Large-scale Linear Programming Research Articles

An efficient outcome-space branch-and-bound algorithm for solving a class of large-scale linear multiplicative programs

An efficient outcome-space branch-and-bound algorithm for solving a class of large-scale linear multiplicative programs

Optimizing air quality monitoring spatial layout by maximizing the coverage of the population in Beijing–Tianjin–Hebei and surrounding areas

The spatial layout of the air quality monitoring network (AQMN) is crucial for objective, accurate, and comprehensive air quality assessment. The current technical standard specified the minimum quantity requirements for air quality monitoring sites, but there were no standards to specify the spatial of monitoring sites. This study proposed a novel framework to evaluate and optimize the spatial layout of AQMN. First, this study proposed three indicators to evaluate the performance of the current AQMN. They were monitoring area repetition rate, population coverage rate, and correlations. The assessment of AQMN in Beijing–Tianjin–Hebei and surroundings areas (BTHs) showed the overall monitoring area repetition rate and population coverage rate was 81.07 % and 35.5 %, respectively, which means the current AQMN in BTHs has very high monitoring repeatability and limited population coverage. Secondly, a large-scale linear programming model was built to optimize the spatial layout and determine the spatial location of 279 newly added monitoring sites in BTHs according to the Environmental Monitoring 14th Five-Year Plan of China. The optimization results showed that the optimized AQMN covered 97 million additional people, and the population coverage rate increased to 49.5 %. The proposed framework provided a valuable tool to evaluate and optimize AQMN and could be a potential solution for developing new technical standards of AQMN.

Virtual Power Plant and Microgrid Control Integration for Campus Environments with Wind Solar Pumped Hydro and Biomass Energy

This paper provides a comprehensive exploration of integrating renewable energy sources, focusing on wind, solar, Pumped Hydro and biogas, into power systems. Virtual Power Plants (VPPs) with an Energy Management System (EMS) offer a promising solution for managing these distributed resources. The VPP model, utilizing mixed integer linear programming (MILP) in GAMS and solved with CPLEX, optimizes net daily profit within a 24-hour interval. This approach efficiently allocates resources, maximizes profitability, and ensures stable energy supply, addressing the challenges posed by variable renewable energy outputs. The MILP model is formulated using linear equations and inequalities within GAMS, with the CPLEX solver being effective for handling large-scale linear programming tasks. The model is based on real-world data, including market variability, weather forecasts, and demand patterns, and can be optimized for better decision-making in VPP operations, ultimately maximizing profits while ensuring a stable and sustainable power supply.

Virtual Power Plant and Microgrid Control Integration for Campus Environments with Wind Solar Pumped Hydro and Biomass Energy

This paper provides a comprehensive exploration of integrating renewable energy sources, focusing on wind, solar, Pumped Hydro and biogas, into power systems. Virtual Power Plants (VPPs) with an Energy Management System (EMS) offer a promising solution for managing these distributed resources. The VPP model, utilizing mixed integer linear programming (MILP) in GAMS and solved with CPLEX, optimizes net daily profit within a 24-hour interval. This approach efficiently allocates resources, maximizes profitability, and ensures stable energy supply, addressing the challenges posed by variable renewable energy outputs. The MILP model is formulated using linear equations and inequalities within GAMS, with the CPLEX solver being effective for handling large-scale linear programming tasks. The model is based on real-world data, including market variability, weather forecasts, and demand patterns, and can be optimized for better decision-making in VPP operations, ultimately maximizing profits while ensuring a stable and sustainable power supply.

A Reinforcement-Learning-Based Multiple-Column Selection Strategy for Column Generation

Column generation (CG) is one of the most successful approaches for solving large-scale linear programming (LP) problems. Given an LP with a prohibitively large number of variables (i.e., columns), the idea of CG is to explicitly consider only a subset of columns and iteratively add potential columns to improve the objective value. While adding the column with the most negative reduced cost can guarantee the convergence of CG, it has been shown that adding multiple columns per iteration rather than a single column can lead to faster convergence. However, it remains a challenge to design a multiple-column selection strategy to select the most promising columns from a large number of candidate columns. In this paper, we propose a novel reinforcement-learning-based (RL) multiple-column selection strategy. To the best of our knowledge, it is the first RL-based multiple-column selection strategy for CG. The effectiveness of our approach is evaluated on two sets of problems: the cutting stock problem and the graph coloring problem. Compared to several widely used single-column and multiple-column selection strategies, our RL-based multiple-column selection strategy leads to faster convergence and achieves remarkable reductions in the number of CG iterations and runtime.

