Transcription of Convex Optimization — Boyd & Vandenberghe 1. Introduction
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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 consumpti
Least-squares minimize kAx−bk2 2 solving least-squares problems • analytical solution: x⋆ = (ATA)−1ATb • reliable and efficient algorithms and software • computation time proportional to n2k(A∈ Rk×n); less if structured • a mature technology using least-squares • least-squares problems are easy to recognize
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