Two nonlinear formulations

Problem Description

From: http://math.stackexchange.com/questions/1652150/turn-off-the-ovens-an-optimization-problem:

I am not at all sure the physics make sense, but lets fox on the mathematics of this model.

Formulation A

An MINLP formulation could be:

The idea is:

• If y_i=0 (oven is turned off) we force x_i=0 (equation Off). Also we have loss_i=0 as we multiplied the loss function by y_i.
• If y_i=1 (oven is turned on), we let the solver decide on x_i. In almost all cases it will choose a value > 0 as it wants to be somewhat close to a_i.
• We don’t have to model explicitly x_i=0 => y_i=0. If x_i=0 then the objective would have a preference for y_i=0 as this will improve the objective.

This model is no longer quadratic. For small instances like this we can use a global solver without a problem.

Formulation B

If we can make the problem quadratic and convex we can solve the problem with many more solvers. So here is a formulation that does just that:

Here we have:

• If y_i=0 (oven is turned off) we force x_i=0 (equation Off).  The slack s_i is left floating. This will cause the algorithm to choose a slack such that the loss is zero.
• If y_i=1 (oven is turned on) we force s_i=0 (equations On1,On2).

As this model is convex all MIQP solvers can handle this.