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IEEE International Conference on Acoustics Speech and Signal Processing

An alternating optimization algorithm for the MIMO secrecy capacity problem under sum power and per-antenna power constraints

Qiang Li, Mingyi Hong, Hoi-To Wai, Wing‐Kin Ma, Ya‐Feng Liu, Zhi-Quan Tom Luo

Published May 1, 2013• 8 citations
Publisher

Abstract

This paper considers transmit covariance optimization for a multi-input multi-output (MIMO) Gaussian wiretap channel. Specifically, we aim to maximize the MIMO secrecy capacity by judiciously designing the transmit covariance under the sum power and per-antenna power constraints. The MIMO secrecy capacity maximization (SCM) problem is nonconvex, and so far there is no tractable solution available. We propose an alternating optimization (AO) approach to handle the SCM problem. In particular, our development consists of two steps: First, we show that the SCM problem can be reexpressed to a form that can be conveniently processed by AO. Second, we develop a custom-designed fast algorithm for each AO iteration. Interestingly, with this fast implementation, the overall AO algorithm can be viewed as performing iterative reweighting and water-filling. Finally, the convergence of the proposed algorithm to a stationary solution of SCM is shown, and numerical results are provided to demonstrate its efficacy.

DOI
10.1109/icassp.2013.6638483

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This paper considers transmit covariance optimization for a multi-input multi-output (MIMO) Gaussian wiretap channel. Specifically, we aim to maximize the MIMO secrecy capacity by judiciously designing the transmit covariance under the sum power and per-antenna power constraints. The MIMO secrecy capacity maximization (SCM) problem is nonconvex, and so far there is no tractable solution available. We propose an alternating optimization (AO) approach to handle the SCM problem. In particular, our development consists of two steps: First, we show that the SCM problem can be reexpressed to a form that can be conveniently processed by AO. Second, we develop a custom-designed fast algorithm for each AO iteration. Interestingly, with this fast implementation, the overall AO algorithm can be viewed as performing iterative reweighting and water-filling. Finally, the convergence of the proposed algorithm to a stationary solution of SCM is shown, and numerical results are provided to demonstrate its efficacy.
This paper considers transmit optimization in multi-input multi-output (MIMO) wiretap channels, wherein we aim at maximizing the secrecy capacity or rate of an MIMO channel overheard by one or multiple eavesdroppers. Such optimization problems are nonconvex, and appear to be difficult especially in the multi-eavesdropper scenario. In this paper, we propose an alternating optimization (AO) approach to tackle these secrecy optimization problems. We first consider the secrecy capacity maximization (SCM) problem in the single eavesdropper scenario. An AO algorithm is derived through a judicious SCM reformulation. The algorithm conducts some kind of reweighting and water-filling in an alternating fashion, and thus is computationally efficient to implement. We also prove that the AO algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point of the SCM problem. Then, we turn our attention to the multiple eavesdropper scenario, where the artificial noise (AN)-aided secrecy rate maximization (SRM) problem is considered. Although the AN-aided SRM problem has a more complex problem structure than the previous SCM, we show that AO can be extended to deal with the former, wherein the problem is handled by solving convex problems in an alternating fashion. Again, the resulting AO method is proven to have KKT point convergence guarantee. For fast implementation, a custom-designed AO algorithm based on smoothing and projected gradient is also derived. The secrecy rate performance and computational efficiency of the proposed algorithms are demonstrated by simulations.
In this paper, we consider the problem of optimizing the transmit co-variance matrix for a multiple-input multiple-output (MIMO) Gaussian wiretap channel. The scenario of interest consists of a transmitter, a legitimate receiver, and multiple non-cooperating eavesdroppers that are all equipped with multiple antennas. Specifically, we design the transmit covariance matrix by maximizing the secrecy rate under a total power constraint, which is a non-convex difference of convex functions (DC) programming problem. We develop an algorithm, termed alternating matrix POTDC algorithm, based on alternating optimization of the eigenvalues and the eigenvectors of the transmit covariance matrix. The proposed alternating matrix POTDC method provides insights into the non-convex nature of the problem and is very general, i.e., additional constraints on the co-variance matrix can easily be incorporated. The secrecy rate performance of the proposed algorithm is demonstrated by simulations.
This paper presents novel numerical approaches to finding the secrecy capacity of the multiple-input multiple-output (MIMO) wiretap channel subject to multiple linear transmit covariance constraints, including sum power constraint, per antenna power constraints and interference power constraint. An analytical solution to this problem is not known and existing numerical solutions suffer from slow convergence rate and/or high per-iteration complexity. Deriving computationally efficient solutions to the secrecy capacity problem is challenging since the secrecy rate is expressed as a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for some special cases. In this paper we propose two low-complexity methods to compute the secrecy capacity along with a convex reformulation for degraded channels. In the first method we capitalize on the accelerated DC algorithm which requires solving a sequence of convex subproblems, for which we propose an efficient iterative algorithm where each iteration admits a closed-form solution. In the second method, we rely on the concave-convex equivalent reformulation of the secrecy capacity problem which allows us to derive the so-called partial best response algorithm to obtain an optimal solution. Notably, each iteration of the second method can also be done in closed form. The simulation results demonstrate a faster convergence rate of our methods compared to other known solutions. We carry out extensive numerical experiments to evaluate the impact of various parameters on the achieved secrecy capacity.
This paper presents two different low-complexity methods for obtaining the secrecy capacity of multiple-input multiple-output (MIMO) wiretap channel subject to a sum power constraint (SPC). The challenges in deriving computationally efficient solutions to the secrecy capacity problem are due to the fact that the secrecy rate is a difference of convex functions (DC) of the transmit covariance matrix, for which its convexity is only known for the degraded case. In the first method, we capitalize on the accelerated DC algorithm, which requires solving a sequence of convex subproblems. In particular, we show that each subproblem indeed admits a water-filling solution. In the second method, based on the equivalent convex-concave reformulation of the secrecy capacity problem, we develop a so-called partial best response algorithm (PBRA). Each iteration of the PBRA is also done in closed form. Simulation results are provided to demonstrate the superior performance of the proposed methods.
We consider a secure transmission including a transmitter, a receiver and an eavesdropper, each being equipped with multiple antennas. The aim is to develop a low-complexity and scalable method to find a globally optimal solution to the problem of secrecy rate maximization under a total power constraint at the transmitter. In principle, the original formulation of the problem is nonconvex. However, it can be equivalently translated into finding a saddle point of a minimax convex-concave program. An existing approach finds the saddle point using the Newton method, whose computational cost increases quickly with the number of transmit antennas, making it unsuitable for large scale antenna systems. To this end, we propose an iterative algorithm based on alternating optimization, which is guaranteed to converge to a saddle point, and thus achieves a globally optimal solution to the considered problem. In particular, each subproblem of the proposed iterative method admits a closed-form solution. We analytically show that the iteration cost of our proposed method is much cheaper than that of the known solution. As a result, numerical results demonstrate that the proposed method remarkably outperforms the existing one in terms of the overall run time.
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 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.