Convex Quadratic Optimization Problem Research Articles

Robust Adaptive Beamforming for Interference Suppression Based on SNR

Robust adaptive beamforming (RAB) can be used to suppress interference signals while retaining the desired signals received by a sensor array. However, desired signal self-cancellation and model mismatch can affect RAB performance. In this paper, a novel interference-plus-noise covariance matrix (INCM) reconstruction method is proposed for RAB to solve these problems. The proposed method divides the desired signal into two ranges according to the input signal-to-noise ratio (SNR), namely low SNR and high SNR. In the low SNR range, INCM reconstruction directly uses the same sample covariance matrix as the sample matrix inversion (SMI) beamformer to retain the advantages of the traditional SMI algorithm. In the high SNR range, the eigenvalues of the sample covariance matrix are used to estimate the interference power and noise power. The optimized interference steering vector (SV) is obtained by solving a quadratic convex optimization problem in an interference subspace. The INCM is reconstructed from the interference SVs, interference power, and noise power. The reconstructed INCM is then used to correct the desired signal SV via maximizing the beamformer output power. This is achieved by solving a quadratically constrained quadratic programming (QCQP) problem. Analysis and simulation results are presented which demonstrate that the proposed method performs well under a variety of mismatch conditions.

Solving general convex quadratic multi-objective optimization problems via a projection neurodynamic model

Solving general convex quadratic multi-objective optimization problems via a projection neurodynamic model

On the convex hull of convex quadratic optimization problems with indicators

We consider the convex quadratic optimization problem in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^{n}$$\\end{document} with indicator variables and arbitrary constraints on the indicators. We show that a convex hull description of the associated mixed-integer set in an extended space with a quadratic number of additional variables consists of an (n+1)×(n+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n+1) \ imes (n+1)$$\\end{document} positive semidefinite constraint (explicitly stated) and linear constraints. In particular, convexification of this class of problems reduces to describing a polyhedral set in an extended formulation. While the vertex representation of this polyhedral set is exponential and an explicit linear inequality description may not be readily available in general, we derive a compact mixed-integer linear formulation whose solutions coincide with the vertices of the polyhedral set. We also give descriptions in the original space of variables: we provide a description based on an infinite number of conic-quadratic inequalities, which are “finitely generated.” In particular, it is possible to characterize whether a given inequality is necessary to describe the convex hull. The new theory presented here unifies several previously established results, and paves the way toward utilizing polyhedral methods to analyze the convex hull of mixed-integer nonlinear sets.

RLSS: real-time, decentralized, cooperative, networkless multi-robot trajectory planning using linear spatial separations

Trajectory planning for multiple robots in shared environments is a challenging problem especially when there is limited communication available or no central entity. In this article, we present Real-time planning using Linear Spatial Separations, or RLSS: a real-time decentralized trajectory planning algorithm for cooperative multi-robot teams in static environments. The algorithm requires relatively few robot capabilities, namely sensing the positions of robots and obstacles without higher-order derivatives and the ability of distinguishing robots from obstacles. There is no communication requirement and the robots’ dynamic limits are taken into account. RLSS generates and solves convex quadratic optimization problems that are kinematically feasible and guarantees collision avoidance if the resulting problems are feasible. We demonstrate the algorithm’s performance in real-time in simulations and on physical robots. We compare RLSS to two state-of-the-art planners and show empirically that RLSS does avoid deadlocks and collisions in forest-like and maze-like environments, significantly improving prior work, which result in collisions and deadlocks in such environments.

Robust Mainlobe Interference Suppression Based on Joint Oblique Projection for Dual Polarization Conformal Array

In this paper, a robust mainlobe interference suppression algorithm is proposed in a space-polarization joint domain based on a dual-polarized conformal array. An oblique projection filter can eliminate the interference while retaining the desired signal information. However, little deviation in signal parameters will lead to significant performance degradation. To solve such problems, the joint steering vector of the desired signal is corrected by solving the quadratic convex optimization problem. Parameters of interferences are first estimated by using multiple signal classification (MUSIC) joint spectrum, and then, the interference covariance matrix can be reconstructed individually to further estimate the interference steering vector. After that, the joint oblique projection operator can be established by the estimated steering vectors. Consequently, an antijamming beamformer can be obtained by the joint oblique projection filter to achieve robustness on the interference parameter estimation deviation and desired signal mismatch. Simulation experiments verify the effectiveness of the proposed method.

