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

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Lagrange Dual Optimization (LDO) technique is a powerful tool for solving constrained optimization problems in 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, the authors 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.