Lecture 2 Piecewise-linear optimization
Lecture 2 Piecewise-linear optimization
Lecture 2 Piecewise-linear optimization
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
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