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Lecture 2 Piecewise-linear optimization

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.

• accept optimization problem in standard notation (max, k·k 1, . . . ) • recognize problems that can be converted to LPs • express the problem in the input format required by a specific LP solver examples of modeling packages • AMPL, GAMS • CVX, YALMIP (MATLAB) • CVXPY, Pyomo, CVXOPT (Python) Piecewise-linear optimization 2–23

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