Necessary and Sufficient Conditions for Rank-One-Generated Cones

A closed convex conic subset [Formula: see text] of the positive semidefinite (PSD) cone is rank-one generated (ROG) if all its extreme rays are by matrices. The ROG property closely related to exactness program (SDP) relaxations nonconvex quadratically constrained quadratic programs (QCQPs) text]. We consider case where obtained as intersection PSD with finitely many homogeneous linear matrix inequalities and constraints identify sufficient conditions that guarantee ROG. Our general framework allows us recover a number well-known results from literature. In two inequalities, we also establish necessity our conditions. This extends one few settings literature—the inequality S-lemma—where an explicit characterization for exists. Finally, show how on cones can be translated into inhomogeneous SDP hull descriptions in original space QCQP. close applications these results; specifically, perspective reformulation simple mixed-binary set via toolkit. Funding: work was supported Office Naval Research [Grant N00014-19-1-2321] National Science Foundation (NSF) Division Civil, Mechanical Manufacturing Innovation CMMI-1454548] Frank A. Helen E. Risch Faculty Development Chair. Part this done while second author visiting Simons Institute Theory Computing, which partially DIMACS/Simons Collaboration Bridging Continuous Discrete Optimization through NSF Computing Communication Foundations CCF-1740425].