Joint Bandwidth and Power Allocation with Admission Control in Wireless Multi-User Networks With and Without Relaying

Equal allocation of bandwidth and/or power may not be efficient for wireless multi-user networks with limited bandwidth and power resources. Joint bandwidth and power allocation strategies for wireless multi-user networks with and without relaying are proposed in this paper for (i) the maximization of the sum capacity of all users; (ii) the maximization of the worst user capacity; and (iii) the minimization of the total power consumption of all users. It is shown that the proposed allocation problems are convex and, therefore, can be solved efficiently. Moreover, the admission control based joint bandwidth and power allocation is considered. A suboptimal greedy search algorithm is developed to solve the admission control problem efficiently. The conditions under which the greedy search is optimal are derived and shown to be mild. The performance improvements offered by the proposed joint bandwidth and power allocation are demonstrated by simulations. The advantages of the suboptimal greedy search algorithm for admission control are also shown.Comment: 30 pages, 5 figures, submitted to IEEE Trans. Signal Processing in June 201

On k-Convex Polygons

We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard problem. We give a characterization of \mbox{$2$-convex} polygons, a particularly interesting class, and show how to recognize them in \mbox{$O(n \log n)$} time. A description of their shape is given as well, which leads to Erd\H{o}s-Szekeres type results regarding subconfigurations of their vertex sets. Finally, we introduce the concept of generalized geometric permutations, and show that their number can be exponential in the number of \mbox{$2$-convex} objects considered.Comment: 23 pages, 19 figure

GMRES-Accelerated ADMM for Quadratic Objectives

We consider the sequence acceleration problem for the alternating direction method-of-multipliers (ADMM) applied to a class of equality-constrained problems with strongly convex quadratic objectives, which frequently arise as the Newton subproblem of interior-point methods. Within this context, the ADMM update equations are linear, the iterates are confined within a Krylov subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its ability to accelerate convergence. The basic ADMM method solves a $\kappa$-conditioned problem in $O(\sqrt{\kappa})$ iterations. We give theoretical justification and numerical evidence that the GMRES-accelerated variant consistently solves the same problem in $O(\kappa^{1/4})$ iterations for an order-of-magnitude reduction in iterations, despite a worst-case bound of $O(\sqrt{\kappa})$ iterations. The method is shown to be competitive against standard preconditioned Krylov subspace methods for saddle-point problems. The method is embedded within SeDuMi, a popular open-source solver for conic optimization written in MATLAB, and used to solve many large-scale semidefinite programs with error that decreases like $O(1/k^{2})$, instead of $O(1/k)$, where $k$ is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on Optimization (SIOPT