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In healthcare applications, there is a growing need to develop machine learning models that use data from a single source, such as that from a wrist wearable device, to monitor physical activities, assess health risks, and provide immediate health recommendations or interventions. However, the limitation of using single-source data often compromises the model's accuracy, as it fails to capture the full scope of human activities. While a more comprehensive dataset can be gathered in a lab setting using multiple sensors attached to various body parts, this approach is not practical for everyday use due to the impracticality of wearing multiple sensors. To address this challenge, we introduce a transfer learning framework that optimizes machine learning models for everyday applications by leveraging multi-source data collected in a laboratory setting. We introduce a novel metric to leverage the inherent relationship between these multiple data sources, as they are all paired to capture aspects of the same physical activity. Through numerical experiments, our framework outperforms existing methods in classification accuracy and robustness to noise, offering a promising avenue for the enhancement of daily activity monitoring.

We consider assortment and inventory planning problems with dynamic stockout-based substitution effects and no replenishment. We consider two settings: 1. Customers can see all available products when they arrive, which is commonly seen in physical stores. 2. The seller can choose to offer a subset of available products to each customer, which is typical on online platforms. Both settings are known to be computationally challenging, and the current approximation algorithms for the two settings are quite different. We develop a unified algorithm framework under the MNL choice model for both settings. Our algorithms improve on the state-of-the-art algorithms in terms of approximation guarantee, runtime, and the ability to manage uncertainty in the total number of customers and handle more complex constraints. In the process, we establish various novel properties of dynamic assortment planning (under the MNL choice) that may be useful more broadly.

Energy cost is increasingly crucial in the modern computing industry with the wide deployment of large-scale machine learning models and language models. For the firms that provide computing services, low energy consumption is important both from the perspective of their own market growth and the government's regulations. In this paper, we study the energy benefits of quantum computing vis-a-vis classical computing. Deviating from the conventional notion of quantum advantage based solely on computational complexity, we redefine advantage in an energy efficiency context. Through a Cournot competition model constrained by energy usage, we demonstrate quantum computing firms can outperform classical counterparts in both profitability and energy efficiency at Nash equilibrium. Therefore quantum computing may represent a more sustainable pathway for the computing industry. Moreover, we discover that the energy benefits of quantum computing economies are contingent on large-scale computation. Based on real physical parameters, we further illustrate the scale of operation necessary for realizing this energy efficiency advantage., Comment: 23 pages, many figures

In practice, incentive providers (i.e., principals) often cannot observe the reward realizations of incentivized agents, which is in contrast to many principal-agent models that have been previously studied. This information asymmetry challenges the principal to consistently estimate the agent's unknown rewards by solely watching the agent's decisions, which becomes even more challenging when the agent has to learn its own rewards. This complex setting is observed in various real-life scenarios ranging from renewable energy storage contracts to personalized healthcare incentives. Hence, it offers not only interesting theoretical questions but also wide practical relevance. This paper explores a repeated adverse selection game between a self-interested learning agent and a learning principal. The agent tackles a multi-armed bandit (MAB) problem to maximize their expected reward plus incentive. On top of the agent's learning, the principal trains a parallel algorithm and faces a trade-off between consistently estimating the agent's unknown rewards and maximizing their own utility by offering adaptive incentives to lead the agent. For a non-parametric model, we introduce an estimator whose only input is the history of principal's incentives and agent's choices. We unite this estimator with a proposed data-driven incentive policy within a MAB framework. Without restricting the type of the agent's algorithm, we prove finite-sample consistency of the estimator and a rigorous regret bound for the principal by considering the sequential externality imposed by the agent. Lastly, our theoretical results are reinforced by simulations justifying applicability of our framework to green energy aggregator contracts., Comment: 72 pages, 6 figures. arXiv admin note: text overlap with arXiv:2304.07407

