arXiv subjects

Amelia Regan

Publications and source records attributed to Amelia Regan.

16 recordsLinked to original sources

arXiv2024

A New Class of Compact Formulations for Vehicle Routing Problems

This paper introduces a novel compact mixed integer linear programming (MILP) formulation and a discretization discovery-based solution approach for the Vehicle Routing Problem with Time Windows (VRPTW). We aim to solve the optimization problem efficiently by constraining the linear programming (LP) solutions to use only flows corresponding to time and capacity-feasible routes that are locally elementary (prohibiting cycles of customers localized in space). We employ a discretization discovery algorithm to refine the LP relaxation iteratively. This iterative process alternates between two steps: (1) increasing time/capacity/elementarity enforcement to increase the LP objective, albeit at the expense of increased complexity (more variables and constraints), and (2) decreasing enforcement without decreasing the LP objective to reduce complexity. This iterative approach ensures we produce an LP relaxation that closely approximates the optimal MILP objective with minimal complexity, facilitating an efficient solution via an off-the-shelf MILP solver. The effectiveness of our method is demonstrated through empirical evaluations on classical VRPTW instances. We showcase the efficiency of solving the final MILP and multiple iterations of LP relaxations, highlighting the decreased integrality gap of the final LP relaxation. We believe that our approach holds promise for addressing a wide range of routing problems within and beyond the VRPTW domain.

math.OC↗

arXiv2023

Graph Master and Local Area Routes for Efficient Column Generation for the Capacitated Vehicle Routing Problem with Time Windows

In this research we consider the problem of accelerating the convergence of column generation (CG) for the weighted set cover formulation of the capacitated vehicle routing problem with time windows (CVRPTW). We adapt two new techniques, Local Area (LA) routes and Graph Master (GM) to these problems. LA-routes rely on pre-computing all lowest cost elementary sub-routes, called LA-arcs, where all customers but the final customer are localized in space. LA-routes are constructed by concatenating LA-arcs where the final customer in a given LA-arc is the first customer in the subsequent LA-arc. To construct the lowest reduced cost elementary route during the pricing step of CG we apply a Decremental State Space Relaxation/time window discretization method over time, remaining demand, and customers visited; where the edges in the associated pricing graph are LA-arcs. To accelerate the convergence of CG we use an enhanced GM approach. We map each route generated during pricing to a strict total ordering of all customers, that respects the ordering of customers in the route; and somewhat preserves spatial locality. Each such strict total ordering is then mapped to a multi-graph where each node is associated with a tuple of customer, capacity remaining, and time remaining. Nodes are connected by feasible LA-arcs when the first/last customers in the LA-arc are less/greater than each intermediate customer (and each other) with respect to the total order. The multi-graph of a given route can express that route and other "related" routes; and every path from source to sink describes a feasible elementary route. Solving optimization over the restricted master problem over all multi-graphs is done efficiently by constructing the relevant nodes/edges on demand.

math.OC↗

arXiv2023

Local Area Routes and Valid Inequalities for Efficient Vehicle Routing

In this research we introduce Local Area (LA) routes for improving the efficiency and tightness of column generation (CG) methods for solving vehicle routing problems (VRP). LA-routes rely on pre-computing the lowest cost elementary sub-route (called an LA-arc) for each tuple consisting of the following: (1) a (first) customer where the LA-arc begins, (2) a distant customer (from the first) where the LA-arc ends, and (3) a set of intermediate customers near the first customer. LA-routes are constructed by concatenating LA-arcs where the final customer in a given LA-arc is the first customer in the subsequent LA-arc. A Decremental State Space Relaxation method is applied over LA-routes to construct the lowest reduced cost elementary route during the pricing step of CG. LA-route based solvers can be used to efficiently tighten the standard set cover VRP using a variant of subset row inequalities, which do not alter the structure of pricing. We incorporate LA-arcs into a novel CG stabilization scheme. Specifically each column generated during pricing is mapped to an ordered list of customers consistent with that column. An LA-arc is consistent with an ordering if the first/last customer in the arc come before/after all other customers in the LA-arc in the associated ordering respectively. Each such ordering is then mapped to a multi-graph where nodes correspond to (customer/demand) and edges correspond to LA-arcs consistent with that ordering. Hence any path from source to sink on the multi-graph is a feasible elementary route. The ordering for a column places customers spatially nearby in nearby positions on the ordering so that routes can be generated so as to permit spatially nearby customers to be visited without traveling far away first. We solve the restricted master problem over these graphs, which has special structure allowing for fast solution.

