We consider the Robust Standard Quadratic Optimization Problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is extremized over the standard simplex. Following most approaches, we model the uncertainty sets by ellipsoids, polyhedra, or spectrahedra, more precisely, by intersections of sub-cones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of uncertainty sets, and present conditions under which the RStQP reduces to a single Standard Quadratic Optimization Problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.
Citation
Bomze, I. M., Kahr, M., Leitner, M. (2018). Trust your data or not - StQP remains StQP: Community Detection via Robust Standard Quadratic Optimization. Tech. report (1), Vienna, Austria, April 2018.