This work is dedicated to the algorithm design in a competitive framework, with the primary goal of learning a stable equilibrium. We consider the dynamic price competition between two firms operating within an opaque marketplace, where each firm lacks information about its competitor. The demand follows the multinomial logit (MNL) choice model, which depends on the consumers' observed price and their reference price, and consecutive periods in the repeated games are connected by reference price updates. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy for the single-period game, to simultaneously capture the long-run market equilibrium and stability. We propose the online projected gradient ascent algorithm (OPGA), where the firms adjust prices using the first-order derivatives of their log-revenues that can be obtained from the market feedback mechanism. Despite the absence of typical properties required for the convergence of online games, such as strong monotonicity and variational stability, we demonstrate that under diminishing step-sizes, the price and reference price paths generated by OPGA converge to the unique SNE, thereby achieving the no-regret learning and a stable market. Moreover, with appropriate step-sizes, we prove that this convergence exhibits a rate of $\mathcal{O}(1/t)$.

We develop the first active learning method in the predict-then-optimize framework. Specifically, we develop a learning method that sequentially decides whether to request the "labels" of feature samples from an unlabeled data stream, where the labels correspond to the parameters of an optimization model for decision-making. Our active learning method is the first to be directly informed by the decision error induced by the predicted parameters, which is referred to as the Smart Predict-then-Optimize (SPO) loss. Motivated by the structure of the SPO loss, our algorithm adopts a margin-based criterion utilizing the concept of distance to degeneracy and minimizes a tractable surrogate of the SPO loss on the collected data. In particular, we develop an efficient active learning algorithm with both hard and soft rejection variants, each with theoretical excess risk (i.e., generalization) guarantees. We further derive bounds on the label complexity, which refers to the number of samples whose labels are acquired to achieve a desired small level of SPO risk. Under some natural low-noise conditions, we show that these bounds can be better than the naive supervised learning approach that labels all samples. Furthermore, when using the SPO+ loss function, a specialized surrogate of the SPO loss, we derive a significantly smaller label complexity under separability conditions. We also present numerical evidence showing the practical value of our proposed algorithms in the settings of personalized pricing and the shortest path problem.

Moment-based distributionally robust optimization (DRO) provides an optimization framework to integrate statistical information with traditional optimization approaches. Under this framework, one assumes that the underlying joint distribution of random parameters runs in a distributional ambiguity set constructed by moment information and makes decisions against the worst-case distribution within the set. Although most moment-based DRO problems can be reformulated as semidefinite programming (SDP) problems that can be solved in polynomial time, solving high-dimensional SDPs is still time-consuming. Unlike existing approximation approaches that first reduce the dimensionality of random parameters and then solve the approximated SDPs, we propose an optimized dimensionality reduction (ODR) approach. We first show that the ranks of the matrices in the SDP reformulations are small, by which we are then motivated to integrate the dimensionality reduction of random parameters with the subsequent optimization problems. Such integration enables two outer and one inner approximations of the original problem, all of which are low-dimensional SDPs that can be solved efficiently. More importantly, these approximations can theoretically achieve the optimal value of the original high-dimensional SDPs. As these approximations are nonconvex SDPs, we develop modified Alternating Direction Method of Multipliers (ADMM) algorithms to solve them efficiently. We demonstrate the effectiveness of our proposed ODR approach and algorithm in solving two practical problems. Numerical results show significant advantages of our approach on the computational time and solution quality over the three best possible benchmark approaches. Our approach can obtain an optimal or near-optimal (mostly within 0.1%) solution and reduce the computational time by up to three orders of magnitude.

Motivated by a number of real-world applications from domains like healthcare and sustainable transportation, in this paper we study a scenario of repeated principal-agent games within a multi-armed bandit (MAB) framework, where: the principal gives a different incentive for each bandit arm, the agent picks a bandit arm to maximize its own expected reward plus incentive, and the principal observes which arm is chosen and receives a reward (different than that of the agent) for the chosen arm. Designing policies for the principal is challenging because the principal cannot directly observe the reward that the agent receives for their chosen actions, and so the principal cannot directly learn the expected reward using existing estimation techniques. As a result, the problem of designing policies for this scenario, as well as similar ones, remains mostly unexplored. In this paper, we construct a policy that achieves a low regret (i.e., square-root regret up to a log factor) in this scenario for the case where the agent has perfect-knowledge about its own expected rewards for each bandit arm. We design our policy by first constructing an estimator for the agent's expected reward for each bandit arm. Since our estimator uses as data the sequence of incentives offered and subsequently chosen arms, the principal's estimation can be regarded as an analogy of online inverse optimization in MAB's. Next we construct a policy that we prove achieves a low regret by deriving finite-sample concentration bounds for our estimator. We conclude with numerical simulations demonstrating the applicability of our policy to real-life setting from collaborative transportation planning., Comment: 50 pages, 4 figures

