Aritra Konar

Feasible Point Pursuit and Successive Approximation of Non-Convex QCQPs

Quadratically constrained quadratic programs (QCQPs) have a wide range of applications in signal processing and wireless communications. Non-convex QCQPs are NP-hard in general. Existing approaches relax the non-convexity using semi-definite relaxation (SDR) or linearize the non-convex part and solve the resulting convex problem. However, these techniques are seldom successful in even obtaining a feasible solution when the QCQP matrices are indefinite. In this letter, a new feasible point pursuit successive convex approximation (FPP-SCA) algorithm is proposed for non-convex QCQPs. FPP-SCA linearizes the non-convex parts of the problem as conventional SCA does, but adds slack variables to sustain feasibility, and a penalty to ensure slacks are sparingly used. When FPP-SCA is successful in identifying a feasible point of the non-convex QCQP, convergence to a Karush-Kuhn-Tucker (KKT) point is thereafter ensured. Simulations show the effectiveness of our proposed algorithm in obtaining feasible and near-optimal solutions, significantly outperforming existing approaches.

Hidden Convexity in QCQP with Toeplitz-Hermitian Quadratics

Quadratically Constrained Quadratic Programming (QCQP) has a broad spectrum of applications in engineering. The general QCQP problem is NP-Hard. This article considers QCQP with Toeplitz-Hermitian quadratics, and shows that it possesses hidden convexity: it can always be solved in polynomial-time via Semidefinite Relaxation followed by spectral factorization. Furthermore, if the matrices are circulant, then the QCQP can be equivalently reformulated as a linear program, which can be solved very efficiently. An application to parametric power spectrum sensing from binary measurements is included to illustrate the results.

Parametric Frugal Sensing of Power Spectra for Moving Average Models

Wideband spectrum sensing is a fundamental component of cognitive radio and other applications. A novel frugal sensing scheme was recently proposed as a means of crowdsourcing the task of spectrum sensing. Using a network of scattered low-end sensors transmitting randomly filtered power measurement bits to a fusion center, a non-parametric approach to spectral estimation was adopted to estimate the ambient power spectrum. Here, a parametric spectral estimation approach is considered within the context of frugal sensing. Assuming a Moving-Average (MA) representation for the signal of interest, the problem of estimating admissible MA parameters, and thus the MA power spectrum, from single bit quantized data is formulated. This turns out being a non-convex quadratically constrained quadratic program (QCQP), which is NP-Hard in general. Approximate solutions can be obtained via semi-definite relaxation (SDR) followed by randomization; but this rarely produces a feasible solution for this particular kind of QCQP. A new Sequential Parametric Convex Approximation (SPCA) method is proposed for this purpose, which can be initialized from an infeasible starting point, and yet still produce a feasible point for the QCQP, when one exists, with high probability. Simulations not only reveal the superior performance of the parametric techniques over the globally optimum solutions obtained from the non-parametric formulation, but also the better performance of the SPCA algorithm over the SDR technique.

Fast Approximation Algorithms for a Class of Non-convex QCQP Problems Using First-Order Methods

A number of important problems in engineering can be formulated as non-convex quadratically constrained quadratic programming (QCQP). The general QCQP problem is NP-Hard. In this paper, we consider a class of non-convex QCQP problems that are expressible as the maximization of the point-wise minimum of homogeneous convex quadratics over a “simple” convex set. Existing approximation strategies for such problems are generally incapable of achieving favorable performance-complexity tradeoffs. They are either characterized by good performance but high complexity and lack of scalability, or low complexity but relatively inferior performance. This paper focuses on bridging this gap by developing high performance, low complexity successive non-smooth convex approximation algorithms for problems in this class. Exploiting the structure inherent in each subproblem, specialized first-order methods are used to efficiently compute solutions. Multicast beamforming is considered as an application example to showcase the effectiveness of the proposed algorithms, which achieve a very favorable performance-complexity tradeoff relative to the existing state of the art.

A fast approximation algorithm for single-group multicast beamforming with large antenna arrays

Multicast beamforming utilizes multiple transmit antennas and subscriber channel state information at the transmitter to direct power towards a group of subscribers while limiting interference caused to others. Unfortunately, the natural max-min-fair multicast beamforming formulation is a nonconvex Quadratically Constrained Quadratic Programming (QCQP) problem. This paper proposes a high performance, low complexity Successive Convex Approximation (SCA) algorithm for max-min-fair multicast beamforming with per antenna power constraints in a massive MIMO setting. The proposed approach is based on iterative approximation of the non-convex problem by a sequence of non-smooth, convex optimization problems, and using an inexact version of the Alternating Direction Method of Multipliers for efficiently computing solutions of each SCA subproblem via proximal operator evaluations. Simulations reveal that the algorithm achieves a very favorable performance-complexity tradeoff relative to the existing state-of-the-art.

