Exact two steps SOCP/SDP formulation for a modified conic trust region subproblem

In this paper, we study a modified version of the conic trust region subproblem which arises within a class of nonlinear programming algorithms. First using a variant of S-Lemma, we give an SOCP/SDP formulation which gives its optimal objective value. Then using the parametrization approach of Dinkelbach and the known exact SOCP/SDP relaxation of the extended trust region subproblem, we find its optimal solution. Finally, some preliminary numerical results are given.