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In this paper, we consider the problem of minimizing a nonconvex quadratic function, subject to two quadratic inequality constraints. As an application, such a quadratic program plays an important role in the trust region method for nonlinear optimization; such a problem is known as the Celis, Dennis, and Tapia (CDT) subproblem in the literature. The Lagrangian dual of the CDT subproblem is a semidefinite program (SDP), hence convex and solvable. However, a positive duality gap may exist between the CDT subproblem and its Lagrangian dual because the CDT subproblem itself is nonconvex. In this paper, we present a necessary and sufficient condition to characterize when the CDT subproblem and its Lagrangian dual admits no duality gap (i.e., the strong duality holds). This necessary and sufficient condition is easy verifiable and involves only one (any) optimal solution of the SDP relaxation for the CDT subproblem. Moreover, the condition reveals that it is actually rare to render a positive duality gap for the CDT subproblems in general. Moreover, if the strong duality holds, then an optimal solution for the CDT problem can be retrieved from an optimal solution of the SDP relaxation, by means of a matrix rank-one decomposition procedure. The same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the Lagrangian multipliers at a KKT point. Furthermore, we show that the condition is numerically easy to work with approximatively.

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In this paper we present a polynomial-time procedure to find a low rank solution for a system of

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This paper considers a two-way relay network, where two legitimate users exchange messages through several cooperative relays in the presence of an eavesdropper, and the Channel State Information (CSI) of the eavesdropper is imperfectly known. The Amplify-and-Forward (AF) relay protocol is used. We design the relay beamforming weights to minimize the total relay transmit power, while requiring the Signal-to-Noise-Ratio (SNRs) of the legitimate users to be higher than the given thresholds and the achievable rate of the eavesdropper to be upper-bounded. Due to the imperfect CSI, a robust optimization problem is summarized. A novel iterative algorithm is proposed, where the line search technique is applied, and the feasibility is preserved during iterations. In each iteration, two Quadratically-Constrained Quadratic Programming (QCQP) subproblems and a one-dimensional subproblem are optimally solved. The optimality property of the robust optimization problem is analyzed. Simulation results show that the proposed algorithm performs very close to the non-robust model with perfect CSI, in terms of the obtained relay transmit power; it achieves higher secrecy rate compared to the existing work. Numerically, the proposed algorithm converges very quickly, and more than 85% of the problems are solved optimally.

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