Convex Optimization Formulation Research Articles

A Highly Scalable Method for Extractive Text Summarization Using Convex Optimization

The paper describes a convex optimization formulation of the extractive text summarization problem and a simple and scalable algorithm to solve it. The optimization program is constructed as a convex relaxation of an intuitive but computationally hard integer programming problem. The objective function is highly symmetric, being invariant under unitary transformations of the text representations. Another key idea is to replace the constraint on the number of sentences in the summary with a convex surrogate. For solving the program we have designed a specific projected gradient descent algorithm and analyzed its performance in terms of execution time and quality of the approximation. Using the datasets DUC 2005 and Cornell Newsroom Summarization Dataset, we have shown empirically that the algorithm can provide competitive results for single document summarization and multi-document query-based summarization. On the Cornell Newsroom Summarization Dataset, it ranked second among the unsupervised methods tested. For the more challenging task of multi-document query-based summarization, the method was tested on the DUC 2005 Dataset. Our algorithm surpassed the other reported methods with respect to the ROUGE-SU4 metric, and it was at less than 0.01 from the top performing algorithms with respect to ROUGE-1 and ROUGE-2 metrics.

Towards Discovering What Patterns Trigger What Labels

In many real applications, especially those involving data objects with complicated semantics, it is generally desirable to discover the relation between patterns in the input space and labels corresponding to different semantics in the output space. This task becomes feasible with MIML (Multi-Instance Multi-Label learning), a recently developed learning framework, where each data object is represented by multiple instances and is allowed to be associated with multiple labels simultaneously. In this paper, we propose KISAR, an MIML algorithm that is able to discover what instances trigger what labels. By considering the fact that highly relevant labels usually share some patterns, we develop a convex optimization formulation and provide an alternating optimization solution. Experiments show that KISAR is able to discover reasonable relations between input patterns and output labels, and achieves performances that are highly competitive with many state-of-the-art MIML algorithms.

A Convex Formulation for Learning from Crowds

Recently crowdsourcing services are often used to collect a large amount of labeled data for machine learning, since they provide us an easy way to get labels at very low cost and in a short period. The use of crowdsourcing has introduced a new challenge in machine learning, that is, coping with the variable quality of crowd-generated data. Although there have been many recent attempts to address the quality problem of multiple workers, only a few of the existing methods consider the problem of learning classifiers directly from such noisy data. All these methods modeled the true labels as latent variables, which resulted in non-convex optimization problems. In this paper, we propose a convex optimization formulation for learning from crowds without estimating the true labels by introducing personal models of the individual crowd workers. We also devise an efficient iterative method for solving the convex optimization problems by exploiting conditional independence structures in multiple classifiers. We evaluate the proposed method against three competing methods on synthetic data sets and a real crowdsourced data set and demonstrate that the proposed method outperforms the other three methods.

Targeted pandemic containment through identifying local contact network bottlenecks.

Decision-making about pandemic mitigation often relies upon simulation modelling. Models of disease transmission through networks of contacts-between individuals or between population centres-are increasingly used for these purposes. Real-world contact networks are rich in structural features that influence infection transmission, such as tightly-knit local communities that are weakly connected to one another. In this paper, we propose a new flow-based edge-betweenness centrality method for detecting bottleneck edges that connect nodes in contact networks. In particular, we utilize convex optimization formulations based on the idea of diffusion with p-norm network flow. Using simulation models of COVID-19 transmission through real network data at both individual and county levels, we demonstrate that targeting bottleneck edges identified by the proposed method reduces the number of infected cases by up to 10% more than state-of-the-art edge-betweenness methods. Furthermore, the proposed method is orders of magnitude faster than existing methods.

Extension of ADMMAlgorithmin Solving Optimal Control Model Governed by Partial Differential Equation

This paper presents an Algorithm for the numerical solution of the Optimal Control model constrained by Partial Differential Equation using the Alternating Direction Method of Multipliers (ADMM) accelerated with a parameter factor in the sense of Nesterov. The ADMM tool wasapplied to a partial differential equation-governed optimization problem of the one-dimensional heat equation type. The constraint and objective functions of the optimal control model were discretized using the Crank-Nicolson and Composite Simpson’s Methods respectively into a derived discrete convex optimization form amenable to the ADMM. The primal-dual residuals were derived to ascertain the rate of convergence of themodel for increasing iterates. An existing example was used to test the efficiency and degree of accuracy of the algorithm and the results were favorable when compared the existing method.

