Linear Programming Solution Research Articles

Multi-Horizon Model Predictive Control for Energy Management in Zero-Emission High-Speed Passenger Vessels

Abstract High-speed passenger vessels (HSPVs) play an important role in coastal regions due to their high energy consumption and emissions. This study develops an optimal energy management system (EMS) for a zero-emission hybrid power system, powered by a fuel cell (FC) and a battery energy storage system (BESS). The global optimal solution for loading the FC and operating the BESS is computed using a linear programming (LP) approach, assuming a priori knowledge of the complete operational profile. Since the global solution is not realizable in an onboard EMS, the main objective of this paper is to develop an algorithm for real-time application. A multi-horizon model predictive control (MH-MPC) algorithm is applied to a model of the power system to solve the optimization problem with a prediction mechanism over the entire route. This approach is benchmarked against the global LP solution to evaluate the effectiveness of the proposed EMS. The results highlight the potential of the MH-MPC approach for achieving a near-optimal EMS in a zero-emission HSPV.

Combining Precision Boosting with LP Iterative Refinement for Exact Linear Optimization

This article studies a combination of the two state-of-the-art algorithms for the exact solution of linear programs (LPs) over the rational numbers in practice, that is, without any roundoff errors or numerical tolerances. By integrating the method of precision boosting inside an LP iterative refinement loop, the combined algorithm is able to leverage the strengths of both methods: the speed of LP iterative refinement, in particular, in the majority of cases when a double-precision floating-point solver is able to compute approximate solutions with small errors, and the robustness of precision boosting whenever extended levels of precision become necessary. We compare the practical performance of the resulting algorithm with both pure methods on a large set of LPs and mixed-integer programs (MIPs). The results show that the combined algorithm solves more instances than a pure LP iterative refinement approach while being faster than pure precision boosting. When embedded in an exact branch-and-cut framework for MIPs, the combined algorithm is able to reduce the number of failed calls to the exact LP solver to zero while maintaining the speed of the pure LP iterative refinement approach. History: Accepted by Antonio Frangioni, Area Editor for Design and Analysis of Algorithms: Continuous. Funding: The work for this article has been conducted within the Research Campus Modal funded by the German Federal Ministry of Education and Research (BMBF) [Grants 05M14ZAM and 05M20ZBM].

A Vehicle-to-Grid planning framework incorporating electric vehicle user equilibrium and distribution network flexibility enhancement

The rapid surge in electric vehicle (EV) adoption, coupled with advancements in charging technologies, emphasizes the critical necessity for expanding EV recharging infrastructure. Simultaneously, the Distribution Network (DN) encounters escalating challenges in meeting charging demand during peak traffic periods. Consequently, there is a mounting demand for the deployment of innovative Vehicle-to-Grid (V2G) technologies to augment the DN’s flexibility in power dispatch and alleviate travel costs for EV users. Hence, this paper proposes an EV-user-equilibrium-(UE)-constrained V2G planning framework that enhances flexibility in the DN. The framework aims to ascertain the optimal placement and capacity of EV charging stations (EVCSs) and V2G charging piles within the Transportation Network (TN). It takes into account the equilibrium condition stemming from competitive EV charging and routing behaviors alongside the optimal expansion of DN energy resources to accommodate the electricity supplied by the V2G piles. This study commences by analyzing EV drivers’ travel decisions, considering the influence of charging and V2G pile locations and sizes. Subsequently, we tackle the Traffic Assignment Problem with User Equilibrium (TAP-UE) model to characterize the steady-state traffic flow distribution of EVs. Following this, we formulate the optimization model for the Coordinated Power and Transportation Network (CPTN), which encompasses the optimal expansion of DN facilities and traffic flow regulation under UE conditions. To mitigate the computational complexity associated with the V2G planning model, we introduce a series of linearization methods to obtain a manageable Mixed-Integer Linear Programming (MILP) solution. Finally, to validate the efficacy of our proposed planning framework, we apply it to two test systems, including a real-world case study. Through these case studies, we explore the necessity and potential benefits of V2G technologies.

A topology and sizing optimisation method for lightweight sandwich structures subject to dynamic and static constraints

A static-dynamic topology-sizing optimisation method is presented. The solution is based on a sequential Mixed-Integer Linear Programming solution and aims to minimise the mass of a structure subjected to concurrent constraints on static and dynamic response. It is shown that the classical problem of the dynamics of lightweight sandwich structures may be mitigated through core topology and face sheet thickness combinations, retaining the static load carrying capacity while presenting stringent dynamic properties at a low mass penalty.A numerical example, in the form of a load carrying sandwich beam which is excited at different frequencies, is used to demonstrate the method.

