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In this work, we study the so-called extended Celis–Dennis–Tapia subproblem that are classical trust region subproblems with an additional quadratic constraint. For this class of problems, we propose a family of relaxations, and study some of its key properties, for instance, convexity issues, exactness of the relaxation, sufficient condition for exactness. The relaxation proposed is obtained adapting to our case a methodology recently developed for trust region problems with an additional conic constraint. We also find sufficient conditions for exactness of the proposed relaxation and discuss the relations between them. Finally, some numerical experiments are presented for testing the proposed relaxation and the sufficient conditions for its tightness.

On the exactness of a simple relaxation for the extended Celis–Dennis–Tapia subproblem