Lagrange Multiplier Optimization for Optimal Spectrum Balancing of DSL with Logarithmic Complexity

Lagrange dual optimization technique (LDO) is a powerful tool for solving constrained optimization problems in multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing systems (OFDM) and is generally considered to be optimal in the literature. LDO relaxes a constrained problem into an unconstrained dual problem using Lagrange multipliers. To solve the dual problem, the optimal value of the Lagrange multipliers should be found. The Lagrange multipliers are usually determined in an iterative process and reducing the number of iterations is of crucial importance to obtain systems with manageable computational complexity. In this paper, we show that for the LDO to be optimal in optimal spectrum balancing of DSL, the joint rate and power region (JRPR) should be strictly convex. Moreover, we propose a new LDO based algorithm with two basic advantages. Firstly, the computational complexity of the algorithm is logarithmic in the desired precision. Secondly, the algorithm can be used to find the optimal solution even for the cases with non-strictly convex JRPR. Our results can potentially be generalized to a wider range of optimization problems in the context of MIMO-OFDM and other complex separable systems.