Parabolic Approximation & Relaxation for MINLP

TL;DR

This paper introduces a novel quadratic approximation method for solving MINLP problems by globally approximating constraint epigraphs with paraboloids, leveraging MILP models to accelerate solution processes especially for problems with trigonometric or exponential functions.

Contribution

The paper presents a new approach to approximate MINLP constraints using paraboloids via MILP, enabling pre-computed lookup tables and faster solution times.

Findings

Significantly accelerates MINLP solution process.

Effective for problems with trigonometric and exponential functions.

Leverages MIQCP advancements for improved MINLP solving.

Abstract

We propose an approach based on quadratic approximations for solving general Mixed-Integer Nonlinear Programming (MINLP) problems. Specifically, our approach entails the global approximation of the epigraphs of constraint functions by means of paraboloids, which are polynomials of degree two with univariate quadratic terms, and relies on a Lipschitz property only. These approximations are then integrated into the original problem. To this end, we introduce a novel approach to compute globally valid epigraph approximations by paraboloids via a Mixed-Integer Linear Programming (MIP) model. We emphasize the possibility of performing such approximations a-priori and providing them in form of a lookup table, and then present several ways of leveraging the approximations to tackle the original problem. We provide the necessary theoretical background and conduct computational experiments on instances of the MINLPLib. As a result, this approach significantly accelerates the solution process of MINLP problems, particularly those involving many trigonometric or few exponential functions. In general, we highlight that the proposed technique is able to exploit advances in Mixed-Integer Quadratically-Constrained Programming (MIQCP) to solve MINLP problems.