Sunday, July 24, 2016

Non-convex QP as a MIP: performance

In this post I showed how a small non-convex QP could be reformulated as a MIP using KKT conditions plus a special (linear) objective. Will this work in practice? How does this compare performance-wise with global solvers? Let’s see if we can get a feeling for this.

Duplicating the problem

One easy way to make the small test problem bigger is to duplicate the problem:

Objective \(c'x_1 + x'_1 Q x_1\) \(+c'x_2 + x'_2 Q x_2\) \(\cdots\) \(+c'x_K + x'_K Q x_K\)
Constraints \(A x_1 \le b\)
\(\ell \le x_1 \le u\)
\(A x_2 \le b\)
\(\ell \le x_2 \le u\)
\(\ddots\)
\(A x_K \le b\)
\(\ell \le x_K \le u\)
This way we can make the model easily much bigger by introducing a set K. Notice that solvers have a difficult time to see through this trick. They are not recognizing they are solving a tiny problem that is just repeated K times. Actually these independent blocks often make the problem tough to solve (I don't know exactly why, but experience shows this is the case).
Modeling

Basically we need to add an index K to the variables \(x_i\), and to the constraints. Finally we add a summation to the objective. The new model looks like:

image

For the KKT model we can do something similar.

Results

Here are some results with K=5 and K=10. I stopped after 1000 seconds (if not proven optimal by then, I report the gap).

image

Note: The global QP solver GloMiQo behaves the same as Antigone.

At least for this model the MIP formulation is doing very well. The Cplex MIP solver beats the Cplex global QP solver by a large margin!

Yalmip

Yalmip has reported similar results produced by its KKTQP method.