Transcription of Lecture 2 Piecewise-linear optimization
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L. VandenbergheEE236A (Fall 2013-14) Lecture 2 Piecewise-linear optimization Piecewise-linear minimization 1- and -norm approximation examples modeling software2 1 Linear and affine functionslinear function:a functionf:Rn Ris linear iff( x+ y) = f(x) + f(y) x, y Rn, , Rproperty:fis linear if and only iff(x) =aTxfor someaaffine function:a functionf:Rn Ris affine iff( x+ (1 )y) = f(x) + (1 )f(y) x, y Rn, Rproperty:fis affine if and only iff(x) =aTx+bfor somea,bPiecewise-linear optimization2 2 Piecewise-linear functionf:Rn Ris (convex)piecewise-linearif it can be expressed asf(x) = maxi=1.
1-norm minimization • xˆ∈ Rn is unknown signal, known to be very sparse • we make linear measurements y =Axˆwith A ∈ Rm×n, m < n estimation by ℓ 1-norm minimization: compute estimate by solving minimize kxk 1 subject to Ax =y estimate is signal with smallest ℓ 1-norm, consistent with measurements equivalent LP (variables x, u ∈ Rn)
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