Adaptive powered descent guidance based on multi-phase pseudospectral convex optimization

The real-time and fuel-optimal powered descent guidance is one of the critical technologies for planetary landing. This work focuses on the powered descent guidance algorithm based on pseudospectral convex optimization and aims to increase the accuracy and adaptability of the algorithm. The entire powered descent process is divided into phases according to the thrust switching times, and the pseudospectral method is adopted to discretize the optimization problem. Convexification process is performed to transform the optimization problem into a convex programming problem, and a successive solving process algorithm is proposed. For the established multi-phase pseudospectral convex optimization formulation, the indirect method is adopted to derive and determine the optimal phase division from switching function, and the adaptive multi-phase pseudospectral convex optimization method is proposed. Numerical examples demonstrate that the proposed method has excellent accuracy and adaptability.

Artifact rejection and missing data imputation in cerebral blood flow velocity signals via trace norm minimization

Objective: Many physiological signals are degraded by significant corruptions that limit their usefulness. One example is cerebral blood flow velocity (CBFV) signals, measured by transcranial Doppler, which are susceptible to large errors from patient motion. In this paper, we propose a method to remove artifacts and impute sections of missing data in these signals. Approach: The method exploits the low-order dynamical relationship between CBFV, arterial blood pressure and, where available, intracranial pressure. It enhances the measured signals by fitting them to a low-order dynamical model, using convex regularization terms that improve robustness to large deviations and missing data. The method is based on a convex optimization formulation and utilizes recent work in trace norm approximation and subspace system identification. Main results: Simulations demonstrate that the method successfully removes real CBFV artifacts and can impute missing data with reasonable accuracy. Performance was improved when intracranial pressure data was available. Conclusion: The methods presented can be used by researchers to remove artifacts and estimate missing sections in CBFV signals. The general approach may be applied to other biomedical signal processing settings. Significance: This low-order dynamical approach has ongoing applications in noninvasive intracranial pressure estimation.

SynCity: Using open data to create a synthetic city of hourly building energy estimates by integrating data-driven and physics-based methods

Cities officials are increasingly interested in understanding spatial and temporal energy patterns of the built environment to facilitate their city’s transition to a low-carbon future. In this paper, a new Augmented-Urban Building Energy Model (A-UBEM) is proposed that combines data-driven and physics-based simulation methods to produce synthetic hourly load curve estimates for every building within a city—similar to data an hourly smart meter would measure. By using only publicly available data, a generalizable two-step process is implemented—that other cities with similar available data can replicate—using New York City as a case study. Step (1) estimates the annual energy use for every building in the city using supervised machine learning algorithms. Step (2) extends these results and leverages physics-based simulation models through a convex optimization formulation that minimizes the squared difference between the aggregated building demand and the observed city-wide hourly electricity demand. Results from step (1) show that the Random Forest algorithm performs best with a mean log squared error of 0.293, while the convex optimization in step (2) results in a mean training error of 6.11% mean absolute percentage error (MAPE). To validate the stability of the produced load curves, Monte Carlo simulations are conducted, using random subsets of buildings from the city, which produce an out-of-sample error averaging 6.41% MAPE across each simulation. Particle swarm optimization is also explored—using the results from the Monte Carlo simulation—to assess if the model could be improved by relaxing certain constraints, but marginal error reductions are found, further proving the stability of the proposed model. Overall, A-UBEM is a first step towards creating highly granular urban-scale synthetic hourly load curves solely using open data. Such load curves are integral for planning sustainable cities and accelerating the adoption of low-carbon distributed energy resources (DERs) and district energy systems.

