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

A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is minimized. The Golomb ruler problem has applications in information theory, astronomy and communications, and it can be seen as a challenge for combinatorial optimization algorithms. Although constructing high quality rulers is well-studied, proving optimality is a far more challenging task. In this paper, we provide a computational comparison of different optimization paradigms, each using a different model (linear integer, constraint programming and quadratic integer) to certify that a given Golomb ruler is optimal. We propose several enhancements to improve the computational performance of each method by exploring bound tightening, valid inequalities, cutting planes and branching strategies. We conclude that a certain quadratic integer programming model solved through a Benders decomposition and strengthened by two types of valid inequalities performs the best in terms of solution time for small-sized Golomb ruler problem instances. On the other hand, a constraint programming model improved by range reduction and a particular branching strategy could have more potential to solve larger size instances due to its promising parallelization features

A Framework for Globally Optimizing Mixed-Integer Signomial Programs

Mixed-integer signomial optimization problems have broad applicability in engineering. Extending the Global Mixed-Integer Quadratic Optimizer, GloMIQO (Misener, Floudas in J. Glob. Optim., 2012. doi:10.1007/s10898-012-9874-7), this manuscript documents a computational framework for deterministically addressing mixed-integer signomial optimization problems to ε-global optimality. This framework generalizes the GloMIQO strategies of (1) reformulating user input, (2) detecting special mathematical structure, and (3) globally optimizing the mixed-integer nonconvex program. Novel contributions of this paper include: flattening an expression tree towards term-based data structures; introducing additional nonconvex terms to interlink expressions; integrating a dynamic implementation of the reformulation-linearization technique into the branch-and-cut tree; designing term-based underestimators that specialize relaxation strategies according to variable bounds in the current tree node. Computational results are presented along with comparison of the computational framework to several state-of-the-art solvers. © 2013 Springer Science+Business Media New York