math.OC↗

arXiv2022

Local Area Routes for Vehicle Routing Problems

We consider an approach for improving the efficiency of column generation (CG) methods for solving vehicle routing problems. We introduce Local Area (LA) route relaxations, an alternative/complement to the commonly used ng-route relaxations and Decremental State Space Relaxations (DSSR) inside of CG formulations. LA routes are a subset of ng-routes and a super-set of elementary routes. Normally, the pricing stage of CG must produce elementary routes, which are routes without repeated customers, using processes which can be computationally expensive. Non-elementary routes visit at least one customer more than once, creating a cycle. LA routes relax the constraint of being an elementary route in such a manner as to permit efficient pricing. LA routes are best understood in terms of ng-route relaxations. Ng-routes are routes which are permitted to have non-localized cycles in space; this means that at least one intermediate customer (called a breaker) in the cycle must consider the starting customer in the cycle to be spatially far away. LA routes are described using a set of special indexes corresponding to customers on the route ordered from the start to the end of the route. LA route relaxations further restrict the set of permitted cycles beyond that of ng-routes by additionally enforcing that the breaker must be a located at a special index where the set of special indexes is defined recursively as follows. The first special index in the route is at index 1 meaning that it is associated with the first customer in the route. The k'th special index corresponds to the first customer after the k-1'th special index, that is not considered to be a neighbor of (considered spatially far from) the customer located at the k-1'th special index. We demonstrate that LA route relaxations can significantly improve the computational speed of pricing when compared to the standard DSSR.

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arXiv2022

Principled Graph Management

Graph Generation is a recently introduced enhanced Column Generation algorithm for solving expanded Linear Programming relaxations of mixed integer linear programs without weakening the expanded relaxations which characterize these methods. To apply Graph Generation we must be able to map any given column generated during pricing to a small directed acyclic graph for which any path from source to sink describes a feasible column. This structure is easily satisfied for vehicle routing, crew scheduling and various logistics problems where pricing is a constrained shortest path problem. The construction of such graphs trades off the size/diversity of a subset of columns modeled by the graphs versus the additional computational time required to solve the problems induced by larger graphs. Graph Generation (GG) has two computational bottlenecks. The first is pricing. Pricing in GG and Column Generation (CG) is identical because of the structure of the problems solved. The second bottleneck is the restricted master problem (RMP), which is more computationally intensive in GG than in CG given the same number of columns generated. By design GG converges in fewer iterations than CG, and hence requires fewer calls to pricing. Therefore, when the computation time of GG is dominated by pricing, as opposed to solving the RMP, GG converges much faster than CG in terms of time. However GG need not converge faster than CG when the GG RMP, rather than pricing, dominates computation. In this paper we introduce Principled Graph Management (PGM), which is an algorithm to solve the GG RMP rapidly by exploiting its special structure. We demonstrate the effectiveness of PGM inside a GG solution to the classical Capacitated Vehicle Routing Problem. We demonstrate that PGM solves the GG RMP hundreds of times faster than the baseline solver and that the improvement in speed increases with problem size.

math.OC↗

arXiv2021

Graph Generation: A New Approach to Solving Expanded Linear Programming Relaxations

In this article we introduce Graph Generation, an enhanced Column Generation (CG) algorithm for solving expanded linear programming relaxations of mixed integer linear programs. To apply Graph Generation, we must be able to map any given column to a small directed acyclic graph for which any path from source to sink describes a feasible column. This structure is easily satisfied for vehicle routing and crew scheduling problems; and other such problems where pricing is a resource constrained shortest path problem. Such graphs are then added to the restricted master problem (RMP) when the corresponding column is generated during pricing. The use of Graph Generation does not weaken the linear programming relaxation being solved. At any given iteration of CG enhanced by Graph Generation; the technique permits the RMP to express a much wider set of columns than those generated during pricing, leading to faster convergence of CG. Graph Generation does not change the structure of the CG pricing problem. We show how the method can be applied in a general way, and then demonstrate the effectiveness of our approach on the classical Capacitated Vehicle Routing Problem.

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arXiv2021

Detour Dual Optimal Inequalities for Column Generation with Application to Routing and Location

We consider the problem of accelerating column generation (CG) for logistics optimization problems using vehicle routing as an example. Without loss of generality, we focus on the Capacitated Vehicle Routing Problem (CVRP) via the addition of a new class of dual optimal inequalities (DOI) that incorporate information about detours from the vehicle routes. These inequalities extend the Smooth-DOI recently introduced in the literature for the solution of certain classes of set-covering problems by CG. The Detour-DOI introduced in this article permit low cost swap operations between items on a given active route with items near to other items on that route to estimate (and bound) the values of the dual variables. Smooth-DOI in contrast only permit low cost swap operations between nearby items. The use of Detour-DOI permits a faster convergence of CG without weakening the linear programming relaxation. We then argue that these DOI can also be conveniently applied to single source capacitated facility location problems. These problems have been shown to be equivalent to a broad class of logistics optimization problems that include, for example telecommunication network design and production planning. The importance of developing vastly more efficient column generation solvers cannot be overstated. Detour-DOI, which permit large numbers of columns to be expressed with a finite set of variables, contributes to this important endeavor.

