A characterization of maximal homogeneous-quadratic-free sets

The intersection cut framework was introduced by Balas in 1971 as a method for generating cutting planes in integer optimization. In this framework, one uses a full-dimensional convex S-free set, where S is the feasible region of the integer program, to derive a cut separating S from a non-integral vertex of a linear relaxation of S. Among all S-free sets, it is the inclusion-wise maximal ones that yield the strongest cuts. Recently, this framework has been extended beyond the integer case in order to obtain cutting planes in non-linear settings. In this work, we consider the specific setting when S is defined by a homogeneous quadratic inequality. In this ‘quadratic-free’ setting, every function \(\Gamma : D^m \rightarrow D^n\) $\mathrm{\Gamma }:D^m\to D^n$ , where \(D^k\) $D^k$ is the unit sphere in \(\mathbb {R}^k\) ${\mathbb{R}}^k$ , generates a representation of a quadratic-free set. While not every \(\Gamma \) $\mathrm{\Gamma }$ generates a maximal quadratic free set, it is the case that every full-dimensional maximal quadratic free set is generated by some \(\Gamma \) $\mathrm{\Gamma }$ . Our main result shows that the corresponding quadratic-free set is full-dimensional and maximal if and only if \(\Gamma \) $\mathrm{\Gamma }$ is non-expansive and satisfies a technical condition. This result yields a broader class of maximal S-free sets than previously known. Our result stems from a new characterization of maximal S-free sets (for general S beyond the quadratic setting) based on sequences that ‘expose’ inequalities defining the S-free set.