a fast and cheap approach for strengthening lagrangian bound for the generalized celis-dennis-tapia subproblem

In this paper, we consider the generalized celis-dennis-tapia problem which is the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints, one of which being convex. when there is a positive duality gap, by exploiting an equivalent form of the dual lagrangian problem, we propose to improve the dual bound by adding one or two linear cuts to the lagrangian relaxation. the present work is motivated by and generalizes the results of [14] for the problem with two strictly convex quadratic constraints. our main contribution is to show that one can include the feasible region in a convex set and then follow the approach in [14] to construct the linear cuts based onsupporting hyperplanes of the convex set. numerical experiments are conducted to assess the quality of the proposed bounds.