This paper proposes a stochastic framework to evaluate the performance of public transit systems under short random service suspensions. We aim to derive closed-form formulations of the mean and variance of the queue length and waiting time. A bulk-service queue model is adopted to formulate the queuing behavior in the system. The random service suspension is modeled as a two-state (disruption and normal) Markov process. We prove that headway is distributed as the difference between two compound Poisson exponential random variables. The distribution is used to specify the mean and variance of queue length and waiting time at each station with analytical formulations. The closed-form stability condition of the system is also derived, implying that the system is more likely to be unstable with high incident rates and long incident duration. The proposed model is implemented on a bus network. Results show that higher incident rates and higher average incident duration will increase both the mean and variance of queue length and waiting time, which are consistent with the theoretical analysis. Crowding stations are more vulnerable to random service suspensions. The theoretical results are validated with a simulation model, showing consistency between the two outcomes.

This study proposes a mixed-integer programming formulation to model the individual-based path (IPR) recommendation problem during public transit service disruptions with the objective of minimizing system travel time and respecting passengers' path choice preferences. Passengers' behavior uncertainty in path choices given recommendations is also considered. We model the behavior uncertainty based on the passenger's prior preferences and posterior path choice probability distribution with two new concepts: epsilon-feasibility and Gamma-concentration, which control the mean and variance of path flows in the optimization problem. We show that these two concepts can be seen as a way of approximating the recourse function (expected system travel time) in a two-stage stochastic optimization. It is proved that these two concepts help to bound the difference between the approximated recourse function and the exact one. Additional theoretical analysis shows that epsilon-feasibility and Gamma-concentration can be seen as an approximation of expectation and chance constraints in a typical stochastic optimization formulation, respectively. The proposed IPR problem with behavior uncertainty is solved efficiently with Benders decomposition. The model is implemented in the Chicago Transit Authority (CTA) system with a real-world urban rail disruption as the case study. Results show that the proposed IPR model significantly reduces the average travel times compared to the status quo and outperforms the capacity-based benchmark path recommendation strategy.

Path planning in the multi-robot system refers to calculating a set of actions for each robot, which will move each robot to its goal without conflicting with other robots. Lately, the research topic has received significant attention for its extensive applications, such as airport ground, drone swarms, and automatic warehouses. Despite these available research results, most of the existing investigations are concerned with the cases of robots with a fixed movement speed without considering uncertainty. Therefore, in this work, we study the problem of path-planning in the multi-robot automatic warehouse context, which considers the time-varying and uncertain robots' movement speed. Specifically, the path-planning module searches a path with as few conflicts as possible for a single agent by calculating traffic cost based on customarily distributed conflict probability and combining it with the classic A* algorithm. However, this probability-based method cannot eliminate all conflicts, and speed's uncertainty will constantly cause new conflicts. As a supplement, we propose the other two modules. The conflict detection and re-planning module chooses objects requiring re-planning paths from the agents involved in different types of conflicts periodically by our designed rules. Also, at each step, the scheduling module fills up the agent's preserved queue and decides who has a higher priority when the same element is assigned to two agents simultaneously. Finally, we compare the proposed algorithm with other algorithms from academia and industry, and the results show that the proposed method is validated as the best performance.

