analysis and experiments, it performs substantially better than classical methods for solving some engineering projects, especially for medium-sized or large projects.

Roughly speaking, the augmented Lagrangian method uses a sequence of iterate point of unconstrained optimization problems, which are constructed by utilizing the Lagrangian multipliers, to approximate the optimal solution of the original problem. Toward this end, we must ensure that the zero dual gap property holds between primal and dual problems. Therefore, saddle point theory received much attention, due to its equivalence with zero dual gap property. It is well known that, for convex programming problems, the zero dual gap holds by using the above classical Lagrangian function. However, the nonzero duality gap may appear for nonconvex optimization problems. The main reason is that the classical Lagrangian function is linear with respect to the Lagrangian multiplier. To overcome this drawback, various types of nonlinear Lagrangian functions and augmented Lagrangian functions have been developed in recent years. For example, Hestenes [1] and Powell [2] independently proposed augmented Lagrangian methods for solving equality constrained problems by incorporating the quadratic penalty term in the classical Lagrangian function. This was extended by Rockafellar [3] to the constrained optimization problem with both equality and inequality constraints. A convex augmented function and the corresponding augmented Lagrangian with zero duality gap property were introduced by Rockafellar and Wets in [4]. This was further extended by Huang and Yang by removing the convexity assumption imposed on the augmented functions as in [4];see [5,6] for the details. Wang et al. [7] proposed two classes of augmented Lagrangian functions, which are simpler than those given in [4,5], and discussed the existence of saddle points. For other kinds of augmented Lagrangian methods refer to [816]; for saddle points theory and multiplier methods, refer to [17-20]. It should be noted that the sufficient conditions given in the above papers for the existence of local saddle points of augmented Lagrangian functions all require the standard …