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Sum-rate maximization in two-way amplify-and-forward (AF) multiple-input multiple-output (MIMO) relaying belongs to the class of difference-of-convex functions (DC) programming problems. DC programming problems occur also in other signal processing applications and are typically solved using different modifications of the branch-and-bound method which, however, does not have any polynomial time complexity guarantees. In this paper, we develop two efficient polynomial time algorithms for the sum-rate maximization in two-way AF MIMO relaying. The first algorithm guarantees to find at least a Karush-Kuhn-Tucker (KKT) solution. There is a strong evidence, however, that such a solution is actually globally optimal. The second algorithm that is based on the generalized eigenvectors shows the same performance as the first one with reduced computational complexity. The objective function of the problem is represented as a product of quadratic fractional ratios and parameterized so that its convex part (versus the concave part) contains only one (or two) optimization variables. One of the algorithms is called POlynomial Time DC (POTDC) and is based on semi-definite programming (SDP) relaxation, linearization, and an iterative Newton-type search over a single parameter. The other algorithm is called RAte-maximization via Generalized EigenvectorS (RAGES) and is based on the generalized eigenvectors method and an iterative search over two (or one, in its approximate version) optimization variables. We derive an upper-bound for the optimal value of the corresponding optimization problem and show by simulations that this upper-bound is achieved by both algorithms. It provides an evidence that the algorithms find a global optimum. The proposed methods are also superior to other state-of-the-art algorithms.

Sum-Rate Maximization in Two-Way AF MIMO Relaying: Polynomial Time Solutions to a Class of DC Programming Problems