Quantum computing is expected to have transformative influences on many domains, but its practical deployments on industry problems are underexplored. We focus on applying quantum computing to operations management problems in industry, and in particular, supply chain management. Many problems in supply chain management involve large state and action spaces and pose computational challenges on classic computers. We develop a quantized policy iteration algorithm to solve an inventory control problem and demonstrative its effectiveness. We also discuss in-depth the hardware requirements and potential challenges on implementing this quantum algorithm in the near term. Our simulations and experiments are powered by \texttt{IBM Qiskit} and the \texttt{qBraid} system., Comment: 6 pages, 5 figures

Inventory management problems with periodic and controllable resets occur in the context of managing water storage in the developing world and retailing limited-time availability products. In this paper, we consider a set of sequential decision problems in which the decision-maker must not only balance holding and shortage costs but discard all inventory before a fixed number of decision epochs, with the option for an early inventory reset. Finding optimal policies using dynamic programming for these problems is particularly challenging since the resulting value functions are non-convex. Moreover, this structure cannot be easily analyzed using existing extended definitions, such as $K$-convexity. Our key contribution is to present sufficient conditions that ensure the optimal policy has an easily interpretable structure that generalizes the well-known $(s, S)$ policy from the operations literature. Furthermore, we demonstrate that the optimal policy has a four-threshold structure under these rather mild conditions. We then conclude with computational experiments, thereby illustrating the policy structures that can be extracted in several inventory management scenarios.

We study Concave Constrained Markov Decision Processes (Concave CMDPs) where both the objective and constraints are defined as concave functions of the state-action occupancy measure. We propose the Variance-Reduced Primal-Dual Policy Gradient Algorithm (VR-PDPG), which updates the primal variable via policy gradient ascent and the dual variable via projected sub-gradient descent. Despite the challenges posed by the loss of additivity structure and the nonconcave nature of the problem, we establish the global convergence of VR-PDPG by exploiting a form of hidden concavity. In the exact setting, we prove an $O(T^{-1/3})$ convergence rate for both the average optimality gap and constraint violation, which further improves to $O(T^{-1/2})$ under strong concavity of the objective in the occupancy measure. In the sample-based setting, we demonstrate that VR-PDPG achieves an $\widetilde{O}(\epsilon^{-4})$ sample complexity for $\epsilon$-global optimality. Moreover, by incorporating a diminishing pessimistic term into the constraint, we show that VR-PDPG can attain a zero constraint violation without compromising the convergence rate of the optimality gap. Finally, we validate the effectiveness of our methods through numerical experiments.

Many real-world optimization problems involve uncertain parameters with probability distributions that can be estimated using contextual feature information. In contrast to the standard approach of first estimating the distribution of uncertain parameters and then optimizing the objective based on the estimation, we propose an integrated conditional estimation-optimization (ICEO) framework that estimates the underlying conditional distribution of the random parameter while considering the structure of the optimization problem. We directly model the relationship between the conditional distribution of the random parameter and the contextual features, and then estimate the probabilistic model with an objective that aligns with the downstream optimization problem. We show that our ICEO approach is asymptotically consistent under moderate regularity conditions and further provide finite performance guarantees in the form of generalization bounds. Computationally, performing estimation with the ICEO approach is a non-convex and often non-differentiable optimization problem. We propose a general methodology for approximating the potentially non-differentiable mapping from estimated conditional distribution to the optimal decision by a differentiable function, which greatly improves the performance of gradient-based algorithms applied to the non-convex problem. We also provide a polynomial optimization solution approach in the semi-algebraic case. Numerical experiments are also conducted to show the empirical success of our approach in different situations including with limited data samples and model mismatches.

The exploration/exploitation trade-off is an inherent challenge in data-driven adaptive control. Though this trade-off has been studied for multi-armed bandits (MAB's) and reinforcement learning for linear systems; it is less well-studied for learning-based control of nonlinear systems. A significant theoretical challenge in the nonlinear setting is that there is no explicit characterization of an optimal controller for a given set of cost and system parameters. We propose the use of a finite-horizon oracle controller with full knowledge of parameters as a reasonable surrogate to optimal controller. This allows us to develop policies in the context of learning-based MPC and MAB's and conduct a control-theoretic analysis using techniques from MPC- and optimization-theory to show these policies achieve low regret with respect to this finite-horizon oracle. Our simulations exhibit the low regret of our policy on a heating, ventilation, and air-conditioning model with partially-unknown cost function., Comment: 16 pages, 2 figures

