Dear Friends(adsbygoogle = window.adsbygoogle || []).push({});

I have a question about linear programming. It would be great to have your comments or suggestions.

Consider the followings

\begin{equation}

\\

Y = [y_{1}, y_{2}, \cdots, y_{n}]

\\

G = [g_{1}, g_{2}, \cdots, g_{n}]

\\

\textbf{X} =

\begin{pmatrix}

0 & x_{1,2} & \cdots & x_{1,n} \\

x_{2,1} & 0 & \cdots & x_{2,n} \\

\vdots & \vdots & \ddots & \vdots \\

x_{n,1} & x_{n,2} & \cdots & 0

\end{pmatrix}

\\

\end{equation}

##\textbf{X}## is not symmetric. Here I show the ##i##th row of matrix ##\textbf{X}## by subscript ##(i,:)##, e.g., ##X_{(2,:)} = [x_{2,1}, 0, \cdots, x_{2,n}]## is the second row vector of matrix ##\textbf{X}## and subscript ##(:,i)## shows the ##i##th column vector.

The problem is

\begin{equation}

\begin{aligned}

& \min &Y = \sum_{j=1}^{n} y_{j}\\

& \text{subject to}

& X_{(i,:)} G^T \geq A \\

&& X_i \geq 0

\end{aligned}

\end{equation}

in which ##i## is fixed, say ##i=2##, matrix ##G## is known, elements of ##\textbf{X}## are also known except the ##i##th row and ##X_{(i,:)}## should be found. Each of the elements in ##Y_i## is a function of the column vector in ##\textbf{X}## of the same index , i.e.,

$$

y_{j} = f(\textbf{X}(:,j))

$$

I googled and found that the function ##f## that I am working with is a piecewise linear function! Actually here the problem is not to find the minimum of a piecewise linear function, but to find the minimum of sum over multiple piecewise linear functions!

Since I am totally new in the filed of linear programming I don't know how such a problem should be solved!

Does anybody have any comments or know some references about how I could solve this?

Thanks in advance.

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# Piecewise linear programming

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