## Sum of piecewise linear functions

I have a problem minimizing sum of piecewise linear functions. Given $f:R\rightarrow R$ with $f(x) = \sum_{i=1}^n \max (a_ix,b_ix+c_i)$ and $a_i,c_i <0; b_i>0$ find an equation that $a_i,b_i, c_i$ must satisfy in order to assure that argmin of $f$ is $x_0$.

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Teese will not be equations but inequalities. Consider $n=1$ then $n=2$ and you easily figure out the pattern. – Alexandre Eremenko Jan 11 at 19:32