LP Solver Research Articles

Location-aware job scheduling for IoT systems using cloud and fog

The rapid growth of Internet of Things (IoT) applications generates vast volumes of high-speed data streams from numerous devices. Cloud computing solutions often handle and manage this data; however, for certain applications, the latency introduced by transmitting data from edge devices to the cloud may be unacceptable. This issue is exacerbated by the bandwidth constraints of public networks, which become significant barriers in large-scale IoT implementations. Consequently, effective resource management, service management, data storage, and power management require more robust infrastructure and complex protocols. An "intelligent gateway" based on fog computing can enhance the efficient utilization of cloud and network resources. Resource planning and management in a fog-cloud environment significantly impact system performance, particularly latency. This problem is known to be NP-hard. This paper addresses the challenge of lifetime optimization for scheduling data-intensive tasks in fog and cloud-based IoT systems. Initially, we propose an Integer Linear Programming (ILP) optimization model to frame the problem. Subsequently, we introduce a heuristic method named Data-Locality Aware Job Scheduling in Fog-Cloud (DLSFC), which is derived from the proposed formulation. The effectiveness of DLSFC is evaluated across various system characteristics, with results demonstrating that DLSFC achieves solutions within 85 % of the ideal outcome provided by the LP solver.

A fluid–particle decomposition approach to matching market design for crowdsourced delivery systems

This paper considers a crowdsourced delivery (CSD) system that effectively utilizes the existing trips to fulfill parcel delivery as a matching problem between CSD drivers and delivery tasks. This matching problem has two major challenges. First, it is a large-scale combinatorial optimization problem that is hard to solve in a reasonable computational time. Second, the evaluation of the objective function for socially optimal matching contains the utility of drivers for performing the tasks, which is generally unobservable private information. To address these challenges, this paper proposes a hierarchical distribution mechanism of CSD tasks that decomposes the matching problem into the task partition (master problem) and individual task-driver matching within smaller groups of drivers (sub-problems). We incorporate an auction mechanism with truth-telling and efficiency into the sub-problems so that the drivers’ perceived utilities are revealed through their bids. Furthermore, we formulate the master problem as a fluid model based on continuously approximated decision variables. By exploiting the random utility framework, we analytically represent the objective function of the problem using continuous variables, without explicitly knowing the drivers’ utilities. The numerical experiment shows that the proposed approach solved large-scale matching problems at least 100 times faster than a naive LP solver and approximated the original objective value with errors of less than 1%. The code of this work is available at https://github.com/yuki-oyama/fluid-particle-csd.

Initial Evaluation of the Merit of Guar as a Dairy Forage Replacement Crop during Drought-Induced Water Restrictions

Dairy is the largest sector of the agricultural industry in New Mexico, but the sector’s need for high quality forage in the face of ongoing drought and climate change causes concern. The economic merit of using guar to replace alfalfa under imposed constraints in irrigation water availability is evaluated. Nutritional quality of guar (Cyamopsis tetragonoloba L.) grown at the NMSU Agricultural Science Center at Clovis was estimated using Near Infrared Spectroscopy (NIRS) analysis. Results show that 45.6 percent of analyzed samples were comparable to Supreme or Premium alfalfa (Medicago sativa) hay using Relative Feed Value index, and 23.7% met this criteria using crude protein. While alfalfa uses 4.4 acre-feet of water (approximately 5400 cubic meters) per season, the guar sampled used 5 inches (500 cubic meters) of applied irrigation. Microsoft Excel Version 2404 LP Solver was used to identify an optimum crop mix of alfalfa and guar under different levels of irrigation water constraints. With increasing levels in modeled irrigation restrictions, the use of guar increased the potential forage production by 3% under a 0.4 acre-foot restriction up to almost 59% under a 1.5 acre-feet irrigation water limit. Our results merit additional research, including forage trials to determine the impact of guar forage on milk production and further agronomic research into growing guar to maximize forage yield.

Scaling Package Queries to a Billion Tuples via Hierarchical Partitioning and Customized Optimization

A package query returns a package---a multiset of tuples---that maximizes or minimizes a linear objective function subject to linear constraints, thereby enabling in-database decision support. Prior work has established the equivalence of package queries to Integer Linear Programs (ILPs) and developed the SketchRefine algorithm for package query processing. While this algorithm was an important first step toward supporting prescriptive analytics scalably inside a relational database, it struggles when the data size grows beyond a few hundred million tuples or when the constraints become very tight. In this paper, we present Progressive Shading, a novel algorithm for processing package queries that can scale efficiently to billions of tuples and gracefully handle tight constraints. Progressive Shading solves a sequence of optimization problems over a hierarchy of relations, each resulting from an ever-finer partitioning of the original tuples into homogeneous groups until the original relation is obtained. This strategy avoids the premature discarding of high-quality tuples that can occur with SketchRefine. Our novel partitioning scheme, Dynamic Low Variance, can handle very large relations with multiple attributes and can dynamically adapt to both concentrated and spread-out sets of attribute values, provably outperforming traditional partitioning schemes such as kd-tree. We further optimize our system by replacing our off-the-shelf optimization software with customized ILP and LP solvers, called Dual Reducer and Parallel Dual Simplex respectively, that are highly accurate and orders of magnitude faster.

