I have this problem:

\begin{align} \min_{\mathbf{w}} & \sum_{i=1}^N c_i P_i(w_i)\\ s.t & \notag\\ & \sum_{i =1 }^N w_i = w \\ & 0 \leq w_i \leq w_{max},~~\forall i \in 1, ..., N \end{align}

where $P_i(w)$ is a piecewise linear convex function. Is there a closed formula to solve this? If not, what is the most relevant algorithm?


1 Answer 1


If $P_i(w)$ is a piecewise linear, convex function, then it can be written as the maximum of a number of linear functions, $P_i(w)=\max \{L_{i1}(w),\ldots,L_{iJ_i}(w)\}$. Then, the optimization problem allows for the reformulation $$ \min_{\mathbf w,\mathbf x} \sum_{i=1}^N c_ix_i = \mathbf c^T \mathbf x, \\ \sum_{i=1}^N w_i = w \\ 0\le w_i \le w_{max}, \qquad i=1,\ldots,N \\ x_i \ge L_{ij}(w) \qquad j=1,\ldots,J_i. $$ This is a linear program with linear constraints that can be solved using, for example, the simplex algorithm. There is, in general, no closed-form solution, however.

  • $\begingroup$ Thank you for your answer! I have come to the same conclusion myself. $\endgroup$
    – burnedWood
    Jan 22, 2015 at 20:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.