Convex Optimization Overview (cnt’d)

Convex Optimization Overview (cnt d)Chuong B. DoNovember 29, 2009 During last week s section, we began our study ofconvex Optimization , the study ofmathematical Optimization problems of the form,minimizex Rnf(x)subject tox C.(1)In a Convex Optimization problem,x Rnis a vector known as theoptimization variable,f:Rn Ris aconvex functionthat we want to minimize, andC Rnis aconvex setdescribing the set of feasible solutions. From a computational perspective, Convex optimiza-tion problems are interesting in the sense that any locally optimal solution will always beguaranteed to be globally optimal. Over the last several decades, general purpose methodsfor solving Convex Optimization problems have become increasingly reliable and these lecture notes, we continue our foray into the field of Convex Optimization .

1.2 Primal and dual problems To show the relationship between the Lagrangian and the original convex optimization prob-lem (OPT), we introduce the notions of the “primal”and “dual problems” associated with a

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