- #1
smehdi
- 16
- 0
Dear Friends
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.
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.