Advancements in the computation of enclosures for multi-objective optimization problems

A central goal for multi-objective optimization problems is to compute their nondominated sets. In most cases these sets consist of infinitely many points and it is not a practical approach to compute them exactly. One solution to overcome this problem is to compute an enclosure, a special kind of coverage, of the nondominated set. For that computation one often makes use of so-called local upper bounds. In this paper we present a generalization of this concept. For the first time, this allows to apply a warm start strategy to the computation of an enclosure. We also show how this generalized concept allows to remove empty areas of an enclosure by deleting certain parts of the lower and upper bound sets which has not been possible in the past. We demonstrate how to apply our ideas to the box approximation algorithm, a general framework to compute an enclosure, as recently used in the solver called BAMOP. We show how that framework can be simplified and improved significantly, especially concerning its practical numerical use. In fact, we show for selected numerical instances that our new approach is up to eight times faster than the original one. Hence, our new framework is not only of theoretical but also of practical use, for instance for continuous convex or mixed-integer quadratic optimization problems.

Low-Complexity WSRMax Precoder Design Using the Dual Coordinate Ascent Method

We propose a low-complexity precoder design for weighted sum rate maximization (WSRMax) in multiuser multiple-input-multiple-output systems with per-antenna power constraints (PAPCs). The proposed design is based on the alternating optimization framework and a fast dual coordinate ascent method in the inner loop. Specifically, the original nonconvex WSRMax problem is first approximated as a sequence of convex quadratic optimization problems subject to PAPCs by adopting alternating optimization. Then, each convex quadratic problem is solved by a dual coordinate ascent method. The proposed dual coordinate ascent method includes no turning parameters, such as the step size in the projected gradient method, which offers both stability and fast convergence. Simulation results show that the dual coordinate ascent method converges faster than existing methods, which significantly improves the total computational efficiency of alternating optimization.

Mathbf {2times 2}-Convexifications for convex quadratic optimization with indicator variables

In this paper, we study the convex quadratic optimization problem with indicator variables. For the {2times 2} case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the {2times 2} case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including the optimal perspective relaxation and the optimal rank-one relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems.

AN INTERIOR-POINT ALGORITHM FOR SIMPLICIAL CONE CONSTRAINED CONVEX QUADRATIC OPTIMIZATION

In this paper, we are concerned with the numerical solution of simplicial cone constrained convex quadratic optimization (SCQO) problems. A reformulation of the K.K.T optimality conditions of SCQOs as an equivalent linear complementarity problem with $\mathcal{P}$-matrix ($\mathcal{P}$-LCP) is considered. Then, a feasible full-Newton step interior-point algorithm (IPA) is applied for solving SCQO via $\mathcal{P}$-LCP. For the completeness of the study, we prove that the proposed algorithm is well-defined and converges locally quadratic to an optimal of SCQOs. Moreover, we obtain the currently best well-known iteration bound for the algorithm with short-update method, namely,$ \mathcal{O}(\sqrt{n}\log\frac{n}{\epsilon })$. Finally, we present a various set of numerical results to show its efficiency.

On second-order conic programming duals for robust convex quadratic optimization problems

In this paper, we deal with second-order conic programming (SOCP) duals for a robust convex quadratic optimization problem with uncertain data in the constraints. We first introduce a SOCP dual problem for this robust convex quadratic optimization problem with polytopic uncertain sets. Then, we obtain a zero duality gap result between this robust convex quadratic optimization problem and its dual problem in terms of a new robust type characteristic cone constraint qualification. We also construct a SOCP dual problem for this robust convex quadratic optimization problem with norm-constrained uncertain sets and obtain the corresponding zero duality gap result between them. Moreover, some numerical examples are given to explain the obtained results.

An event-triggered iteratively reweighted convex optimization approach to multi-period portfolio selection

This paper addresses multi-period portfolio selection driven by events. We define an event-triggering function to mimic fund managers to activate sequential portfolio rebalancing and maximize Sharpe and Sortino ratios in the Markowitz’s return–risk framework. At first, the multi-period portfolio selection problem is formulated with a variable weight as a series of biconvex optimization problems with a surrogated objective function to maximize the Sharpe ratio or Sortino ratio. In each period, the portfolio optimization problem is further reformulated as an iteratively reweighted convex quadratic optimization problem. The multi-period portfolio selection problem is then solved sequentially based on a defined event-trigger function and a quadratic optimizer. The experiments are done on eight world stock markets datasets to show the capabilities of the proposed approach in calculating market indices and equal-weighted portfolios in terms of Sharpe and Sortino ratios.