Column-Randomized Linear Programs: Performance Guarantees and Applications

Solving Large-Scale Linear Programs by Randomly Sampling Columns Large-scale linear programs (LPs) play a pivotal role in various applications, including the classic cutting-stock problem and the vehicle routing problem. The standard solution approach for these LPs, namely, column generation (CG), is often computationally intractable because of the NP-hard nature of the corresponding subproblem in many applications. In “Column-Randomized Linear Programs: Performance Guarantees and Applications,” Akchen and Mišić introduce a randomized method that involves first sampling a set of columns from the original LP and subsequently solving an LP composed of the sampled columns, termed the column-randomized LP. The authors analyze the optimality gap of the column-randomized LPs and establish conditions under which the gap is small. Empirical findings demonstrate the effectiveness of the column-randomized LP approach, showcasing its advantages over the CG approach in two practical applications: the cutting-stock problem and nonparametric choice model estimation.

Infeasibility Detection with Primal-Dual Hybrid Gradient for Large-Scale Linear Programming

Infeasibility Detection with Primal-Dual Hybrid Gradient for Large-Scale Linear Programming

Solving Resource Allocation Problem in Wifi Network by Dantzig-Wolfe Decomposition Algorithm

Online advertising and advancements is a recent trend in marketing technology, in this context we consider a new form of contract which allows advertisers to specify in the Wifi system. Based on the structure of the system, we have to organize and manage resource allocation such that the guaranteed display is satisfied. We introduce a new mathematical model and develop an optimization framework that aims to optimize “fairness” of allocation each campaign over its targeted location. Because of large scale problem, the Dantzig-Wolfe decomposition is proposed for solving it. Dantzig-Wolfe decomposition is a technique for dealing with large scale linear programming and modified to solve linear integer programming, nonlinear programming. Especially, it is used mostly in linear programming when its size is very large, and its structure is appropriate. The technique has been successfully applied in a variety of contexts. In this paper, we introduce a new model of a resource allocation problem in Wifi network and represent Dantzig-Wolfe decomposition for solving this problem by dividing the number of advertisement impressions when users access the Wifi network. The numerical simulation shows the efficiency of our proposed method.

A Column Generation Scheme for Distributionally Robust Multi-Item Newsvendor Problems

This paper studies a distributionally robust multi-item newsvendor problem, where the demand distribution is unknown but specified with a general event-wise ambiguity set. Using the event-wise affine decision rules, we can obtain a conservative approximation formulation of the problem, which can typically be further reformulated as a linear program. In order to efficiently solve the resulting large-scale linear program, we develop a column generation-based decomposition scheme and speed up the computational efficiency by exploiting a special column selection strategy and stopping early based on a Karush-Kuhn-Tucker condition test. Focusing on the Wasserstein ambiguity set and the event-wise mean absolute deviation set, a computational study demonstrates both the computational efficiency of the proposed algorithm, which significantly outperforms a commercial solver and a Benders decomposition method, and the out-of-sample superiority of distributionally robust solutions relative to their sample average approximation counterparts. History: Accepted by Nicola Secomandi, Area Editor for Stochastic Models & Reinforcement Learning. Funding: This work was supported by the Natural Sciences and Engineering Research Council of Canada [492997-2016, RGPIN-2016-05208], the National Natural Science Foundation of China [71972012], Alliance de recherche numérique du Canada, and Canada Research Chairs [CRC-2018-00105]. It was also supported by Groupe d’études et de recherche en analyse des décisions (GERAD). Finally, this research was enabled in part by support provided by Digital Research Alliance of Canada ( https://alliancecan.ca/en ). Supplemental Material: The software that supports the findings of this study is available within the paper and its supplemental information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0010 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0010 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Seismic vulnerability prediction of masonry aggregates: Iterative Finite element Upper Bound limit analysis approximating no tensile resistance

