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. audience: the book is suitable for researchers in mathematical programming and optimization and postgraduate students in applied mathematics.