Exactness Conditions for Semidefinite Programming Relaxations of Generalization of the Extended Trust Region Subproblem

Rujun Jiang () and Duan Li ()
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Rujun Jiang: School of Data Science, Fudan University, 200433 Shanghai, P.R. China
Duan Li: School of Data Science, City University of Hong Kong, Hong Kong, P. R. China

Mathematics of Operations Research, 2023, vol. 48, issue 3, 1235-1253

Abstract: The extended trust region subproblem (ETRS) of minimizing a quadratic objective over the unit ball with additional linear constraints has attracted a lot of attention in the last few years because of its theoretical significance and wide spectra of applications. Several sufficient conditions to guarantee the exactness of its semidefinite programming (SDP) relaxation have been recently developed in the literature. In this paper, we consider a generalization of the extended trust region subproblem (GETRS), in which the unit ball constraint in the ETRS is replaced by a general, possibly nonconvex, quadratic constraint, and the linear constraints are replaced by a conic linear constraint. We derive sufficient conditions for guaranteeing the exactness of the SDP relaxation of the GETRS under mild assumptions. Our main tools are two classes of convex relaxations for the GETRS based on either a simultaneous diagonalization transformation of the quadratic forms or linear combinations of the quadratic forms. We also compare our results to the existing sufficient conditions in the literature. Finally, we extend our results to derive a new sufficient condition for the exactness of the SDP relaxation of general diagonal quadratically constrained quadratic programs, where each quadratic function is associated with a diagonal matrix.

Keywords: Primary: 90C20; 90C22; 90C26; secondary: 90C30; quadratically constrained quadratic programming; extended trust region subproblem; semidefinite programming; second order cone programming (search for similar items in EconPapers)
Date: 2023
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