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arXiv2021

Clustering of Time Series Data with Prior Geographical Information

Time Series data are broadly studied in various domains of transportation systems. Traffic data area challenging example of spatio-temporal data, as it is multi-variate time series with high correlations in spatial and temporal neighborhoods. Spatio-temporal clustering of traffic flow data find similar patterns in both spatial and temporal domain, where it provides better capability for analyzing a transportation network, and improving related machine learning models, such as traffic flow prediction and anomaly detection. In this paper, we propose a spatio-temporal clustering model, where it clusters time series data based on spatial and temporal contexts. We propose a variation of a Deep Embedded Clustering(DEC) model for finding spatio-temporal clusters. The proposed model Spatial-DEC (S-DEC) use prior geographical information in building latent feature representations. We also define evaluation metrics for spatio-temporal clusters. Not only do the obtained clusters have better temporal similarity when evaluated using DTW distance, but also the clusters better represents spatial connectivity and dis-connectivity. We use traffic flow data obtained by PeMS in our analysis. The results show that the proposed Spatial-DEC can find more desired spatio-temporal clusters.

cs.LG↗

arXiv2021

Family Column Generation: A Principled Stabilized Column Generation Approach

We tackle the problem of accelerating column generation (CG) approaches to set cover formulations in operations research. At each iteration of CG we generate a dual solution that approximately solves the LP over all columns consisting of a subset of columns in the nascent set. We refer to this linear program (LP) as the Family Restricted Master Problem (FRMP), which provides a tighter bound on the master problem at each iteration of CG, while preserving efficient inference. For example, in the single source capacitated facility location problem (SSCFLP) the family of a column $l$ associated with facility $f$ and customer set $N_l$ contains the set of columns associated with $f$ and the customer set that lies in the power set of $N_l$. The solution to FRMP optimization is attacked with a coordinate ascent method in the dual. The generation of direction of travel corresponds to solving the restricted master problem over columns corresponding to the reduced lowest cost column in each family given specific dual variables based on the incumbent dual, and is easily generated without resolving complex pricing problems. We apply our algorithm to the SSCFLP and demonstrate improved performance over two relevant baselines.

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arXiv2021

Multi-Robot Routing with Time Windows: A Column Generation Approach

Robots performing tasks in warehouses provide the first example of wide-spread adoption of autonomous vehicles in transportation and logistics. The efficiency of these operations, which can vary widely in practice, are a key factor in the success of supply chains. In this work we consider the problem of coordinating a fleet of robots performing picking operations in a warehouse so as to maximize the net profit achieved within a time period while respecting problem- and robot-specific constraints. We formulate the problem as a weighted set packing problem where the elements in consideration are items on the warehouse floor that can be picked up and delivered within specified time windows. We enforce the constraint that robots must not collide, that each item is picked up and delivered by at most one robot, and that the number of robots active at any time does not exceed the total number available. Since the set of routes is exponential in the size of the input, we attack optimization of the resulting integer linear program using column generation, where pricing amounts to solving an elementary resource-constrained shortest-path problem. We propose an efficient optimization scheme that avoids consideration of every increment within the time windows. We also propose a heuristic pricing algorithm that can efficiently solve the pricing subproblem. While this itself is an important problem, the insights gained from solving these problems effectively can lead to new advances in other time-widow constrained vehicle routing problems.

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arXiv2021

B-ETS: A Trusted Blockchain-based Emissions Trading System for Vehicle-to-Vehicle Networks