How to prestress compliant mechanisms for a targeted stiffness adjustment

The paper deals with compliant mechanisms with a variable stiffness behavior. Devices with such a behavior are getting more and more attention in recent years, especially in contact applications such as human-robotic-interactions or haptic interfaces. In the work presented here the variable stiffness behavior is achieved by the targeted application of prestressing forces. An optimization based method is presented to identify suitable force application points. Furthermore, the amount and direction of force is determined in order to cause a specific change in stiffness. The method consists of a convex optimization formulation and is based on modal parameters. The method is intended to work for any given design of a compliant mechanism. In this publication the method is tested and evaluated using two compliant mechanisms—the cartwheel hinge and a shape-adaptive airfoil profile.

Enhanced Fairness and Scalability of Power Control Schemes in Multi-Cell Massive MIMO

This paper studies the transmit power optimization in multi-cell massive multiple-input multiple-output (MIMO) systems. Network-wide max-min fairness (NW-MMF) and network-wide proportional fairness (NW-PF) are two well-known power control schemes in the literature. The NW-MMF focus on maximizing the fairness among users at the cost of penalizing users with good channel conditions. On the other hand, the NW-PF focuses on maximizing the sum SE, thereby ignoring fairness, but gives some extra attention to the weakest users. However, both of these schemes suffer from a scalability issue which means that for large networks, it is highly probable that one user has a very poor channel condition, pushing the spectral efficiency (SE) of all users towards zero. To overcome the scalability issue of NW-MMF and NW-PF, we propose a novel power control scheme that is provably scalable. This scheme maximizes the geometric mean (GM) of the per-cell max-min SE. To solve this new optimization problem, we prove that it can be rewritten in a convex optimization form and then solved using standard tools. The simulation results highlight the benefits of our model which is balancing between NW-PF and NW-MMF.

Fractional Dynamic Caching: A Collaborative Design of Storage and Backhaul

To reduce the average file delivery time under limited backhaul bandwidth and cache storage, we propose a fractional dynamic caching scheme, that coordinates the utilization of the backhaul and the cache storage. In the proposed scheme, only part of each file is pre-fetched and stored in the static storage segment of small base stations (SBSs), while remaining part of the file will be fetched into the dynamic storage segment when requested by users. We aim to minimize the average file delivery time by investigating the optimal prefetching policy, i.e., the size of the pre-fetched part of each file and the size of the dynamic storage. We formulate the file delivery time minimization problem as a convex optimization formulation. We compute the closed-form of the optimal pre-fetching policy when the wireless rate is constant. Then, a heuristic pre-fetching policy is derived, using the conditional average wireless rate. To overcome the issue that file popularity may be unknown or time-varying, we further provide an approximation of the heuristic pre-fetching policy which is irrelevant to the file popularity. We also discuss on the potential application of the proposed caching scheme in scenarios with heterogeneous file sizes. Numerical results show that the proposed scheme can substantially improve the backhaul utilization efficiency, and reduce the average file delivery time. The heuristic pre-fetching policy has similar performance with the optimal pre-fetching policy yet with much lower complexity.

Optimal Commissions and Subscriptions in Networked Markets

Problem definition: We consider a platform that charges commission rates and subscription fees to sellers and buyers for facilitating transactions but does not directly control the transaction prices, which are endogenously determined. Buyers and sellers are divided into types, and we represent the compatibility between different types using a bipartite network. Traders are heterogeneous in terms of their valuations, and different types have possibly different value distributions. Buyers may have additional value for trading with some seller types. The platform chooses commissions/subscriptions to maximize its revenues. Academic/practical relevance: Two salient features of most online platforms are that they do not dictate the transaction prices, and they use commissions/subscriptions for extracting revenues. We shed light on how these commissions/subscriptions should be set in networked markets. Methodology: Using tools from convex optimization and combinatorial optimization, we obtain tractable methods for computing the optimal commissions/subscriptions and provide insights into the platform’s revenues, buyer/seller surplus, and welfare. Results: We provide a tractable convex optimization formulation to obtain the revenue-maximizing commissions/subscriptions, and establish that, typically, different types should be charged different commissions/subscriptions depending on their network positions. We establish that the latter result holds even when the traders on each side have identical value distributions, and in this setting we provide lower and upper bounds on the platform’s revenues in terms of the supply-demand imbalance across the network. Motivated by simpler schemes used in practice, we show that the revenue loss can be unbounded when all traders on the same side are charged the same commissions/subscriptions, and bound the revenue loss in terms of the supply-demand imbalance across the network. Charging only buyers or only sellers leads to at least half of the optimal revenues, when different types on the same side can be charged differently. Managerial implications: Our results highlight the suboptimality of commonly used payment schemes and showcase the importance of accounting for the compatibility between different user types.

