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

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 as well in other signal processing applications and are typically solved using different modifications of the branch-and-bound method. This method, however, does not have any polynomial time complexity guarantees. In this paper, we show that a class of DC programming problems, to which the sum-rate maximization in two-way MIMO relaying belongs, can be solved very efficiently in polynomial time, and develop two algorithms. The objective function of the problem is represented as a product of quadratic 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 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 also derive an upper-bound for the optimal values of the corresponding optimization problem and show by simulations that this upper-bound can be achieved by both algorithms. The proposed methods for maximizing the sum-rate in the two-way AF MIMO relaying system are shown to be superior to other state-of-the-art algorithms.Comment: 35 pages, 10 figures, Submitted to the IEEE Trans. Signal Processing in Nov. 201

Globally Optimal Energy-Efficient Power Control and Receiver Design in Wireless Networks

The characterization of the global maximum of energy efficiency (EE) problems in wireless networks is a challenging problem due to the non-convex nature of investigated problems in interference channels. The aim of this work is to develop a new and general framework to achieve globally optimal solutions. First, the hidden monotonic structure of the most common EE maximization problems is exploited jointly with fractional programming theory to obtain globally optimal solutions with exponential complexity in the number of network links. To overcome this issue, we also propose a framework to compute suboptimal power control strategies characterized by affordable complexity. This is achieved by merging fractional programming and sequential optimization. The proposed monotonic framework is used to shed light on the ultimate performance of wireless networks in terms of EE and also to benchmark the performance of the lower-complexity framework based on sequential programming. Numerical evidence is provided to show that the sequential fractional programming framework achieves global optimality in several practical communication scenarios.Comment: Accepted for publication in the IEEE Transactions on Signal Processin

Global Optimization for a Class of Nonlinear Sum of Ratios Problem

We present a branch and bound algorithm for globally solving the sum of ratios problem. In this problem, each term in the objective function is a ratio of two functions which are the sums of the absolute values of affine functions with coefficients. This problem has an important application in financial optimization, but the global optimization algorithm for this problem is still rare in the literature so far. In the algorithm we presented, the branch and bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of this problem belong. Convergence of the algorithm is shown. At last, some numerical examples are given to vindicate our conclusions

Optimal exact designs of experiments via Mixed Integer Nonlinear Programming

Optimal exact designs are problematic to find and study because there is no unified theory for determining them and studyingtheir properties. Each has its own challenges and when a method exists to confirm the design optimality, it is invariablyapplicable to the particular problem only.We propose a systematic approach to construct optimal exact designs by incorporatingthe Cholesky decomposition of the Fisher Information Matrix in a Mixed Integer Nonlinear Programming formulation. Asexamples, we apply the methodology to find D- and A-optimal exact designs for linear and nonlinear models using global orlocal optimizers. Our examples include design problems with constraints on the locations or the number of replicates at theoptimal design points

An interior point algorithm for minimum sum-of-squares clustering

Copyright @ 2000 SIAM PublicationsAn exact algorithm is proposed for minimum sum-of-squares nonhierarchical clustering, i.e., for partitioning a given set of points from a Euclidean m-space into a given number of clusters in order to minimize the sum of squared distances from all points to the centroid of the cluster to which they belong. This problem is expressed as a constrained hyperbolic program in 0-1 variables. The resolution method combines an interior point algorithm, i.e., a weighted analytic center column generation method, with branch-and-bound. The auxiliary problem of determining the entering column (i.e., the oracle) is an unconstrained hyperbolic program in 0-1 variables with a quadratic numerator and linear denominator. It is solved through a sequence of unconstrained quadratic programs in 0-1 variables. To accelerate resolution, variable neighborhood search heuristics are used both to get a good initial solution and to solve quickly the auxiliary problem as long as global optimality is not reached. Estimated bounds for the dual variables are deduced from the heuristic solution and used in the resolution process as a trust region. Proved minimum sum-of-squares partitions are determined for the rst time for several fairly large data sets from the literature, including Fisher's 150 iris.This research was supported by the Fonds National de la Recherche Scientifique Suisse, NSERC-Canada, and FCAR-Quebec