Trade-offs in biomethane production by substrate mixture optimization under German market conditions

BackgroundNew regulations and market conditions in Germany affect the profitability of biomethane upgrading as a repowering option for existing biogas plants following on-site CHP utilization. These conditions present trade-off challenges between higher sustainability requirements, maintaining production capacity and new revenue opportunities. Optimization methods, such as linear programming (LP), are essential for determining the ideal substrate mixture and profitable solutions amidst multiple market conditions, plant-specific process constraints, and substrate properties.MethodsWe updated a substrate mixture optimization model within an assessment framework for the repowering of existing biogas plants (BGPs), which focuses on the operator’s perspective. By integrating multiple German biomethane markets for various BGPs, we assessed changes in the substrate mixture, GHG emissions, contribution margins, and constraint parameters to derive conclusions for operators and future framework design.ResultsIntegrating market revenues and constraints can increase contribution margins by 12–55%. Additional gains can be achieved by considering multiple markets simultaneously but limited to a few BGPs. The plant-specific LP solution space and used benchmark market are decisive. The former limits the potential of high substrate-specific contribution margins, which has a significantly higher impact than the relation between plant-specific characteristics and process constraints. The advanced fuel market is currently the lead market for biomethane, incentivizing GHG-emission extensive substrates, decreasing gas production and GHG emissions but increasing levelized cost of energy (LCOE) and partially CO2 abatement costs.ConclusionsThe key to improve profitability and to supply an increasing biomethane demand while fulfilling new requirements is a large LP solution space. Increasing market options, substrate availability, and digestion system capacity achieve this on the operator’s side. Policy makers could reduce normative requirements such as the maize cap or double counting of advanced fuels and favor high but uniform GHG requirements. Operators can prepare robustly for the future substrate mixture by adding digester volume and pre-treatment tech, ensuring long-term and diverse substrate availability, and contracts with flexible components. Although current market conditions can improve specific GHG emissions, they do not necessarily increase manure usage when other options, such as straw, are viable. Other regulatory support systems will be required to do so.

Maximum Consensus Floating Point Solutions for Infeasible Low-Dimensional Linear Programs with Convex Hull as the Intermediate Representation

Maximum Consensus Floating Point Solutions for Infeasible Low-Dimensional Linear Programs with Convex Hull as the Intermediate Representation

Long-term upgrade strategies in multiband and multifiber optical transport networks

Flexible-grid optical transport networks (OTNs) are currently undergoing a massive scale upgrade to bandwidth variable transceivers (BVTs) to cater to the exponential Internet traffic growth. Network operators seeking to plan future capacity in a scalable manner can also add additional bands of operations or additional fibers in the OTN. We propose two strategies for planning multiband upgrades in OTNs, which use an integer linear programming (ILP) solution and a heuristic solution with a reinforcement learning (RL)-based extension. Network simulation studies on three differently sized OTNs show that the heuristic and its RL-based extension perform extremely close to the optimal ILP-based solution while reducing execution times drastically. Finally, a techno-economic study based on what we believe to be a novel cost model highlights the cost-per-bit-per-second savings through carefully planned upgrades.

Pollutant transport model with large time-scale and estimation of land-use export coefficients at a watershed level.

The pollutant transport equilibrium in a watershed can be analyzed on a large time scale, and land-use export coefficients can be calculated directly under certain hydrologic and transport conditions, by ignoring hydrologic and transport processes at small space and time scales on hydrologic response units. In this study, the water environment system of a watershed was deconstructed into three parts (source, source-sink, and runoff transport) to construct a pollutant transportation equilibrium model on a large time scale. A watershed with an annual source-sink accumulation of zero was defined as a completely transported watershed; therefore, we derived a completely transported equilibrium equation. The problem of seeking the land export coefficient was converted into a problem of seeking the optimal solution of linear programming, which can be estimated according to the variation in pollutant output processes. The feasibility of the solution can be analyzed using multi-year stochastic rainfall processes. The model was used to analyze the transport equilibrium of chemical oxygen demand (COD), total nitrogen (TN), and total phosphorus (TP) upstream of the monitored cross-sections in a watershed, which covered 3145.66 km2. The land export coefficients were calculated according to the model. The model calculations indicated that the watershed was completely transported during perennial years. The calculated export coefficients of COD, TN, and TP for farmland, primary vegetation, and urban land were within the range of general empirical values. The calculated maximum accumulations of COD, TN, and TP were 0.19 × 107, 0.063 × 107, and 0.049 × 106 kg, respectively, for perennial rainfall. PRACTITIONER POINTS: A completely transported watershed was defined, and a model of pollutant transportation equilibrium with large time-scale was constructed. A problem of seeking the optimal solution of a linear programming was designed to estimate the land export coefficient of COD, TN, and TP. The runoff transport and accumulation processes of COD, TN, and TP in a watershed was analyzed.

