Convex Optimization — Boyd & Vandenberghe 1. Introduction

Convex Optimization Boyd & Vandenberghe1. Introduction mathematical Optimization least-squares and linear programming Convex Optimization example course goals and topics nonlinear Optimization brief history of Convex optimization1 1 Mathematical Optimization (mathematical) Optimization problemminimizef0(x)subject tofi(x) bi, i= 1,..,m x= (x1,..,xn): Optimization variables f0:Rn R: objective function fi:Rn R,i= 1,..,m: constraint functionsoptimal solutionx has smallest value off0among all vectors thatsatisfy the constraintsIntroduction1 2 Examplesportfolio Optimization variables: amounts invested in different assets constraints: budget, investment per asset, minimum return objective: overall risk or return variancedevice sizing in electronic circuits variables: device widths and lengths constraints: manufacturing limits, timing requirements,maximum area objective: power consumptiondata fitting variables: model parameters constraints: prior information, parameter limits objective: measure of

Operations that preserve convexity practical methods for establishing convexity of a set C 1. apply definition x1,x2 ∈ C, 0 ≤ θ≤ 1 =⇒ θx1 +(1−θ)x2 ∈ C 2. show that Cis obtained from simple convex sets (hyperplanes, halfspaces, norm balls, . . . ) by operations that preserve convexity • intersection • affine functions

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