# Visible points, the separation problem, and applications to MINLP

## TL;DR

This paper introduces a novel technique for generating tighter cutting planes in mixed-integer nonlinear programming by utilizing the concept of visible points and reverse polars, improving the validity and strength of cuts.

## Contribution

The paper develops a new approach based on visible points and reverse polars to produce valid, tighter cuts for MINLP, with characterizations for quadratic and polynomial constraints.

## Key findings

- Visible points' reverse polar matches that of the feasible region.
- Characterization of visible points for single non-convex constraints.
- Extended formulations for polynomial relaxations.

## Abstract

In this paper we introduce a technique to produce tighter cutting planes for mixed-integer non-linear programs. Usually, a cutting plane is generated to cut off a specific infeasible point. The underlying idea is to use the infeasible point to restrict the feasible region in order to obtain a tighter domain. To ensure validity, we require that every valid cut separating the infeasible point from the restricted feasible region is still valid for the original feasible region. We translate this requirement in terms of the separation problem and the reverse polar. In particular, if the reverse polar of the restricted feasible region is the same as the reverse polar of the feasible region, then any cut valid for the restricted feasible region that separates the infeasible point, is valid for the feasible region. We show that the reverse polar of the visible points of the feasible region from the infeasible point coincides with the reverse polar of the feasible region. In the special where the feasible region is described by a single non-convex constraint intersected with a convex set we provide a characterization of the visible points. Furthermore, when the non-convex constraint is quadratic the characterization is particularly simple. We also provide an extended formulation for a relaxation of the visible points when the non-convex constraint is a general polynomial. Finally, we give some conditions under which for a given set there is an inclusion-wise smallest set, in some predefined family of sets, whose reverse polars coincide.