Error and Energy Aware Sensor Deployment for RSS Localization—A Linear Quadratic Regulator Formulation

In network localization, sensor deployment positions relative to the points of interest (PoIs) and the fusion center (FC) greatly affect the overall localization error and energy performance. To account for both aspects, sensor deployment is formulated as an optimization problem in which the cost function combines both error and energy terms. Given a set of candidate positions, the problem is then to determine where sensors should be deployed such that the cost function is minimized. One of the challenges associated with this problem is NP-hardness, nonlinearity, and nonconvexity. Moreover, the localization error is submodular in relation to the number of deployed sensors. Thus, in addition to deployment positions, the order of deployment is an important factor to consider. To solve this problem, two suboptimal sequential low-complexity deployment algorithms are proposed. In the first algorithm, deployment is modeled as a constrained linear quadratic regulator (LQR) problem. Using this formulation, deployment is formulated as a constrained convex quadratic optimization problem that can be easily solved. To further reduce computational complexity, a second greedy algorithm is proposed. In this algorithm, a local cost function combining a sensor’s transmission energy and the localization errors at the PoIs is used to make the deployment decision. Simulation results show the effectiveness of the proposed algorithms in comparison to other methods.

A graph-based decomposition method for convex quadratic optimization with indicators

In this paper, we consider convex quadratic optimization problems with indicator variables when the matrix Q defining the quadratic term in the objective is sparse. We use a graphical representation of the support of Q, and show that if this graph is a path, then we can solve the associated problem in polynomial time. This enables us to construct a compact extended formulation for the closure of the convex hull of the epigraph of the mixed-integer convex problem. Furthermore, motivated by inference problems with graphical models, we propose a novel decomposition method for a class of general (sparse) strictly diagonally dominant Q, which leverages the efficient algorithm for the path case. Our computational experiments demonstrate the effectiveness of the proposed method compared to state-of-the-art mixed-integer optimization solvers.

Covariance Matrix Reconstruction via a Subspace Method and Spatial Spectral Estimation for Robust Adaptive Beamforming

Several robust adaptive beamforming (RAB) algorithms based on interference plus noise covariance matrix (INCM) reconstruction have been recently proposed, which can enhance the robustness of beamforming algorithms when certain mismatches occur in the model. However, some approaches ignore the resolution of the Capon spectral estimator (CSE), leading to reconstruction errors. This paper proposes a novel RAB algorithm formulated using the subspace projection method and spatial spectral estimation (SSE), which is named INCM-SSE. First, without using the CSE, the subspace projection matrix (SPM) is obtained through the integral of the angular sector where the signal of interest (SOI) is located. Subsequently, after estimating the direction of arrival (DOA) of incident signals using the multiple signal classification (MUSIC) algorithm, we project the sample covariance matrix (SCM) onto the SPM to eliminate the SOI influence. Then, the estimation method of interference powers is derived. Moreover, the INCM is reconstructed based on the estimated powers and steering vector (SV) of interferences. The SV of the SOI is optimized by solving a quadratic convex optimization problem. The INCM-SSE algorithm not only employs SSE to improve the angular resolution but also reduces the influence of the SOI component by using SPM. Simulation results indicate that the proposed method is robust against various types of mismatches, thus achieving superior overall performance.

Analysis of Information Search around the Time of Childbirth: Estimating Probability Distributions of Search Dates via Mathematical Optimization

Many mothers have considerable anxiety about pregnancy, childbirth, and childcare. For such mothers, searching for information on the Internet is an effective means of dissolving their anxieties. We consider the problem of estimating, for each search word, a distribution of search dates with respect to children’s birth dates. Most of the empirical distributions have unimodal or bimodal shapes, and some of them are asymmetric about extremal points and rise or fall sharply. We propose nonparametric estimation methods based on mathematical optimization models for such probability distributions. Our unimodal and bimodal optimization models automatically estimate the optimal extremal points and can be extended to multimodal distributions. These models are formulated as mixed-integer convex quadratic optimization problems, which can be solved exactly using optimization software. Experimental results using real-world and synthetic datasets demonstrate that our methods are effective by comparison to conventional moving average and kernel estimation methods.