Secure Information Transmissions in Wireless-Powered Cognitive Radio Networks for Internet of Medical Things

In this paper, we consider the issue of the secure transmissions for the cognitive radio-based Internet of Medical Things (IoMT) with wireless energy harvesting. In these systems, a primary transmitter (PT) will transmit its sensitive medical information to a primary receiver (PR) by a multi-antenna-based secondary transmitter (ST), where we consider that a potential eavesdropper may listen to the PT’s sensitive information. Meanwhile, the ST also transmits its own information concurrently by utilizing spectrum sharing. We aim to propose a novel scheme for jointly designing the optimal parameters, i.e., energy harvesting (EH) time ratio and secure beamforming vectors, for maximizing the primary secrecy transmission rate while guaranteeing secondary transmission requirement. For solving the nonconvex optimization problem, we transfer the problem into convex optimization form by adopting the semidefinite relaxation (SDR) method and Charnes–Cooper transformation technique. Then, the optimal secure beamforming vectors and energy harvesting duration can be obtained easily by utilizing the CVX tools. According to the simulation results of secrecy transmission rate, i.e., secrecy capacity, we can observe that the proposed protocol for the considered system model can effectively promote the primary secrecy transmission rate when compared with traditional zero-forcing (ZF) scheme, while ensuring the transmission rate of the secondary system.

A Convex Solution of the $H_\infty$-Optimal Controller Synthesis Problem for Multidelay Systems

Optimal controller synthesis is a bilinear problem and hence difficult to solve in a computationally efficient manner. We are able to resolve this bilinearity for systems with delay by first convexifying the problem in infinite dimensions, i.e., formulating the $H_\infty$-optimal state-feedback controller synthesis problem for distributed-parameter systems as a linear operator inequality, which a form of convex optimization with operator variables. Next, we use positive matrices to parameterize positive “complete quadratic” operators, allowing the controller synthesis problem to be solved using semidefinite programming (SDP). We then use the solution to this SDP to calculate the feedback gains and provide effective methods for real-time implementation. Finally, we use several test cases to verify that the resulting controllers are optimal to several decimal places as measured by the minimal achievable closed-loop $H_\infty$ norm, and as compared against controllers designed using high-order Padé approximations.

Joint Learning of Measurement Matrix and Signal Reconstruction via Deep Learning

In this work, we propose an automatic sensing and reconstruction scheme based on deep learning within the compressive sensing (CS) framework. Classical CS utilizes pre-determined linear projections in the form of random measurements and convex optimization with a known sparsity basis to reconstruct signals. Here, we develop a data-driven approach to learn both the measurement matrix and the inverse reconstruction scheme for a given class of signals, such as images. The developed deep learning approach paves the way for end-to-end learning and reconstruction of signals with the aid of cascaded fully connected and multistage convolutional layers with a weighted loss function in an adversarial learning framework. Results obtained over the CIFAR-10 image database show that the proposed deep learning architectures provide higher peak signal-to-noise ratio (PSNR) levels, and, hence, learn better measurement matrices than that of randomly selected, specifically designed to reduce average coherence with a given basis, or state-of-the-art data driven approaches. The learned measurement matrices achieve higher PSNR compared to random or designed matrices not only when they are utilized in the proposed data-driven approach but also when used in $\ell _1$ based recovery. The reconstruction performance on the test dataset improves as more training samples are utilized. Quantitative results for sparsity level analysis, incremental measurement design, and various training scenarios are provided.

Distributionally Robust Markovian Traffic Equilibrium

In a Markovian traffic equilibrium model, users move toward their destinations by a sequence of successive link choices using a discrete choice model at each node, taking congestion into account. Although a convex optimization formulation is available to compute the equilibrium flows for a continuous distribution of link utilities, practical applications have thus far been mainly restricted to the multinomial logit model and its variants. In this paper, we relax the assumption of a complete joint distribution of link utilities to only knowledge on the marginal distributions and propose a new convex optimization formulation for a distributionally robust Markovian traffic equilibrium. The formulation is provably efficiently solvable and has the flexibility of allowing for general marginal distributions, thus capturing different types of nonidentical, skewed, and heavy-tailed distributions at the link level.