Risk-assessment of carbon-dioxide recycling in a gas-fired power plant using CVaR-based convex optimization

With the rise of renewable resources’ penetration, the use of clean excessive renewable energy as the primary source of a Power to Gas (PtG) system for the reduction of CO2 pollution is quite advantageous. This paper presents a mixed integer linear formulation for modeling the optimal operation of a gas-fired power plant (GFPP)-PtG system that utilizes an excessive renewable resource, wind power, as the primary mover of the proposed PtG system for capturing and recycling the produced CO2 pollution. CO2 and CH4 gas storages are considered in the proposed PtG system to boost the operational efficiency of the system. Additionally, the scenario generation method models both wind power generation and electrical demand. The effective conditional value at risk (CVaR) method, which is a convex risk modeling approach, models the risks imposed by the generated scenarios for wind power and electrical demand. Due to the convexity of the proposed formulation for the PtG system and the risk modeling method, the ultimate model can be simply solved by using any available LP solver. To evaluate the outcomes of an optimization problem based on CVaR, a comparative analysis is conducted with chance constraints programming, which demonstrates the optimality and computational efficiency of the CVaR results due to convexity. The effectiveness of the proposed model is approved by modeling a test system based on real data, and the obtained numerical results are discussed comprehensively.

R2T: Instance-optimal Truncation for Differentially Private Query Evaluation with Foreign Keys

Answering SPJA queries under differential privacy (DP), including graph pattern counting under node-DP as an important special case, has received considerable attention in recent years. The dual challenge of foreign-key constraints and self-joins is particularly tricky to deal with, and no existing DP mechanisms can correctly handle both. For the special case of graph pattern counting under node-DP, the existing mechanisms are correct (i.e., satisfy DP), but they do not offer nontrivial utility guarantees or are very complicated and costly. In this paper, we propose the first DP mechanism for answering arbitrary SPJA queries in a database with foreign-key constraints. Meanwhile, it achieves a fairly strong notion of optimality, which can be considered as a small and natural relaxation of instance optimality. Finally, our mechanism is simple enough that it can be easily implemented on top of any RDBMS and an LP solver. Experimental results show that it offers order-of-magnitude improvements in terms of utility over existing techniques, even those specifically designed for graph pattern counting.

Facilities Location in the Supply Chain Network Using an Iterated Local Search Algorithm

The supply chain design problem is one of the important problems in supply chain management. In this problem, the goal is to locate several facilities for distribution centers in different geographical areas to cover customer demands. In the model presented in this paper, customer demand is met through distribution centers, while these distribution centers supply products from manufacturing plants. In this paper, a mathematical model of integer linear programming is presented that, in addition to the optimal supply chain network design, considers the facility's location. This model aims to minimize the costs of sending products from the factory to distribution centers, from distribution centers to customers, and establishing distribution centers. To evaluate the proposed model, different network samples were created randomly on small, medium and large scales for experiments. In order to perform the experiments, a linear programming solver (LP Solver) and an iterated local search algorithm were used to compare the performance of each of these methods to find the optimal solution. These experiments show the superiority of the iterated local search algorithm compared to LP Solver in achieving the optimal solution in the shortest runtime for all instances.

Exact Complexity Certification of Suboptimal Branch-and-Bound Algorithms for Mixed-Integer Linear Programming

In this paper, we present a method to exactly certify the computational complexity of standard suboptimal branch-and-bound (B&B) algorithms for computing suboptimal solutions to mixed-integer linear programming (MILP) problems. Three well-known approaches for suboptimal B&B are considered. This work shows that it is possible to exactly certify the computational complexity also when these approaches are used. Moreover, it also enables to compute exact bounds on the level of suboptimality actually to be obtained online, also for methods previously without any such guarantees. It additionally provides a novel deeper insight into how they affect the performance of the B&B algorithm in terms of the required computation time and memory storage. The exact bounds on the online worst-case computational complexity (e.g., the accumulated number of LP solver iterations or size of the B&B tree) and the worst-case suboptimality computed with the proposed method are very relevant for real-time applications such as Model Predictive Control (MPC) for hybrid systems. The numerical experiments confirm the correctness of the proposed method, and they demonstrate the usefulness of the certification method for certification of a standard online B&B-based MILP solver employing the three considered suboptimal techniques.