An extended multi-model regression approach for compressive strength prediction and optimization of a concrete mixture

Due to the significant delay and cost associated with experimental tests, a model-based evaluation of concrete compressive strength is of high value, both for the purpose of strength prediction as well as the mixture optimization. In contrast to the prior recent studies employing a single regression model, in this paper, we present a combined multi-model framework where the regression methods based on artificial neural network, random forest regression and polynomial regression are jointly implemented for compressive strength prediction with a higher accuracy. The outcomes of the individual regression models are combined via a linear weighting strategy and optimized over the training data set as a quadratic convex optimization problem. It is worth mentioning that due to the convexity of the formulated problem, the globally optimum weighting strategy is obtained via standard numerical solvers. Afterwards, employing the obtained regression model, a multi-objective genetic algorithm-based method is proposed for mixture optimization under practical constraints, where a Pareto front of the cost-CS trade-off has been obtained employing the available data set. Numerical evaluations show that the proposed multi-model regression achieves a significantly higher prediction accuracy, i.e., approximately 18% reduction in the obtained prediction mean squared error, without weight optimization, and roughly 30% reduction in the obtained prediction mean squared error with an optimized combination following a convex quadratic optimization, compared to the best single model regression method employing a multi-layered artificial neural network.

Implementation of an embedded model predictive controller for a novel medical oxygen concentrator

Medical Oxygen Concentrators (MOCs) produce high purity oxygen from ambient air for medical therapies, and can range in size from large stationary units to small ultra-portable devices. These devices use a complex Rapid Pressure Swing Adsorption (RPSA) cyclic process which is subject to many process disturbances. Feedback control is required to operate a MOC reliably in different operating conditions and in presence of disturbances. Recently, a multivariable Model Predictive Controller (MPC) for a single-bed MOC device was developed in simulation, and we now present an implementation of this MPC for a lab-scale MOC device. The single-bed MOC uses a four-step RPSA cycle which can be controlled in real-time by adjusting the four cycle step durations to control the product oxygen concentration and product storage tank pressure. The MPC uses a linear model, identified using experimental sub-space system identification techniques, and an embedded convex quadratic optimization problem for making control decisions. This work presents the implementation of this MPC algorithm using Raspberry Pi hardware for the single-bed MOC device. The complete closed-loop system is evaluated using various set point tracking and disturbance case studies.

The equivalence of optimal perspective formulation and Shor's SDP for quadratic programs with indicator variables

In this paper, we compare the strength of the optimal perspective relaxation and Shor's SDP relaxation for convex quadratic optimization problems with indicator variables and prove that they are equivalent.

Model Predictive Control for Blending Processes in Cement Plants

In this paper, we discuss model predictive control applied to blending processes. Blending processes are ubiquitous in the chemical process industries since reactants usually need to be mixed before entering a reactor. Many times, the blending is trivial as pure streams of reactants are mixed. We consider non-trivial blending problems in which non-pure streams with the reactants are to be mixed. The motivating example is the blending problem that occurs in cement production. The raw mix for the cement kiln must have a specified chemical composition. This composition is obtained by mixing piles with different chemical compositions and economic value such that the raw mix meets specifications in the cheapest possible way. We formulate the blending problem as a nonlinear optimization problem that can be approximated well as a convex quadratic optimization problem. We implement the corresponding nonlinear and linear model predictive controllers (NMPC, LMPC) using a continuous-time transfer function description that is realized as a discrete-time linear state space model. The controller obtains feedback by combining regularly sampled online measurements and irregularly sampled laboratory measurements using a time variant dynamic Kalman filter with memory. Numerical simulations demonstrate that the NMPC and LMPC have similar performance.

An Efficient Linear Detection Scheme Based on L-BFGS Method for Massive MIMO Systems

For massive multiple-input multiple-output (MIMO) systems, minimum mean square error (MMSE) detection is near-optimal, but requires high-complexity matrix inversion. To avoid matrix inversion, we formulate MMSE detection as a strictly convex quadratic optimization problem, which can be solved iteratively by the recognized most efficient Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. According to special properties of massive MIMO systems, we propose a novel limited-memory BFGS (L-BFGS) scheme for MMSE detection with one correction search, unit step length, and simplified initialization, which can greatly reduce the storage and computation cost compared to BFGS method. Simulation results finally verify the effectiveness of the proposed scheme.