Fast feasibility pursuit for non-convex QCQPS via first-order methods

Quadratically Constrained Quadratic Programming (QCQP) is NP-hard in its general non-convex form, but it frequently arises in engineering design and applications ranging from state estimation to beamforming and clustering. Several polynomial-time approximation algorithms exist for non-convex QCQP problems (QCQPs), but their success hinges upon the ability to find at least one feasible point - which is also hard for a general problem instance. In this paper, we present a framework for computing feasible points of general non-convex QCQPs using simple first-order methods. Our approach features low computational and memory requirements, which makes it well-suited for application on large-scale problems. Experiments indicate the empirical effectiveness of our approach, despite currently lacking theoretical guarantees.

Non-convex consensus ADMM for satellite precoder design

Owing to the rapidly increasing traffic demands on satellite connectivity, the current exclusive frequency allocation is becoming obsolete. Instead, aggressive frequency reuse and interference mitigation techniques are promising ideas that both industry and academia are investigating. This paper proposes an optimization precoding technique for dealing with the multibeam interference due to the aggressive frequency reuse. In contrast to general multiuser multiple input multiple output (MIMO) schemes, multibeam satellite precoding techniques call for frame-by-frame quadratically constrained quadratic optimization of a large number of variables. We focus on the multigroup multicast beamforming optimization problem, and we propose to adopt a consensus-based alternating direction method of multipliers (C-ADMM) approach, in order to mitigate complexity. The proposed C-ADMM approach is shown to exhibit comparable optimization performance at considerably lower complexity relative to the prior state-of-art for the formulation considered.

First-Order Methods for Fast Feasibility Pursuit of Non-convex QCQPs

Quadratically constrained quadratic programming (QCQP) is NP-hard in its general non-convex form, yet it frequently arises in various engineering applications. Several polynomial-time approximation algorithms exist for non-convex QCQP problems (QCQPs), but their success hinges upon the ability to find at least one feasible point—which is also hard for a general problem instance. In this paper, we present a heuristic framework for computing feasible points of general non-convex QCQPs using simple first-order methods. Our approach features low computational and memory requirements, which makes it well suited for application to large-scale problems. While a priori it may appear that these benefits come at the expense of technical sophistication, rendering our approach too simple to even merit consideration for a non-convex and NP-hard problem, we provide compelling empirical evidence to the contrary. Experiments on synthetic as well as real-world instances of non-convex QCQPs reveal the surprising effectiveness of first-order methods compared to more established and sophisticated alternatives.

Parametric Frugal sensing of autoregressive power spectra

Estimating the power spectrum of a wide-sense stationary stochastic process is a core component of several signal processing tasks. Distributed spectrum sensing problems naturally emerge in cases where measurements of different realizations of a stochastic process are collected at multiple spatial locations. This paper describes a distributed power spectrum sensing scheme for stochastic processes which are well represented by an autoregressive (AR) process. The sensing model comprises a network of scattered low-end sensors which transmit randomly filtered, one bit quantized power measurements to a fusion center. The problem of AR power spectrum estimation from such binary power measurements is cast as a non-convex optimization problem, and an alternating minimization algorithm is proposed to obtain a stationary point. Simulations showcase the effectiveness of this scheme when the AR parametrization is valid.

Parametric frugal sensing of Moving Average power spectra

Wideband spectrum sensing is one of the core components of cognitive radio. A novel frugal sensing scheme was recently proposed by Mehanna et al, aiming to crowdsource spectrum sensing operations to a network of sensors transmitting randomly filtered power measurement bits to a fusion center (FC). The ambient power spectrum is then estimated at the FC using a non-parametric approach. Here, it is assumed that the primary signal admits a Moving Average (MA) parametrization, and the frugal sensing problem is revisited from a parametric spectral estimation point of view. We show that the problem of estimating admissible MA parameters (and thus the MA power spectrum) from single bit quantized data can be formulated as a non-convex Quadratically Constrained Quadratic Program (QCQP). This is NP-Hard in general, but semidefinite-relaxation (SDR) can be employed to obtain approximate solutions. Simulations reveal the superior performance of the SDR technique over the globally optimal solution obtained from the non-parametric formulation, when the MA assumption is valid.