A computational status update for exact rational mixed integer programming

The last milestone achievement for the roundoff-error-free solution of general mixed integer programs over the rational numbers was a hybrid-precision branch-and-bound algorithm published by Cook, Koch, Steffy, and Wolter in 2013. We describe a substantial revision and extension of this framework that integrates symbolic presolving, features an exact repair step for solutions from primal heuristics, employs a faster rational LP solver based on LP iterative refinement, and is able to produce independently verifiable certificates of optimality. We study the significantly improved performance and give insights into the computational behavior of the new algorithmic components. On the MIPLIB 2017 benchmark set, we observe an average speedup of 10.7x over the original framework and 2.9 times as many instances solved within a time limit of two hours.

Progress in mathematical programming solvers from 2001 to 2020

This study investigates the progress made in LP and MILP solver performance during the last two decades by comparing the solver software from the beginning of the millennium with the codes available today. On average, we found out that for solving LP/MILP, computer hardware got about 20 times faster, and the algorithms improved by a factor of about nine for LP and around 50 for MILP, which gives a total speed-up of about 180 and 1,000 times, respectively. However, these numbers have a very high variance and they considerably underestimate the progress made on the algorithmic side: many problem instances can nowadays be solved within seconds, which the old codes are not able to solve within any reasonable time.

Pseudo polynomial size LP formulation for calculating the least core value of weighted voting games

In this paper, we propose a pseudo polynomial size LP formulation for finding a payoff vector in the least core of a weighted voting game. The numbers of variables and constraints in our formulation are both bounded by O(nW+), where n is the number of players and W+ is the total sum of (integer) voting weights. When we employ our formulation, a commercial LP solver calculates a payoff vector in the least core of practical weighted voting games in a few seconds. We also extend our approach to vector weighted voting games.

Forward-Chaining Partial-Order Planning

Over the last few years there has been a revival of interest in the idea of least-commitment planning with a number of researchers returning to the partial-order planning approaches of UCPOP and VHPOP. In this paper we explore the potential of a forward-chaining state-based search strategy to support partial-order planning in the solution of temporal-numeric problems. Our planner, POPF, is built on the foundations of grounded forward search, in combination with linear programming to handle continuous linear numeric change. To achieve a partial ordering we delay commitment to ordering decisions, timestamps and the values of numeric parameters, managing sets of constraints as actions are started and ended. In the context of a partially ordered collection of actions, constructing the linear program is complicated and we propose an efficient method for achieving this. Our late-commitment approach achieves flexibility, while benefiting from the informative search control of forward planning, and allows temporal and metric decisions to be made - as is most efficient - by the LP solver rather than by the discrete reasoning of the planner. We compare POPF with the approach of constructing a sequenced plan and then lifting a partial order from it, showing that our approach can offer improvements in terms of makespan, and time to find a solution, in several benchmark domains.

Lagrangian Decomposition for Optimal Cost Partitioning

Optimal cost partitioning of classical planning heuristics has been shown to lead to excellent heuristic values but is often prohibitively expensive to compute. Lagrangian decomposition and Lagrangian relaxation are classical tools in mathematical programming that apply to optimization problems with a special block structure. We analyze the application of Lagrangian decomposition to cost partitioning in the context of operator-counting heuristics and interpret Lagrangian multipliers as cost functions for the combined heuristics. This allows us to view the computation of an optimal cost partitioning as an iterative process that can be seeded with any cost partitioning and improves over time. We derive an anytime algorithm to compute an optimal non-negative cost partitioning of abstraction heuristics without involving an LP solver. In each iteration, the computation reduces to independent shortest path problems in all abstractions. Finally, we discuss the extension to general cost functions.

E-aplan: a tool for teaching collaborative aggregate production planning in industrial engineering

En este trabajo se presenta una herramienta informática titulada E–aplan Express (versión 2018), de libre acceso para uso educativo y comercial, para la modelización y resolución de planes de producción agregados generando una planificación de la producción a medio-largo plazo, en base a una demanda prevista en ese periodo. La herramienta E–aplan modela el plan de producción agregado a través de un modelo de programación lineal entera mixta (MILP). El motor de optimización LP solver genera la planificación ajustando todas las variables de optimización con el menor error posible. Por último, se presenta un ejemplo ilustrativo que considera una planificación de la producción agregada colaborativa, en una cadena de suministro de dos eslabones. Se modelan diferentes escenarios para considerar simultáneamente los objetivos de planificación de las dos empresas de la red.

