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Lecture 2 Piecewise-linear optimization

Lecture 2 Piecewise-linear optimization

Lecture 2 Piecewise-linear optimization

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Minimizing a sum of piecewise-<strong>linear</strong> functions<br />

minimize f(x)+g(x) = max<br />

i=1,...,m (aT i x+b i )+ max<br />

i=1,...,p (cT i x+d i )<br />

• cost function is piecewise-<strong>linear</strong>: maximum of mp affine functions<br />

(<br />

f(x)+g(x) = max (ai +c j ) T x+(b i +d j ) )<br />

i=1,...,m<br />

j=1,...,p<br />

• equivalent LP with m+p inequalities<br />

minimize t 1 +t 2<br />

subject to a T i x+b i ≤ t 1 , i = 1,...,m<br />

c T i x+d i ≤ t 2 , i = 1,...,p<br />

note that for fixed x, optimal t 1 , t 2 are t 1 = f(x), t 2 = g(x)<br />

<strong>Piecewise</strong>-<strong>linear</strong> <strong>optimization</strong> 2–5

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