Monitoring Long Term Voltage Instability Due to Distribution and Transmission Interaction Using Unbalanced $\mu$ PMU and PMU Measurements

This paper extends the idea of the Thevenin equivalent into unbalanced 3{\phi} circuits and proposes a 3{\phi} long-term voltage stability indicator (VSI), that can identify critical loads in a system. Furthermore, in order to identify whether the voltage stability limit is due to the transmission network or a distribution network, a transmission-distribution distinguishing index (TDDI) is proposed. The novelty in the proposed indices is that they can account for the unbalance in the lines and loads, enabling them to use unbalanced phasor measurements naturally. This is supported by mathematical derivations and numerical results. A convex optimization formulation to estimate the 3{\phi} Thevenin equivalent using PMU & {\mu}PMU measurements is proposed, making it possible to calculate VSI and TDDI in an online model-free manner. Numerical simulations performed using co-simulation between Pypower and GridlabD are presented for the IEEE 9 bus and the 30 bus transmission networks combined with several modified IEEE 13 node and 37 node distribution networks. These case studies validate the proposed 3{\phi}-VSI and TDDI over a wide range of scenarios and demonstrate the importance of {\mu}PMU measurements in identifying the regions causing long term voltage instability.

Convex Formulations for Fair Principal Component Analysis

Though there is a growing literature on fairness for supervised learning, incorporating fairness into unsupervised learning has been less well-studied. This paper studies fairness in the context of principal component analysis (PCA). We first define fairness for dimensionality reduction, and our definition can be interpreted as saying a reduction is fair if information about a protected class (e.g., race or gender) cannot be inferred from the dimensionality-reduced data points. Next, we develop convex optimization formulations that can improve the fairness (with respect to our definition) of PCA and kernel PCA. These formulations are semidefinite programs, and we demonstrate their effectiveness using several datasets. We conclude by showing how our approach can be used to perform a fair (with respect to age) clustering of health data that may be used to set health insurance rates.

Robust Optimal Design of Energy Efficient Series Elastic Actuators: Application to a Powered Prosthetic Ankle.

Design of rehabilitation and physical assistance robots that work safely and efficiently despite uncertain operational conditions remains an important challenge. Current methods for the design of energy efficient series elastic actuators use an optimization formulation that typically assumes known operational requirements. This approach could lead to actuators that cannot satisfy elongation, speed, or torque requirements when the operation deviates from nominal conditions. Addressing this gap, we propose a convex optimization formulation to design the stiffness of series elastic actuators to minimize energy consumption and satisfy actuator constraints despite uncertainty due to manufacturing of the spring, unmodeled dynamics, efficiency of the transmission, and the kinematics and kinetics of the load. To achieve convexity, we write energy consumption as a scalar convex-quadratic function of compliance. As actuator constraints, we consider peak motor torque, peak motor velocity, limitations due to the speed-torque relationship of DC motors, and peak elongation of the spring. We apply our formulation to the robust design of a series elastic actuator for a powered prosthetic ankle. Our simulation results indicate that a small trade-off between energy efficiency and robustness is justified to design actuators that can operate with uncertainty.

A Novel Fast Semidefinite Programming-Based Approach for Optimal Reactive Power Dispatch

Reactive power planning problem is the key to secure and economic operation of power systems. Optimal management of existing reactive sources leads to loss minimization and economic performance of the system. Because of the nonlinear inter-relation between the physical parameters of the electric grid, this problem is a highly nonlinear and nonconvex constrained optimization problem. The application of semidefinite programming (SDP) to power system problems has recently gained considerable research attention. A recent SDP formulation uses a convex relaxation to the nonconvex optimal power flow problem under some technical conditions. This paper proposes a novel equivalent convex optimization formulation for the optimal reactive power dispatch (ORPD) problem and presents a new framework for finding the global optimum. Numerical results for the IEEE 30-bus and 118-bus test systems show that the proposed scheme obtains the optimum operation point and outperforms various state-of-the-art methods significantly.