We address a dynamic covering location problem of an Unmanned Aerial Vehicle Base Station (UAV-BS), where the location sequence of a single UAV-BS in a wireless communication network is determined to satisfy data demand arising from ground users. This problem is especially relevant in the context of smart grid and disaster relief. The vertical movement ability of the UAV-BS and non-convex covering functions in wireless communication restrict utilizing classical planar covering location approaches. Therefore, we develop new formulations to this emerging problem for a finite time horizon to maximize the total coverage. In particular, we develop a mixed-integer non-linear programming formulation which is non-convex in nature, and propose a Lagrangean Decomposition Algorithm (LDA) to solve this formulation. Due to high complexity of the problem, the LDA is still unable to find good local solutions to large-scale problems. Therefore, we develop a Continuum Approximation (CA) model and show that CA would be a promising approach in terms of both computational time and solution accuracy. Our numerical study also shows that the CA model can be a remedy to build efficient initial solutions for exact solution algorithms.

Connected and automated vehicle (CAV) technology is providing urban transportation managers tremendous opportunities for better operation of urban mobility systems. However, there are significant challenges in real-time implementation, as the computational time of the corresponding operations optimization model increases exponentially with increasing vehicle numbers. Following the companion paper (Chen et al., 2020) which proposes a novel automated traffic control scheme for isolated intersections, this study proposes a network-level real-time traffic control framework for CAVs on grid networks. The proposed framework integrates a rhythmic control (RC) method with an online routing algorithm to realize collisionfree control of all CAVs on a network and achieve superior performance in average vehicle delay, network traffic throughput, and computational scalability. Specifically, we construct a preset network rhythm that all CAVs can follow to move on the network and avoid collisions at all intersections. Based on the network rhythm, we then formulate online routing for the CAVs as a mixed integer linear program, which optimizes the entry times of CAVs at all entrances of the network and their time-space routings in real time. We provide a sufficient condition that the linear programming relaxation of the online routing model yields an optimal integer solution. Extensive numerical tests are conducted to show the performance of the proposed operations management framework under various scenarios. It is illustrated that the framework is capable of achieving negligible delays and increased network throughput. Furthermore, the computational time results are also promising. The CPU time for solving a collision-free control optimization problem with 2,000 vehicles is only 0.3 s on an ordinary personal computer.

We study an urban bike lane planning problem based on the fine-grained bike trajectory data, which is made available by smart city infrastructure such as bike-sharing systems. The key decision is where to build bike lanes in the existing road network. As bike-sharing systems become widespread in the metropolitan areas over the world, bike lanes are being planned and constructed by many municipal governments to promote cycling and protect cyclists. Traditional bike lane planning approaches often rely on surveys and heuristics. We develop a general and novel optimization framework to guide the bike lane planning from bike trajectories. We formalize the bike lane planning problem in view of the cyclists' utility functions and derive an integer optimization model to maximize the utility. To capture cyclists' route choices, we develop a bilevel program based on the Multinomial Logit model. We derive structural properties about the base model and prove that the Lagrangian dual of the bike lane planning model is polynomial-time solvable. Furthermore, we reformulate the route choice based planning model as a mixed integer linear program using a linear approximation scheme. We develop tractable formulations and efficient algorithms to solve the large-scale optimization problem. Via a real-world case study with a city government, we demonstrate the efficiency of the proposed algorithms and quantify the trade-off between the coverage of bike trips and continuity of bike lanes. We show how the network topology evolves according to the utility functions and highlight the importance of understanding cyclists' route choices. The proposed framework drives the data-driven urban planning scheme in smart city operations management.

In practical applications, it is common that wildfire simulators do not correctly predict the evolution of the fire scar. Usually, this is caused due to multiple factors including inaccuracy in the input data such as land cover classification, moisture, improperly represented local winds, cumulative errors in the fire growth simulation model, high level of discontinuity/heterogeneity within the landscape, among many others. Therefore in practice, it is necessary to adjust the propagation of the fire to obtain better results, either to support suppression activities or to improve the performance of the simulator considering new default parameters for future events, best representing the current fire spread growth phenomenon. In this article, we address this problem through a new methodology using Derivative-Free Optimization (DFO) algorithms for adjusting the Rate of Spread (ROS) factors in a fire simulation growth model called Cell2Fire. To achieve this, we solve an error minimization optimization problem that captures the difference between the simulated and observed fire, which involves the evaluation of the simulator output in each iteration as part of a DFO framework, allowing us to find the best possible factors for each fuel present on the landscape. Numerical results for different objective functions are shown and discussed, including a performance comparison of alternative DFO algorithms., Comment: 8 figures, 35 pages