Urban areas are negatively impacted by Carbon Dioxide (CO2 ) and Nitrogen Oxide (NOx) emissions. In order to achieve a cost-effective reduction of greenhouse gas emissions and to combat climate change, the European Union (EU) introduced an Emissions Trading System (ETS) where organizations can buy or receive emission allowances as needed. The current ETS is a centralized one, consisting of a set of complex rules. It is currently administered at the organizational level and is used for fixed-point sources of pollution such as factories, power plants, and refineries. However, the current ETS cannot efficiently cope with vehicle mobility, even though vehicles are one of the primary sources of CO2 and NOx emissions. In this study, we propose a new distributed Blockchain-based emissions allowance trading system called B-ETS. This system enables transparent and trustworthy data exchange as well as trading of allowances among vehicles, relying on vehicle-to-vehicle communication. In addition, we introduce an economic incentive-based mechanism that appeals to individual drivers and leads them to modify their driving behavior in order to reduce emissions. The efficiency of the proposed system is studied through extensive simulations, showing how increased vehicle connectivity can lead to a reduction of the emissions generated from those vehicles. We demonstrate that our method can be used for full life-cycle monitoring and fuel economy reporting. This leads us to conjecture that the proposed system could lead to important behavioral changes among the drivers

cs.MA↗

arXiv2019

Introduction of a Hybrid Monitor to Cyber-Physical Systems

Computing and technology have vastly changed since the introduction of firewall technology in 1988. The internet has grown from a simple network of networks to a cyber and physical entity that encompasses the entire planet. Cyber-physical systems(CPS) now control most of the day to day operations of human civilization from autonomous cars to nuclear energy plants. While phenomenal, this growth spurt has created new security threats. These are threats that cannot be blocked by a firewall for they are not only cyber but cyber-physical. In light of these new cyber-physical threats, this text proposes a security measure that promises to enhance the security of cyber-physical systems. Using theoretical cyber, physical, and cyber-physical attack scenarios, this text will highlight the need for some sort of monitor in cyber-physical systems as an extra security measure. Additionally, this text will illustrate the efficiency of said monitor using a Shannon entropy proof, and a multiple security domain nondeducibility(MSDND) proof.

cs.CR↗

arXiv2019

A convolution recurrent autoencoder for spatio-temporal missing data imputation

When sensors collect spatio-temporal data in a large geographical area, the existence of missing data cannot be escaped. Missing data negatively impacts the performance of data analysis and machine learning algorithms. In this paper, we study deep autoencoders for missing data imputation in spatio-temporal problems. We propose a convolution bidirectional-LSTM for capturing spatial and temporal patterns. Moreover, we analyze an autoencoder's latent feature representation in spatio-temporal data and illustrate its performance for missing data imputation. Traffic flow data are used for evaluation of our models. The result shows that the proposed convolution recurrent neural network outperforms state-of-the-art methods.

cs.LG↗

arXiv2019

A Spatial-Temporal Decomposition Based Deep Neural Network for Time Series Forecasting

Spatial time series forecasting problems arise in a broad range of applications, such as environmental and transportation problems. These problems are challenging because of the existence of specific spatial, short-term and long-term patterns, and the curse of dimensionality. In this paper, we propose a deep neural network framework for large-scale spatial time series forecasting problems. We explicitly designed the neural network architecture for capturing various types of patterns. In preprocessing, a time series decomposition method is applied to separately feed short-term, long-term and spatial patterns into different components of a neural network. A fuzzy clustering method finds cluster of neighboring time series based on similarity of time series residuals; as they can be meaningful short-term patterns for spatial time series. In neural network architecture, each kernel of a multi-kernel convolution layer is applied to a cluster of time series to extract short-term features in neighboring areas. The output of convolution layer is concatenated by trends and followed by convolution-LSTM layer to capture long-term patterns in larger regional areas. To make a robust prediction when faced with missing data, an unsupervised pretrained denoising autoencoder reconstructs the output of the model in a fine-tuning step. The experimental results illustrate the model outperforms baseline and state of the art models in a traffic flow prediction dataset.

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arXiv2013

Online Classification Using a Voted RDA Method

We propose a voted dual averaging method for online classification problems with explicit regularization. This method employs the update rule of the regularized dual averaging (RDA) method, but only on the subsequence of training examples where a classification error is made. We derive a bound on the number of mistakes made by this method on the training set, as well as its generalization error rate. We also introduce the concept of relative strength of regularization, and show how it affects the mistake bound and generalization performance. We experimented with the method using $\ell_1$ regularization on a large-scale natural language processing task, and obtained state-of-the-art classification performance with fairly sparse models.

cs.LG↗

arXiv2013

Thompson Sampling in Dynamic Systems for Contextual Bandit Problems

We consider the multiarm bandit problems in the timevarying dynamic system for rich structural features. For the nonlinear dynamic model, we propose the approximate inference for the posterior distributions based on Laplace Approximation. For the context bandit problems, Thompson Sampling is adopted based on the underlying posterior distributions of the parameters. More specifically, we introduce the discount decays on the previous samples impact and analyze the different decay rates with the underlying sample dynamics. Consequently, the exploration and exploitation is adaptively tradeoff according to the dynamics in the system.

cs.LG↗