Precoder Design for Physical Layer Multicasting

This paper studies the instantaneous rate maximization and the weighted sum delay minimization problems over a K-user multicast channel, where multiple antennas are available at the transmitter as well as at all the receivers. Motivated by the degree of freedom optimality and the simplicity offered by linear precoding schemes, we consider the design of linear precoders using the aforementioned two criteria. We first consider the scenario wherein the linear precoder can be any complex-valued matrix subject to rank and power constraints. We propose cyclic alternating ascent based precoder design algorithms and establish their convergence to respective stationary points. Simulation results reveal that our proposed algorithms considerably outperform known competing solutions. We then consider a scenario in which the linear precoder can be formed by selecting and concatenating precoders from a given finite codebook of precoding matrices, subject to rank and power constraints. We show that under this scenario, the instantaneous rate maximization problem is equivalent to a robust submodular maximization problem which is strongly NP hard. We propose a deterministic approximation algorithm and show that it yields a bicriteria approximation. For the weighted sum delay minimization problem we propose a simple deterministic greedy algorithm, which at each step entails approximately maximizing a submodular set function subject to multiple knapsack constraints, and establish its performance guarantee.Comment: 37 pages, 8 figures, submitted to IEEE Trans. Signal Pro

Vignettes on robust combinatorial optimization

Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2018.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (pages 137-142).In this thesis, we design and analyze algorithms for robust combinatorial optimization in various settings. First, we consider the problem of simultaneously maximizing multiple objectives, all monotone submodular, subject to a cardinality constraint. We focus on the case where the number of objectives is super-constant yet much smaller than the cardinality of the chosen set. We propose several algorithms (including one with the best achievable asymptotic guarantee for the problem). Experiments on synthetic data show that a heuristic based on our more practical and fast algorithm outperforms current practical algorithms in all cases considered. Next, we study the problem of robust maximization of a single monotone submodular function in scenarios where after choosing a feasible set of elements, some elements from the chosen set are adversarially removed. Under some restriction on the number of elements that can be removed, we give the first constant factor approximation algorithms as well as the best possible asymptotic approximation in certain cases. We also give a black box result for the much more general setting of deletion-robust maximization subject to an independence system. Lastly, we consider a robust appointment scheduling problem where the goal is to design simple appointment systems that try to achieve both high server utilization as well as short waiting times, under uncertainty in job processing times. When the order of jobs is fixed and one seeks to find optimal appointment duration for jobs, we give a simple heuristic that achieves the first constant factor (2) approximation. We also give closed form optimal solutions in various special cases that supersede previous work. For the setting where order of jobs is also flexible and under-utilization costs are homogeneous, it was previously shown that an EPTAS exists. We instead focus on simple and practical heuristics, and find a ratio based ordering which is 1.0604 approximate, improving on previous results for similarly practical heuristics.by Rajan Udwani.Ph. D