Nonlinear Relaxation for Convex MINLPs

What is a Convex MINLP?

For the purposes of this article, we define a convex MINLP as a mixed integer nonlinear problem which is convex if the integrality requirements are relaxed.

For example, consider the MINLP problem \(\min\{f(x,y) : g_i(x,y) \leq 0, i\in I\, x\in[x_l, x_u], y \in \{0,1\}\}\). If, by relaxing the integrality requirements on \(y\), i.e. \(\min\{f(x,y) : g_i(x,y) \leq 0, i\in I\, x\in[x_l, x_u], y \in [0,1]\}\), the resulting problem is a convex NLP then, in a slight abuse of terminology, the original MINLP is characterised as convex.

What is a Nonlinear Relaxation?

Octeract Engine has the ability to identify convex MINLP problems. For those problems, there are two methods that can be followed to derive a relaxation. The solution of the relaxation will be a lower bound to the solution of the original.

The first method is the same as it is for non-convex MINLPs: the nonlinear terms of the problem are linearised using guaranteed underestimators and the lower bounding problem becomes an (MI)LP.

The second method is specific to convex MINLPs: the relaxation of the integrality requirements for convex MINLPs will yield a convex NLP. The solution of which is, by definition, lower bounding to the solution of the original MINLP.

Why use a Nonlinear Relaxation?

Oftentimes, the solution of the nonlinear relaxation can be significantly tighter to the original problem. Therefore, the LLB solution can be very good from the beginning of the solution process. This is a feature that linear relaxation cannot necessarily possess.

There are two challenges with this approach:

  1. The NLP, corresponding to the relaxation of the MINLP, requires a convex nonlinear solver to be used. This poses a challenge, especially in large problems as nonlinear solvers may require a large amount of time to derive the solution. Furthermore, if the solver is posed with a time limit, the returned solution may be (severely) suboptimal. Typically, MILP and LP solvers handle large problems better than NLP solvers.

  2. On the contrary to the MILP relaxation of the convex MINLP, the nonlinear relaxation cannot provide any information regarding the integral values of the original problem. If a linear relaxation is used for the original problem, then the solution of the MILP relaxation can be used to hot-start the solution of the MINLP upper bounding problem. This, therefore, may reduce the computational time for the global solution.

When to use a Nonlinear Relaxation

If the problem at hand is a convex MINLP, it is worth attempting to solve the nonlinear relaxation. This is due to the fact that it can provide a lower bounding solution faster, especially if the MILP relaxation is suspected to be very loose.

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