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Unformatted text preview: Lecture 10: Lagrangian Duality February 21, 2007 Lecture 10 Outline Motivation Visualization of PrimalDual Framework PrimalDual Constrained Optimization Problems Dual Function Properties Weak and Strong Duality Examples Convex Optimization 1 Lecture 10 Duality Theory An important part of optimization theory Its implications are far reaching both in theory and practice A powerful tool providing: A basis for the development of a rich class of optimization algorithms A general systematic way for developing bounding strategies (both in continuous and discrete optimization) A basis for sensitivity analysis Convex Optimization 2 Lecture 10 Main Idea and Issues in Duality Theory Associate an equivalent dual problem with a given (primal) problem Methodology applicable to a general constrained optimization problem Investigate: Is there a general relation beetween the primal and its associated dual problem? Under which conditions the primal and the dual problems have the same optimal values ? Under which conditions the primal and dual optimal solutions exist ? What are the relations between primal and dual optimal solutions ? What kind of information the dual optimal solutions provide about the primal problem? Convex Optimization 3 Lecture 10 Geometric Visualization of Duality We illustrate duality using an abstract geometric framework This framework provides insights into: Weak duality Strong duality (zero duality gap) Existence of duality gap Within this setting, we define: A geometric primal problem using an abstract set V R m R A corresponding geometric dual problem using the hyperplanes that support the set V Convex Optimization 4 Lecture 10 Geometric Primal Consider an abstract (nonempty) set V of vectors ( u,w ) R m R The set V intersects the waxis, i.e., (0 ,w ) V for some w R The set V extends north and east: [ North ] For any ( u,w ) V and u R m with u u , we have ( u,w ) V [ East ] For any (...
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 Spring '07
 AngeliaNedich

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