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Lecture 2 Piecewise-linear optimization

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L. VandenbergheEE236A (Fall 2013-14) Lecture 2Piecewise- linear optimization Piecewise-linear minimization 1- and -norm approximation examples modeling software2 1Linear 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 2Piecewise- linear functionf:Rn Ris (convex)piecewise-linearif it can be expressed asf(x) = maxi=1,...,m(aTix+bi)fis parameterized bym n-vectorsaiandmscalarsbixaTix+bif(x)(the termpiecewise-affineis more accurate but less common) Piecewise-linear optimization2 3Piecewise- linear minimizationminimizef(x) = maxi=1.

Approximate linear separation of non-separable sets minimize XN i=1 max{0,1−si(aTvi+b)} • penalty 1−si(aT i vi+b)for misclassifying point vi • can be interpreted as a heuristic for minimizing #misclassified points • a piecewise-linear minimization problem with variables a, b Piecewise-linear optimization 2–21

  Linear

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