An Algorithm for Global Maximization of Secrecy Rates in Gaussian MIMO Wiretap Channels

Optimal signaling for secrecy rate maximization in Gaussian MIMO wiretap channels is considered. While this channel has attracted a significant attention recently and a number of results have been obtained, including the proof of the optimality of Gaussian signalling, an optimal transmit covariance matrix is known for some special cases only and the general case remains an open problem. An iterative custom-made algorithm to find a globally-optimal transmit covariance matrix in the general case is developed in this paper, with guaranteed convergence to a \textit{global} optimum. While the original optimization problem is not convex and hence difficult to solve, its minimax reformulation can be solved via the convex optimization tools, which is exploited here. The proposed algorithm is based on the barrier method extended to deal with a minimax problem at hand. Its convergence to a global optimum is proved for the general case (degraded or not) and a bound for the optimality gap is given for each step of the barrier method. The performance of the algorithm is demonstrated via numerical examples. In particular, 20 to 40 Newton steps are already sufficient to solve the sufficient optimality conditions with very high precision (up to the machine precision level), even for large systems. Even fewer steps are required if the secrecy capacity is the only quantity of interest. The algorithm can be significantly simplified for the degraded channel case and can also be adopted to include the per-antenna power constraints (instead or in addition to the total power constraint). It also solves the dual problem of minimizing the total power subject to the secrecy rate constraint.Comment: accepted by IEEE Transactions on Communication

A Sequential Quadratic Programming Method for Volatility Estimation in Option Pricing

Our goal is to identify the volatility function in Dupire's equation from given option prices. Following an optimal control approach in a Lagrangian framework, we propose a globalized sequential quadratic programming (SQP) algorithm with a modified Hessian - to ensure that every SQP step is a descent direction - and implement a line search strategy. In each level of the SQP method a linear-quadratic optimal control problem with box constraints is solved by a primal-dual active set strategy. This guarantees L1 constraints for the volatility, in particular assuring its positivity. The proposed algorithm is founded on a thorough first- and second-order optimality analysis. We prove the existence of local optimal solutions and of a Lagrange multiplier associated with the inequality constraints. Furthermore, we prove a sufficient second-order optimality condition and present some numerical results underlining the good properties of the numerical scheme.Dupire equation, parameter identification, optimal control, optimality conditions, SQP method, primal-dual active set strategy

Robust Adaptive Beamforming for General-Rank Signal Model with Positive Semi-Definite Constraint via POTDC

The robust adaptive beamforming (RAB) problem for general-rank signal model with an additional positive semi-definite constraint is considered. Using the principle of the worst-case performance optimization, such RAB problem leads to a difference-of-convex functions (DC) optimization problem. The existing approaches for solving the resulted non-convex DC problem are based on approximations and find only suboptimal solutions. Here we solve the non-convex DC problem rigorously and give arguments suggesting that the solution is globally optimal. Particularly, we rewrite the problem as the minimization of a one-dimensional optimal value function whose corresponding optimization problem is non-convex. Then, the optimal value function is replaced with another equivalent one, for which the corresponding optimization problem is convex. The new one-dimensional optimal value function is minimized iteratively via polynomial time DC (POTDC) algorithm.We show that our solution satisfies the Karush-Kuhn-Tucker (KKT) optimality conditions and there is a strong evidence that such solution is also globally optimal. Towards this conclusion, we conjecture that the new optimal value function is a convex function. The new RAB method shows superior performance compared to the other state-of-the-art general-rank RAB methods.Comment: 29 pages, 7 figures, 2 tables, Submitted to IEEE Trans. Signal Processing on August 201