Plug-in electric vehicles (PEVs) can contribute to energy and environmental challenges. Among different types of PEVs, plug-in hybrid electric taxis (PHETs) go in advance. In this study, we provide a spatial and temporal PHET charging demand forecasting method based on one-month global positioning system (GPS)-based taxi travel data in Beijing. Then, using the charging demand forecasting results, a mixed integer linear programming (MILP) model is formulated to plan PHET charging stations in the central area of Beijing. The model minimizes both investment and operation costs of all the PHET charging stations and takes into account the service radius of charging stations, charging demand satisfaction and rational occupation rates of chargers. At last, the test of the planning method is carried out numerically through simulations and the analysis is complemented according to the results.

Increasing penetration of distribution generation (DG) and electric vehicles (EVs) calls for an effective way to estimate the achievable capacity connected to the distribution systems, but the exogenous uncertainties of DG outputs and EV charging loads make it challengeable. This study provides a joint capacity evaluation method with a risk threshold setting function for photovoltaic (PV) generation and EV charging stations (EVCSs). The method is mathematically formulated as a distributionally robust joint chance constrained programming model. And the worst-case conditional value at risk (WC-CVaR) approximation and an iterative algorithm based on semidefinite program (SDP) are used to solve the model. Finally, the method test is carried out numerically on IEEE 33-bus radial distribution system.

Most cities in India do not have water distribution networks that provide water throughout the entire day. As a result, it is common for homes and apartment buildings to utilize water storage systems that are filled during a small window of time in the day when the water distribution network is active. However, these water storage systems do not have disinfection capabilities, and so long durations of storage (i.e., as few as four days) of the same water leads to substantial increases in the amount of bacteria and viruses in that water. This paper considers the stochastic control problem of deciding how much water to store each day in the system, as well as deciding when to completely empty the water system, in order to tradeoff: the financial costs of the water, the health costs implicit in long durations of storing the same water, the potential for a shortfall in the quantity of stored versus demanded water, and water wastage from emptying the system. To solve this problem, we develop a new Binary Dynamic Search (BiDS) algorithm that is able to use binary search in one dimension to compute the value function of stochastic optimal control problems with controlled resets to a single state and with constraints on the maximum time span in between resets of the system.

Sudden cardiac arrest is a significant public health concern. Successful treatment of cardiac arrest is extremely time sensitive, and use of an automated external defibrillator (AED) where possible significantly increases the probability of survival. Placement of AEDs in public locations can improve survival by enabling bystanders to treat victims of cardiac arrest prior to the arrival of emergency medical responders. However, since the exact locations of future cardiac arrests cannot be known a priori, AEDs must be placed strategically in public locations to ensure their accessibility in the event of an out-of-hospital cardiac arrest emergency. In this paper, we propose a data-driven optimization model for deploying AEDs in public spaces while accounting for uncertainty in future cardiac arrest locations. Our approach involves discretizing a continuous service area into a large set of scenarios, where the probability of cardiac arrest at each location is itself uncertain. We model uncertainty in the spatial risk of cardiac arrest using a polyhedral uncertainty set that we calibrate using historical cardiac arrest data. We propose a solution technique based on row-and-column generation that exploits the structure of the uncertainty set, allowing the algorithm to scale gracefully with the total number of scenarios. Using real cardiac arrest data from the City of Toronto, we conduct an extensive numerical study on AED deployment public locations. We find that hedging against cardiac arrest location uncertainty can produce AED deployments that outperform a intuitive sample average approximation by 9 to 15%, and cuts the performance gap with respect to an ex-post model by half. Our findings suggest that accounting for cardiac arrest location uncertainty can lead to improved accessibility of AEDs during cardiac arrest emergencies and the potential for improved survival outcomes., Comment: 55 pages

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.