Consider a linear optimization problem, with absolute values, of the following form:
\begin{equation} \begin{array}{rl} \text{minimize}\ &\mathbf{c'x}+\mathbf{d'y}\\ \text{subject to}\ &\mathbf{Ax}+\mathbf{By}\leq\mathbf{b}\\ &y_i=|x_i|, \end{array} \end{equation}
Assume that all entries of B and d are nonnegative.
I have to provide an example to show that if B has negative entries, the problem may have a local minimum that is not a global minimum, but I have really not idea how to it. Can you help me ?
ps: what will happen if the entries of c and A are negatives ?
Hint:
The set of $x\in\mathbb{R}$ which satisfy $1\leqslant|x|\leqslant2$ can be written as $[-2,-1]\cup[1,2]$. This constraint divides the feasible region into two disconnected components...