Automated Generation of Consistent Graph Models With Multiplicity Reasoning

Advanced tools used in model-based systems engineering (MBSE) frequently represent their models as graphs. In order to test those tools, the automated generation of well-formed (or intentionally malformed) graph models is necessitated which is often carried out by solver-based model generation techniques. In many model generation scenarios, one needs more refined control over the generated unit tests to focus on the more relevant models. Type scopes allow to precisely define the required number of newly generated elements, thus one can avoid the generation of unrealistic and highly symmetric models having only a single type of elements. In this paper, we propose a 3-valued scoped partial modeling formalism, which innovatively extends partial graph models with predicate abstraction and counter abstraction. As a result, well-formedness constraints and multiplicity requirements can be evaluated in an approximated way on incomplete (unfinished) models by using advanced graph query engines with numerical solvers (e.g., IP or LP solvers). Based on the refinement of 3-valued scoped partial models, we propose an efficient model generation algorithm that generates models that are both well-formed and satisfy the scope requirements. We show that the proposed approach scales significantly better than existing SAT-solver techniques or the original graph solver without multiplicity reasoning. We illustrate our approach in a complex design-space exploration case study of collaborating satellites introduced by researchers at NASA JPL.

Optimization of a grid connected hybrid energy system

Nowadays, energy policy makers in Turkey are discussing about transition to regional energy pricing policy due to the illegal usage rates in some regions in the country. A possible transition to regional pricing policy needs a detailed analysis on development of regional energy networks. By developing such networks, outer-dependency of energy can be reduced in regions and energy costs can reduce by using energy generation potential from renewable resources within regional networks. Moreover, the existing energy grid can be considered as a backup option to satisfy shortened demand after generation from renewable resources. In other words, grid-connected hybrid energy systems can be a promising solution to satisfy energy demand at a lower cost. Hybrid energy systems are discovered as a beneficial way to provide sustainable energy supply in regions with renewable energy potential. On the other hand, these kind of energy systems requires an analytic investigation in terms of design and operation, because of the existence of different energy resources. So, optimization of system design and operation decisions are extremely important. In this paper, optimization of a hybrid energy network is considered and a linear programming model to optimize operation decisions in a hybrid energy network for a town is proposed. The proposed model is solved on a numerical example with 10 demand points and 2 generation plants. An operation strategy for the numerical example is obtained after solution with a commercial LP solver. This model can be used a decision support tool for policy makers in development of energy policies.

A transshipment model for logistics management at Indian Oil Corporation

Three refineries of the Indian Oil Corporation procure crude oil from suppliers, at production sites in the Persian Gulf, West Africa, North Africa, West Asia, and India. The crude oil is shipped to two ports in the state of Gujarat, India, by large tankers and medium sized vessels. From these two ports, the crude oil is pumped to the refineriesviapipelines. The refineries have known capacity, that are different for the two types of crude oil. In this paper, the scaled-up problem has been modelled, as a transshipment network. Next, the concrete instance of the problem, has been solved using an LP solver. This was followed by post-optimality analysis of the solution. The flow values on all arcs, and optimal product mix, validate actual decisions. Precise shipping requirements obtained from the solution, are shared in advance with marine transporters to improve supply chain coordination.

Outer approximation with conic certificates for mixed-integer convex problems

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with $\mathcal{K}^*$ cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the $\mathcal{K}^*$ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate $\mathcal{K}^*$ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful $\mathcal{K}^*$ cuts. Our new open source MI-conic solver Pajarito (http://github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear (MILP) solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX's specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of $\mathcal{K}^*$ cuts.

Refuting conjectures in extremal combinatorics via linear programming

We apply simple linear programming methods and an LP solver to refute a number of open conjectures in extremal combinatorics.

Low voltage power grid congestion reduction using a community battery: Design principles, control and experimental validation

By installing a battery storage system in the power grid, Distribution Network Operators (DNOs) can solve congestion problems caused by decentralized renewable generation. This paper provides the necessary theory to use such a community battery for grid congestion reduction, backed up by experimental results. A simple network model was constructed by linearizing the load flow equations using a constant impedance load model. Using this model, an accurate estimate of voltage and overload problems is fed into a receding horizon charge path optimizer. The charge path optimization problem is posed as a linear problem and subsequently solved by an LP solver. The algorithms have been applied and validated on a real-world community battery installation. It was found that the voltages and currents can be controlled to a great degree, increasing the grid capacity significantly. The proposed control framework can be used to safeguard network constraints and is compatible with other battery control goals, such as energy trading or energy independence. Network design formulas are described with which a DNO can quickly estimate the potential (de) stabilization of a community battery on the steady-state voltages and currents in the grid.