Linear programming constraints: multiplication of optimization variables
constraints, linear-programming, matlab, optimization
Solution
Since the problem is not entirely linear, I do not believe you can solve it as-is using the `linprog` function. However, you should be able to reformulate the problem as a mixed integer linear programming problem. Then you would be able to use for example this extension from Matlab Central to solve the problem.
Assuming that `x_in(t)` and `x_out(t)` are non-negative variables with upper bounds `x_in_max` and `x_out_max`, respectively, then you can add the variables `y_in(t)` and `y_out(t)` to your optimization problem and include the following constraints:
(1) y_in(t) and y_out(t) are binary, i.e. 0 or 1
(2) x_in(t) <= x_in_max * y_in(t)
(3) x_out(t) <= x_out_max * y_out(t)
(4) y_in(t) + y_out(t) = 1
Given that `y_in` and `y_out` are binary variables, constraints (2) and (3) relate the `x_` and `y_` variables with each other and ensure that the `x_` variables remain within bounds (fix bounds on the `x_` variables can thus and should be removed from the problem formulation). Constraint (4) ensures that either the `_in` or the `_out` event occurs, but not both at the same time.
Problem
Given an optimization problem with two optimization variables (`x_in(t)`, `x_out(t)`). For any time-step, when `x_in` is non-zero, `x_out` must be zero (and vice versa). Written as a constraint: ``` x_in(t)*x_out(t)=0 ``` How can such a constraint be included in Matlab's `linprog` function?