All You Need is a Paraboloid: Quadratic Cuts for Non-Convex MINLP

It is only half the job to find a good solution for a mathematical optimization problem, as one needs to verify its quality by specifying a dual bound. When it comes to mixed-integer nonlinear programming (MINLP), strong prerequisites such as constraint qualifications appear suitable, but may be difficult to verify computationally. In practice, solvers apply local refinement or convexification strategies to retrieve tight dual bounds. However, these concepts require appropriate big-M formulations, generate new sub-problems, or struggle to represent non-convex characteristics in terms of high accuracy, all of which can lead to long running times. As an alternative, we aim to leverage recent advances in mixed-integer quadratically-constrained programming (MIQCP) and propose a global approximation of constraint functions by paraboloids, \ie, univariate quadratic terms. The approximation is retrieved as a solution to a mixed-integer linear programming (MIP) problem. Further, for each nonlinear constraint function, we solve such MIPs and determine small numbers of paraboloids approximating it from either side. A replacement of the nonlinearities with the corresponding quadratic functions leads to a quadratically-constrained relaxation of the original problem. Solving the MIQCP relaxation then leads to a dual bound whose tightness depends on the approximation guarantee of the paraboloids. In summary, this approach enables solvers that are explicitly tailored for quadratic constraints to solve MINLPs to global optimality.