A novel 3D Finite Element Upper Bound limit analysis approach for the evaluation of the seismic vulnerability of masonry aggregates when the collapse may occur both locally and globally is presented.The model relies on a 3D discretization of the aggregate by means of infinitely resistant hexahedron elements and rigid plastic quadrilateral interfaces, where all plastic dissipation occurs.The interface behavior is assumed isotropic, with friction, cohesion, and tension cutoff. The collapse horizontal acceleration is obtained in the discretized model solving large-scale linear programming problems with standard numerical routines. According to a consolidated literature, the vulnerability prediction should be done with pre-assigned failure mechanisms formed by large macro-blocks mutually roto-translating, and assuming a material unable to withstand tensile stresses without mutual sliding between macro-blocks. However, such assumptions may be responsible for an overestimation of the load-carrying capacity and the determination of a realistic mechanism is neither obvious nor possible in any case. Discretizing the aggregate into 3D elements and interfaces, the failure mechanism is automatically found in the solution of the limit analysis problem, but enforcing the no-tension material hypothesis may be responsible for the activation of parasite sliding between contiguous elements or stalling issues of the numerical algorithm. In the first case, the failure mechanism changes considerably and the computational evaluation of the horizontal acceleration at collapse can be affected by unsafe inaccuracy.The present approach iteratively approximates the no-tension material model at low computational cost, progressively shrinking the failure surface and evaluating (i) collapse accelerations, (ii) failure mechanisms active, (iii) power dissipated on interfaces and (iv) power expended by loads not dependent on the collapse multiplier. By analyzing the failure mechanisms obtained, it is possible to select where spurious sliding is not dominant. Enforcing to zero the power dissipated on interfaces it is possible to estimate the collapse multiplier corresponding to the twofold assumption that (i) parasite sliding is negligible and (ii) the material approximates in a technically acceptable way the no-tension one.The model is first benchmarked on a masonry portal, where a reference solution is available, and then applied to two historical aggregates located in Italy.

Temporally detailed modeling and analysis of global net zero energy systems focusing on variable renewable energy

This study newly develops a recursive-dynamic global energy model with an hourly temporal resolution for electricity and hydrogen balances, aiming to assess the role of variable renewable energy (VRE) in a carbon-neutral world. This model, formulated as a large-scale linear programming model (with 500 million each of variables and constraints), calculates the energy supply for 100 regions by 2050. The detailed temporal resolution enables the model to incorporate the variable output of VRE and system integration options, such as batteries, water electrolysis, curtailment, and the flexible charging of battery electric vehicles. Optimization results suggest that combing various technical options suitable for local energy situations is critical to reducing global CO2 emissions cost-effectively. Not only VRE but also CCS-equipped gas-fired and biomass-fired power plants largely contribute to decarbonizing power supply. The share of VRE in global power generation in 2050 is estimated to be 57% in a cost-effective case. The results also imply economic challenges for an energy system based on 100% renewable energy. For example, the average mitigation cost in 2050 is 69USD/tCO2 in the cost-effective case, while it increases to 139USD/tCO2 in the 100% renewable case. The robustness of this argument is tested by sensitivity analyses.

Toric geometry of entropic regularization

Entropic regularization is a method for large-scale linear programming. Geometrically, one traces intersections of the feasible polytope with scaled toric varieties, starting at the Birch point. We compare this to log-barrier methods, with reciprocal linear spaces, starting at the analytic center. We revisit entropic regularization for unbalanced optimal transport, and we develop the use of optimal conic couplings. We compute the degree of the associated toric variety, and we explore algorithms like iterative scaling.

An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography, and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving some large-scale CMOT problems on synthetic data. The preliminary experiments on the discrete tomography problem also highlight the potential modeling capability of our model.

Convergence and iteration-complexity of a primal-dual majorization-minimization method for large-scale linear programming

Convergence and iteration-complexity of a primal-dual majorization-minimization method for large-scale linear programming

Flexibility potential of aggregated electric vehicle fleets to reduce transmission congestions and redispatch needs: A case study in Austria

Redispatch measures are a key instrument of the Central European electricity market design to prevent congestion of several transmission lines after the market clearing. In addition to generation rescheduling, demand-side flexibility can reduce this redispatch need. This paper mainly aims to analyse the flexibility demand application of electric vehicle fleets in the redispatch market. The developed European electricity market model minimises dispatch- and subsequently redispatch costs whilst using electric vehicle fleets as flexible demand. A novel modelling approach allows the integration of this flexibility into large-scale linear programming models without losing their essential parameters. This case study is examined in Austria with a remarkably high share of electricity from renewable energy sources and different market penetrations of electromobility. Results show that integrating this system as a redispatch measure leads to a reduced curtailment of renewable energies (up to 25%), whilst less additional thermal power plant usage is needed. Furthermore, redispatch cost and the associated CO2 emissions are reduced by 3.3% to 13.9%. By contrast, using this flexibility as a market-based charging strategy raises CO2 emissions and redispatch costs drastically by 186% if 2 million electric vehicles are considered. Especially with the high electrification rate of the transport sector, the provided flexibility potential significantly impacts the electricity market. Further efforts could address the influence of demand-side flexibility not only on redispatch within one control area but also on cross-border redispatch and counter-trading.

