Overview

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.

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9781402076275

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Reviews

From the reviews: Lagrange and penalty functions provide a powerful approach for study of constrained optimization problems. ... The book gives a systematic and unified presentation of many important results that have been obtained in this area during last several years. ... The book develops a unified approach to duality and penalization and to convergence analysis of the first and second order optimality conditions. ... A number of impressive new results on the existence of an exact penalty parameter have been obtained in the book. (Vladimir Gaitsgory, gazette The Australian Mathematical Society, Vol. 32 (4), 2005) In the monograph a whole optimization theory is developed ... . Besides a large number of theoretical statements, results of numerical experiments showing usefulness of the presented approach are also reported ... . It is shown that a much larger class of optimization problems than that of the convex ones allow for a thorough theoretical analysis and deep results. ... The monograph can be recommended to researchers in mathematical optimization being in interested in nonconvex problems. (Stephan Dempe, OR-News, Issue 23, 2005)

From the reviews: <p> Lagrange and penalty functions provide a powerful approach for study of constrained optimization problems. a ] The book gives a systematic and unified presentation of many important results that have been obtained in this area during last several years. a ] The book develops a unified approach to duality and penalization and to convergence analysis of the first and second order optimality conditions. a ] A number of impressive new results on the existence of an exact penalty parameter have been obtained in the book. (Vladimir Gaitsgory, gazette The Australian Mathematical Society, Vol. 32 (4), 2005) <p> In the monograph a whole optimization theory is developed a ] . Besides a large number of theoretical statements, results of numerical experiments showing usefulness of the presented approach are also reported a ] . It is shown that a much larger class of optimization problems than that of the convex ones allow for a thorough theoretical analysis and deep results. a ] The monograph can be recommended to researchers in mathematical optimization being in interested in nonconvex problems. (Stephan Dempe, OR-News, Issue 23, 2005)

From the reviews: Lagrange and penalty functions provide a powerful approach for study of constrained optimization problems. ... The book gives a systematic and unified presentation of many important results that have been obtained in this area during last several years. ... The book develops a unified approach to duality and penalization and to convergence analysis of the first and second order optimality conditions. ... A number of impressive new results on the existence of an exact penalty parameter have been obtained in the book. (Vladimir Gaitsgory, gazette The Australian Mathematical Society, Vol. 32 (4), 2005) In the monograph a whole optimization theory is developed ... . Besides a large number of theoretical statements, results of numerical experiments showing usefulness of the presented approach are also reported ... . It is shown that a much larger class of optimization problems than that of the convex ones allow for a thorough theoretical analysis and deep results. ... The monograph can be recommended to researchers in mathematical optimization being in interested in nonconvex problems. (Stephan Dempe, OR-News, Issue 23, 2005)

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