The Floor Is Lava: Halving Natural Genomes with Viaducts, Piers, and Pontoons.

Whole Genome Duplications (WGDs) are events that double the content and structure of a genome. In some organisms, multiple WGD events have been observed while loss of genetic material is a typical occurrence following a WGD event. The requirement of classic rearrangement models that every genetic marker has to occur exactly two times in a given problem instance, therefore, poses a serious restriction in this context. The Double-Cut and Join (DCJ) model is a simple and powerful model for the analysis of large structural rearrangements. After being extended to the DCJ-Indel model, capable of handling gains and losses of genetic material, research has shifted in recent years toward enabling it to handle natural genomes, for which no assumption about the distribution of markers has to be made. The traditional theoretical framework for studying WGD events is the Genome Halving Problem (GHP). While the GHP is solved for the DCJ model for genomes without losses, there are currently no exact algorithms utilizing the DCJ-Indel model that are able to handle natural genomes. In this work, we present a general view on the DCJ-Indel model that we apply to derive an exact polynomial time and space solution for the GHP on genomes with at most two genes per family before generalizing the problem to an integer linear program solution for natural genomes.

Improved strategies of the Equality Set Projection (ESP) algorithm for computing polytope projection

Abstract. ​​​​​​​This paper proposes an optimization method for the Equality Set Projection algorithm to compute the orthogonal projection of polytopes. However, its computational burden significantly increases for the case of dual degeneracy, which limits the application of the algorithm. Two improvements have been proposed to solve this problem for the Equality Set Projection algorithm: first, a new criterion that does not require a discussion of the uniqueness of the solution in linear programming, which simplifies the algorithm process and reduces the computational cost; and second, an improved method that abandons the calculation of a ridge's equality set to reduce the computational burden in the case of high-dimensional dual degeneracy.

Fleet scheduling for electric towing of aircraft under limited airport energy capacity

Taxiing aircraft using electric vehicles is seen as an effective solution to meet aviation targets of climate neutrality. However, making the transition to electric taxiing operations is expected to significantly increase the electricity demand at airports. In this paper we propose a mixed-integer linear program to schedule electric vehicles for aircraft towing and battery charging, while considering a limit for the supply of energy. The objective of the schedule is to maximize emissions savings. For computational tractability, we develop an Adaptive Large Neighbourhood Search which makes use of multiple local search heuristics to identify scheduling solutions. For daily scheduling with a small fleet size, the developed heuristic achieves solutions with an average 4% gap to the best linear programming solution. The results show that charging the vehicles during daytime is essential to maximize saved emissions: removing charging opportunities for a few hours during the day reduces the performance by an average of 6.4%. In addition, it is found that fast charging leads to low vehicle downtime, unless the battery size exceeds 750kWh, when charging rates over 150kW become unnecessary. Overall, our model provides support for infrastructure planning of airports during the transition to aircraft electric taxiing.

Digital Twin-Assisted, SFC-Enabled Service Provisioning in Mobile Edge Computing

Mobile Edge Computing (MEC) has been identified as a desirable computing paradigm that provides efficient and effective services for various applications, while meeting stringent service delay requirements. Orthogonal to the MEC computing paradigm, Network Function Virtualization (NFV) technology is another enabling technology that provides the network resource management with great flexibility and scalability, where the instances of Virtual Network Functions (VNFs) are deployed in edge servers as Service Function Chains (SFCs) for SFC-enabled services. Although reliable service provisioning in MEC environments is fundamentally important, the deployed VNF instances usually are not reliable, which can be affected by their software implementation, their execution duration, the workload among edge servers, and so on. Empowered by digital twin techniques, the states of VNF instances can be maintained by their digital twins in a real-time manner and their reliability can be accurately predicted through their digital twins. In this paper, we study digital twin-assisted, SFC-enabled reliable service provisioning in MEC networks by exploiting the dynamics of VNF instance reliability. We concentrate on two novel optimization problems of reliable service provisioning: the service cost minimization problem, and the dynamic service admission maximization problem. We first show their NP-hardness. We then formulate an Integer Linear Program (ILP) solution, and devise an approximation algorithm with a constant approximation ratio for the service cost minimization problem. We thirdly provide an ILP solution to the offline version of the dynamic service admission maximization problem. Built upon this offline ILP solution, we also develop an online algorithm with a provable competitive ratio for the problem, by adopting the primal-dual dynamic updating technique. We finally evaluate the performance of the proposed algorithms via simulations. Simulation results demonstrate that the proposed algorithms outperform their comparison benchmarks, and improve the performance of their comparison counterparts by no less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10.2 \%$</tex-math></inline-formula> .