Consensus-based Dantzig-Wolfe decomposition

Dantzig-Wolfe decomposition (DWD) is a classical algorithm for solving large-scale linear programs whose constraint matrix involves a set of independent blocks coupled with a set of linking rows. The algorithm decomposes such a model into a master problem and a set of independent subproblems that can be solved in a distributed manner. In a typical implementation, the master problem is solved centrally. In certain settings, solving the master problem centrally is undesirable or infeasible, such as in the case of decentralized storage of data, or when independent agents who are responsible for the subproblems desire privacy of information. In this paper, we propose a fully distributed DWD algorithm which relies on solving the master problem using a consensus-based Alternating Direction Method of Multipliers (ADMM) method. We derive error bounds on the optimality gap and feasibility violation of the proposed approach. We provide preliminary computational results for our algorithm using a Message Passing Interface implementation on a delivery planning problem, the multi-commodity network flow problem, and synthetic instances where we obtain high quality solutions. An open-source implementation of the algorithm is available.

Linear programming with a feasible direction interior point technique for smooth optimization

We propose an adaptation of the Feasible Direction Interior Points Algorithm (FDIPA) of J. Herskovits, for solving large-scale linear programs. At each step, the solution of two linear systems with the same coefficient matrix is determined. This step involves a significant computational effort. Reducing the solution time of linear systems is, therefore, a way to improve the performance of the method. The linear systems to be solved are associated with definite positive symmetric matrices. Therefore, we use Split Preconditioned Conjugate Gradient (SPCG) method to solve them, together with an Incomplete Cholesky preconditioner using Matlab’s ICHOL function. We also propose to use the first iteration of the conjugate gradient, and to presolve before applying the algorithm, in order to reduce the computational cost. Following, we then provide mathematica proof that show that the iterations approach Karush–Kuhn–Tucker points of the problem under reasonable assumptions. Finally, numerical evidence show that the method not only works in theory but is also competitive with more advanced methods.

Solution of Large-Scale Linear Programming Problem by Using Computer Technique

Linear programming (LP) is an important part of applied mathematics. This method has found its applications to important areas of product mix, blending, and diet problems. Steel, chemical, food processing industries and Oil refineries industry are also using LP with considerable success. But in practical LP can be very large. In this paper, our intent is to formulate an LP model of some large-scale real-life-oriented problems and to apply computer techniques for solving these problems. Starting with the graphical procedure which provides an ample amount of understanding into some fundamental concepts, the simple procedure of solving LP problems is developed. Finally, a special class of LP problem, namely Transportation is taken up and solved. We also solved the simplex system by using FORTRAN programming.

An Efficient Data Driven Model for Generation Expansion Planning with Short Term Operational Constraints

Generation expansion planning (GEP) models have been useful aids for long-term planning. Recent growth in intermittent renewable generation has increased the need to represent the capability for non-renewables to respond to rapid changes in daily loads, leading research to bring unit commitment (UC) features into GEPs. Such GEP+UC models usually contain discrete variables which, along with many details, make computation times impractically long for analysts who need to develop, debug, modify and use the GEP for many alternative runs. We propose a GEP with generation aggregated by technology type, and with the minimal UC content necessary to represent the limitations on generation to respond to rapid changes in demand, i.e., ramp-up and ramp-down constraints, with ramp limits estimated from historical data on maximum rates of change of each generation type. We illustrate with data for the province of Ontario in Canada; the GEP is a large scale linear program that solves in less than one hour on modest computing equipment, with credible solutions.

A computationally efficient algorithm for computing convex hull prices

Electricity markets worldwide allow participants to bid non-convex production offers. While non-convex offers can more accurately reflect a resource’s capabilities, they create challenges for market clearing processes. For example, system operators may be required to execute side payments to participants whose costs are not covered through energy sales as determined via traditional locational marginal pricing schemes. Convex hull pricing minimizes this and other types of side payments while providing uniform (i.e., locationally and temporally consistent) prices. Computing convex hull prices involves solving either a large-scale linear program or the Lagrangian dual of the corresponding non-convex scheduling problem. Further, the former approach requires explicit descriptions of market participants’ convex hulls.While linear programs for computing convex hull prices are large, their structure is naturally decomposable by generators. Here, we propose and empirically analyze a Benders decomposition approach to computing convex hull prices that leverages recent advances in convex hull formulations for thermal generating units. We demonstrate across a large set of test instances that our decomposition approach only requires modest computational effort, obtaining solutions at least an order of magnitude faster than the equivalent large-scale linear programming approach. Overall, we provide a computationally feasible method for computing convex hull prices for industrial scale market clearing problems, enabling the possibility of practical adoption of this advanced pricing mechanism.