Polynomial Solvability of Variants of the Trust-Region Subproblem

We consider an optimization problem of the form min xQx+ cx s.t. ‖x− μh‖ ≤ rh, h ∈ S, ‖x− μh‖ ≥ rh, h ∈ K, x ∈ P, where P ⊆ R is a polyhedron defined by m inequalities and Q is general and the μh ∈ R and the rh quantities are given; a strongly NP-hard problem. In the case |S| = 1, |K| = 0 and m = 0 one obtains the classical trust-region subproblem which is polynomially solvable, and has been the focus of much interest because of applications to combinatorial optimization and nonlinear programming We prove that for each fixed pair |S| and |K| our problem can be solved in polynomial time provided that either (1) |S| > 0 and the number of faces of P that intersect ⋂ h{x ∈ R n : ‖x − μh‖ ≤ rh, h ∈ S} is polynomially bounded, or (2) |S| = 0 and m is bounded.