Existence of solutions for polyhedral convex set optimization problems

Polyhedral convex set optimization problems are the simplest optimization problems with set-valued objective function. Their role in set optimization is comparable to the role of linear programs in scalar optimization. Vector linear programs and multiple objective linear programs provide proper subclasses. In this article, we choose a solution concept for arbitrary polyhedral convex set optimization problems out of several alternatives, show existence of solutions and characterize the existence of solutions in different ways. Two known results are obtained as particular cases, both with proofs being easier than the original ones: The existence of solutions of bounded polyhedral convex set optimization problems and a characterization of the existence of solutions of vector linear programs.

Unimodality of Parametric Linear Programming Solutions and Efficient Quantile Estimation

For linear optimization problems with a parametric objective, so-called parametric linear programs (PLP), we show that the optimal decision values are, under few technical restrictions, unimodal functions of the parameter, at least in the two-degrees-of-freedom case. Assuming that the parameter is random and follows a known probability distribution, this allows for an efficient algorithm to determe the quantiles of linear combinations of the optimal decisions. The novel results are demonstrated with probabilistic economic dispatch. For an example setup with uncertain fuel costs, quantiles of the resulting inter-regional power flows are computed. The approach is compared against Monte Carlo and piecewise computation techniques, proving significantly reduced computation times for the novel procedure. This holds especially when the feasible set is complex and/or extreme quantiles are desired. This work is limited to problems with two effective degrees of freedom and a one-dimensional uncertainty. Future extensions to higher dimensions could yield a key tool for the analysis of probabilistic PLPs and, specifically, risk management in energy systems.

Integer Programming Based Optimization of Power Consumption for Data Center Networks

With the quickly developing data centers in smart cities, reducing energy consumption and improving network performance, as well as economic benefits, are essential research topics. In particular, Data Center Networks do not always run at full capacity, which leads to significant energy consumption. This paper experiments with a range of optimization tools to find the optimal solutions for the Integer Linear Programming (ILP) model of network power consumption. The study reports on experiments under three communication patterns (near, long, and random), measuring runtime and memory consumption in order to evaluate the performance of different ILP solvers.While the results show that, for near traffic pattern, most of the tools rapidly converge to the optimal solution, CP-SAT provides the most stable performance and outperforms the other solvers for the long traffic pattern. On the other hand, for random traffic pattern, Gurobi can be considered to be the best choice, since it is able to solve all the benchmark instances under the time limit and finds solutions faster by 1 or 2 orders of magnitude than the other solvers do.

Random Projections for Linear Programming: An Improved Retrieval Phase

One way to solve very large linear programs in standard form is to apply a random projection to the constraints, then solve the projected linear program [ 63 ]. This will yield a guaranteed bound on the optimal value, as well as a solution to the projected linear program. The process of constructing an approximate solution of the original linear program is called solution retrieval. We improve theoretical bounds on the approximation error of the retrieved solution obtained as in Reference [ 42 ] and propose an improved retrieval method based on alternating projections. We show empirical results illustrating the practical benefits of the new approach.

On complexity of finding strong-weak solutions in bilevel linear programming

We consider bilevel linear programs (BLPs) that model hierarchical decision-making settings with two independent decision-makers (DMs), referred to as a leader (an upper-level DM) and a follower (a lower-level DM). BLPs are strongly NP-hard. In general, the follower's rational reaction (i.e., a set that contains optimal solutions of the lower-level problem for a given leader's decision) is not a singleton. If we assume that, for a given leader's decision, the follower always selects a solution from the rational reaction set that is most (least) favorable to the leader, then we obtain the optimistic (pessimistic) model. It is known that the optimistic (pessimistic) model remains NP-hard even if an optimal pessimistic (optimistic) solution of the same BLP is known. One interesting generalization of these two approaches studied in the related literature is to consider the so-called α-strong-weak model, where parameter α∈[0,1] controls the leader's level of conservatism. That is, α provides the probability of the follower's “cooperation” in a sense that, the lower-level's rational decisions that are most and least favorable to the leader are picked with probabilities α and 1−α, respectively. In this note we show that for any fixed α∈(0,1) the problem of finding an optimal α-strong-weak solution remains strongly NP-hard, even when both optimal optimistic and pessimistic solutions for the same BLP are known.

