Lagrange-type Functions in Constrained Non-Convex Optimization

This volume provides a systematic examination of Lagrange-type functions and augmented Lagrangians. Weak duality, zero duality gap property and the existence of an exact penalty parameter are examined. Weak duality allows one to estimate a global minimum. The zero duality gap property allows one to reduce the constrained optimization problem to a sequence of unconstrained problems, and the existence of an exact penalty parameter allows one to solve only one unconstrained problem. By applying Lagrange-type functions, a zero duality gap property for nonconvex constrained optimization problems is established under a coercive condition. It is shown that the zero duality gap property is equivalent to the lower semi-continuity of a perturbation function. In particular, for a type of kth power penalty functions, this book obtains an analytic expression of the least exact penalty parameter and establishes that a fairly small exact penalty parameter can be achieved. As shown by numerical experiments, this property is very important for some global methods of Lipschitz programming, otherwise ill conditioning may occur.