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

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page

Fractional Order Version of the HJB Equation

We consider an extension of the well-known Hamilton-Jacobi-Bellman (HJB) equation for fractional order dynamical systems in which a generalized performance index is considered for the related optimal control problem. Owing to the nonlocality of the fractional order operators, the classical HJB equation, in the usual form, does not hold true for fractional problems. Effectiveness of the proposed technique is illustrated through a numerical example.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN 1555-1423, CODEN: JCNDDM. Submitted 28-June-2018; Revised 15-Sept-2018; Accepted 28-Oct-201

Directly Coupled Observers for Quantum Harmonic Oscillators with Discounted Mean Square Cost Functionals and Penalized Back-action

This paper is concerned with quantum harmonic oscillators consisting of a quantum plant and a directly coupled coherent quantum observer. We employ discounted quadratic performance criteria in the form of exponentially weighted time averages of second-order moments of the system variables. A coherent quantum filtering (CQF) problem is formulated as the minimization of the discounted mean square of an estimation error, with which the dynamic variables of the observer approximate those of the plant. The cost functional also involves a quadratic penalty on the plant-observer coupling matrix in order to mitigate the back-action of the observer on the covariance dynamics of the plant. For the discounted mean square optimal CQF problem with penalized back-action, we establish first-order necessary conditions of optimality in the form of algebraic matrix equations. By using the Hamiltonian structure of the Heisenberg dynamics and related Lie-algebraic techniques, we represent this set of equations in a more explicit form in the case of equally dimensioned plant and observer.Comment: 11 pages, a brief version to be submitted to the IEEE 2016 Conference on Norbert Wiener in the 21st Century, 13-15 July, Melbourne, Australi

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universally optimal among point configurations in Euclidean spaces of dimensions $8$ and $24$, respectively. In other words, they minimize energy for every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians), which is a strong form of robustness not previously known for any configuration in more than one dimension. This theorem implies their recently shown optimality as sphere packings, and broadly generalizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct the optimal auxiliary functions used to attain these bounds, we prove a new interpolation theorem, which is of independent interest. It reconstructs a radial Schwartz function $f$ from the values and radial derivatives of $f$ and its Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers $n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove this theorem, we construct an interpolation basis using integral transforms of quasimodular forms, generalizing Viazovska's work on sphere packing and placing it in the context of a more conceptual theory.Comment: 95 pages, 6 figure