Fast Continuous and Integer L-Shaped Heuristics Through Supervised Learning

We propose a methodology at the nexus of operations research and machine learning (ML) leveraging generic approximators available from ML to accelerate the solution of mixed-integer linear two-stage stochastic programs. We aim at solving problems where the second stage is demanding. Our core idea is to gain large reductions in online solution time, while incurring small reductions in first-stage solution accuracy by substituting the exact second-stage solutions with fast, yet accurate, supervised ML predictions. This upfront investment in ML would be justified when similar problems are solved repeatedly over time—for example, in transport planning related to fleet management, routing, and container yard management. Our numerical results focus on the problem class seminally addressed with the integer and continuous L-shaped cuts. Our extensive empirical analysis is grounded in standardized families of problems derived from stochastic server location (SSLP) and stochastic multi-knapsack (SMKP) problems available in the literature. The proposed method can solve the hardest instances of SSLP in less than 9% of the time it takes the state-of-the-art exact method, and in the case of SMKP, the same figure is 20%. Average optimality gaps are, in most cases, less than 0.1%. History: Accepted by Alice Smith, Area Editor (for this paper) for Design and Analysis of Algorithms–Discrete. Funding: Financial support from the Institut de Valorisation des Données (IVADO) Fundamental Research Project Grants [project entitled “Machine Learning for (Discrete) Optimization”]; Canada Research Chairs; the Natural Sciences and Engineering Research Council of Canada [Collaborative Research and Development Grant CRD-477938-14]; and the Canadian National Railway Company Chair in Optimization of Railway Operations at Université de Montréal is gratefully acknowledged. E. Frejinger holds a Canada Research Chair. Computations were made on the supercomputer Béluga, managed by Calcul Québec and Digital Research Alliance of Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation; the Ministère de l’Économie, de la Science et de l’Innovation du Québec; and the Fonds de Recherche du Québec – Nature et Technologies.

Near-Optimal and Collaborative Service Caching in Mobile Edge Clouds

In this paper, we study the problem of service caching in an MEC network under a service market with multiple network service providers competing for both computation and bandwidth resources in terms of Virtual Machines (VMs) in the MEC network. We first propose an Integer Linear Program (ILP) solution and a randomized rounding algorithm, for the problem without VM sharing among different network service providers. We then devise a distributed and stable game-theoretical mechanism for the problem with VM sharing among network service providers, with the aim to minimize the social cost of all network service providers, through introducing a novel cost sharing model and a coalition formation game. We also analyze the performance guarantee of the proposed mechanism, Strong Price of Anarchy (SPoA). We thirdly consider the cost- and delay-sensitive service caching problem with temporal VM sharing, and propose a mechanism with provable SPoA. We finally evaluate the performance through extensive simulations and a real world test-bed implementation. Experimental results demonstrate that the proposed algorithms outperform existing approaches by achieving at least 40% lower social cost via service caching and resource sharing among different network service providers.

An energy-aware combinatorial auction-based virtual machine scheduling model and heuristics for green cloud computing

Considering the increasing demand for cloud computing, and the financial and environmental impact of the increasing energy consumption trend of data centers, improving energy efficiency is vital for cloud service providers. In this study, an energy-aware virtual machine scheduling model is proposed which is based on the multi-unit nondiscriminatory combinatorial auction. The model includes a powerful bidding language that allows users to declare their complicated virtual machine requests using logical AND and OR relations along with the time constraints. The study also presents the formal definition of the model and the associated optimization problem for determining the optimum schedule and energy-efficient placement of VMs on physical servers. The optimization problem is formulated using integer linear programming and several heuristic solution methods including the Genetic Algorithm are proposed for this problem. The performances of the model and the proposed heuristics are assessed on a comprehensive test suite. The proposed model is estimated to provide approximately a 37% improvement in revenues, and the solution methods are estimated to provide high-quality solutions within only 5% of the optimum which enable the model to be deployed in large-scale clouds.