On the Optimality of the Stationary Solution of Secrecy Rate Maximization for MIMO Wiretap Channel

TL;DR

This paper proves that the stationary solutions found by local optimization methods for the MIMO wiretap channel's secrecy rate maximization are actually globally optimal, supported by analytical proof and numerical validation.

Contribution

It provides an analytical proof that the KKT conditions have a unique solution, ensuring global optimality of stationary points in MIMO secrecy capacity problems.

Findings

Local optimization methods achieve global optimality in practice.

Unique KKT solution exists for both degraded and non-degraded cases.

Proposed low-complexity algorithm effectively finds stationary points.

Abstract

To achieve perfect secrecy in a multiple-input multiple-output (MIMO) Gaussian wiretap channel (WTC), we need to find its secrecy capacity and optimal signaling, which involves solving a difference of convex functions program known to be non-convex for the non-degraded case. To deal with this, a class of existing solutions have been developed but only local optimality is guaranteed by standard convergence analysis. Interestingly, our extensive numerical experiments have shown that these local optimization methods indeed achieve global optimality. In this paper, we provide an analytical proof for this observation. To achieve this, we show that the Karush-Kuhn-Tucker (KKT) conditions of the secrecy rate maximization problem admit a unique solution for both degraded and non-degraded cases. Motivated by this, we also propose a low-complexity algorithm to find a stationary point. Numerical results are presented to verify the theoretical analysis.