On the Sample Complexity and Optimization Landscape for Quadratic Feasibility Problems

We consider the problem of recovering a complex vector $\mathbf{x}\in \mathbb{C}^n$ from $m$ quadratic measurements $\{\langle A_i\mathbf{x}, \mathbf{x}\rangle\}_{i=1}^m$. This problem, known as quadratic feasibility, encompasses the well known phase retrieval problem and has applications in a wide range of important areas including power system state estimation and x-ray crystallography. In general, not only is the the quadratic feasibility problem NP-hard to solve, but it may in fact be unidentifiable. In this paper, we establish conditions under which this problem becomes {identifiable}, and further prove isometry properties in the case when the matrices $\{A_i\}_{i=1}^m$ are Hermitian matrices sampled from a complex Gaussian distribution. Moreover, we explore a nonconvex {optimization} formulation of this problem, and establish salient features of the associated optimization landscape that enables gradient algorithms with an arbitrary initialization to converge to a \emph{globally optimal} point with a high probability. Our results also reveal sample complexity requirements for successfully identifying a feasible solution in these contexts.Comment: 21 page

Non-convex Quadratically Constrained Quadratic Programming: Hidden Convexity, Scalable Approximation and Applications

University of Minnesota Ph.D. dissertation. September 2017. Major: Electrical Engineering. Advisor: Nicholas Sidiropoulos. 1 computer file (PDF); viii, 85 pages.Quadratically Constrained Quadratic Programming (QCQP) constitutes a class of computationally hard optimization problems that have a broad spectrum of applications in wireless communications, networking, signal processing, power systems, and other areas. The QCQP problem is known to be NP–hard in its general form; only in certain special cases can it be solved to global optimality in polynomial-time. Such cases are said to be convex in a hidden way, and the task of identifying them remains an active area of research. Meanwhile, relatively few methods are known to be effective for general QCQP problems. The prevailing approach of Semidefinite Relaxation (SDR) is computationally expensive, and often fails to work for general non-convex QCQP problems. Other methods based on Successive Convex Approximation (SCA) require initialization from a feasible point, which is NP-hard to compute in general. This dissertation focuses on both of the above mentioned aspects of non-convex QCQP. In the first part of this work, we consider the special case of QCQP with Toeplitz-Hermitian quadratic forms and establish that it possesses hidden convexity, which makes it possible to obtain globally optimal solutions in polynomial-time. The second part of this dissertation introduces a framework for efficiently computing feasible solutions of general quadratic feasibility problems. While an approximation framework known as Feasible Point Pursuit-Successive Convex Approximation (FPP-SCA) was recently proposed for this task, with considerable empirical success, it remains unsuitable for application on large-scale problems. This work is primarily focused on speeding and scaling up these approximation schemes to enable dealing with large-scale problems. For this purpose, we reformulate the feasibility criteria employed by FPP-SCA for minimizing constraint violations in the form of non-smooth, non-convex penalty functions. We demonstrate that by employing judicious approximation of the penalty functions, we obtain problem formulations which are well suited for the application of first-order methods (FOMs). The appeal of using FOMs lies in the fact that they are capable of efficiently exploiting various forms of problem structure while being computationally lightweight. This endows our approximation algorithms the ability to scale well with problem dimension. Specific applications in wireless communications and power grid system optimization considered to illustrate the efficacy of our FOM based approximation schemes. Our experimental results reveal the surprising effectiveness of FOMs for this class of hard optimization problems

Many Physical Design Problems are Sparse QCQPs

Physical design refers to mathematical optimization of a desired objective (e.g. strong light--matter interactions, or complete quantum state transfer) subject to the governing dynamical equations, such as Maxwell's or Schrodinger's differential equations. Computing an optimal design is challenging: generically, these problems are highly nonconvex and finding global optima is NP hard. Here we show that for linear-differential-equation dynamics (as in linear electromagnetism, elasticity, quantum mechanics, etc.), the physical-design optimization problem can be transformed to a sparse-matrix, quadratically constrained quadratic program (QCQP). Sparse QCQPs can be tackled with convex optimization techniques (such as semidefinite programming) that have thrived for identifying global bounds and high-performance designs in other areas of science and engineering, but seemed inapplicable to the design problems of wave physics. We apply our formulation to prototypical photonic design problems, showing the possibility to compute fundamental limits for large-area metasurfaces, as well as the identification of designs approaching global optimality. Looking forward, our approach highlights the promise of developing bespoke algorithms tailored to specific physical design problems.Comment: 9 pages, 4 figures, plus references and Supplementary Material

Spectrum sharing backhaul satellite-terrestrial systems via analog beamforming

© 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.Current satellite and terrestrial backhaul systems are deployed in disjoint frequency bands. This fact precludes an efficient use of the spectrum and limits the evolution of wireless backhauling networks. In this paper, we propose an interference mitigation technique in order to allow the spectrum coexistence between satellite and terrestrial backhaul links. This interference reliever is implemented at the terrestrial backhaul nodes, which are assumed to be equipped with multiple antennas. Due to the large bandwidth and huge number of antennas required in these systems, we consider pure analog beamforming. Precisely, we assume a phased array beamforming configuration so that the terrestrial backhaul node can only reduce the interference by changing the phases of each beamforming weight. Two cases are considered: the 18 and 28 GHz band where transmit and receive beamforming optimization problems shall be tackled, respectively. In both cases, the optimization problem results in a nonconvex problem that we propose to solve via two alternative convex approximation methods. These two approaches are evaluated and they present less than 1 dB array gain loss with respect to the upper bound solution. Finally, the spectral efficiency gains of the proposed spectrum sharing scenarios are validated in numerical simulations.Peer ReviewedPostprint (published version