I want to solve a constrained optimisation problem of form:
minimize objective function subject to
constraint1 OR constraint2 or Constraint3
My present methodology is to solve the objective function by consdiering the first constraint only and obtain the optimal function values (say v1). And then solve the objective function considering only the second constraint and obtain optimal function value v2 and then similarly v3.
Then find the $\min\{v1,v2,v3\}$.
Or is there is a better way to solve it.
A random example: $$ \mbox{minimize}\; x + y+ z $$ s.t $$2x + y \le 4 \\ OR -3x + -67y \ge 56 $$
You can do this by using an extra binary variable $\delta$ that equals $1$ if and only if the first constraint is met. Use the following constraints: $$ 2x+y\le 4 +M(1-\delta) \\ -3x -67 y \ge 56 -M \delta \\ \delta \in \{0,1\} $$
$M$ is a large constant. You can see that if $\delta$ is set to $1$ the problem is equivalent to
$$ 2x+y\le 4 \\ -3x -67 y \ge 56 -M \\ $$ in other words the first constraint is required, but the second one is switched off as $56-M$ is very small.
If $\delta = 0$, the problem becomes $$ 2x+y\le 4 +M \\ -3x -67 y \ge 56 $$ and in this case the second constraint is required, and the first one is switched off, as $4+M$ is very large.
The solver will set the best value for $\delta$ and thus automatically consider the best scenario possible.