Exact Solver Research Articles

Exact wave solver for nonparaxial laser beam propagation

Simulations of inertial confinement fusion (ICF) experiments require high-fidelity models for laser beam propagation in a nonuniform plasma with varying index of refraction. We describe a new numerical wave solver that is applicable to centimeter-scale length plasmas encountered in indirect drive ICF applications. The one-way Helmholtz equation (OHE) generalizes the time-harmonic paraxial wave equation to large angles. Here, we present a methodology to numerically evaluate the exact solution to the OHE. This solution is computed by analytically advancing eigenfunctions of the one-way Helmholtz operator along a propagation direction and is applicable to any given index of a refraction profile. We compare our exact method with a commonly used approximate split-step technique for solving the OHE. As a test problem, we consider nonparaxial propagation of Gaussian and speckled beams in a plasma density channel with internal reflection. We find that the split-step approach incurs significant errors compared to the exact solution computed using the novel algorithm.

A Fast Exact Solver with Theoretical Analysis for the Maximum Edge-Weighted Clique Problem

The maximum vertex-weighted clique problem (MVWCP) and the maximum edge-weighted clique problem (MEWCP) are two natural extensions of the fundamental maximum clique problem. In this paper, we systematically study MEWCP and make the following major contributions: (1) We show that MEWCP is NP-hard even when the minimum degree of the graph is n-2, in contrast to MVWCP which is polynomial-time solvable when the minimum degree of the graph is at least n-3. This result distinguishes the complexity of the two problems for the first time. (2) To address MEWCP, we develop an efficient branch-and-bound algorithm called MEWCat with both practical and theoretical performance guarantees. In practice, MEWCat utilizes a new upper bound tighter than existing ones, which allows for more efficient pruning of branches. In theory, we prove a running-time bound of O*(1.4423^n) for MEWCat, which breaks the trivial bound of O*(2^n) in the research line of practical exact MEWCP solvers for the first time. (3) Empirically, we evaluate the performance of MEWCat on various benchmark instances. The experiments demonstrate that MEWCat outperforms state-of-the-art exact solvers significantly. For instance, on 16 DIMACS graphs that the state-of-the-art solver BBEWC fails to solve within 7200 seconds, MEWCat solves all of them with an average time of less than 1000 seconds. On real-world graphs, MEWCat achieves an average speedup of over 36x.

Modeling, landscape analysis, and solving the capacitated single-allocation hub maximal covering problem using the GARVND hybrid algorithm

In this paper, a new variation to the Hub Maximal Covering problem called the Capacitated Single-Allocation Hub Maximal Covering Problem (CSAHMCP) is addressed for the first time and a novel binary-integer mathematical model has been proposed. To analyze the features of the problem’s solution space and to determine the most effective heuristic approach to solve it, a comprehensive Fitness Landscape Analysis is performed which reveals that the problem is of ‘rugged plain’ type and can best be solved using an algorithm with good exploration and exploitation capabilities at the same time. Accordingly, we proposed a new algorithm named GARVND, which is a hybrid of the Genetic Algorithm acting as the main framework for diversification, and a variant of the Variable Neighborhood Descent algorithm called ‘Reduced VND’ acting as a means to improve individual solutions (intensification) by searching in the spaces generated by four types of neighborhood operators. We also introduced two original crossover and mutation operators that can be used for any hub location solution method. The parameters of GARVND are tuned toward less computational times and higher solution quality by using the Taguchi experiment design technique. We solved 114 CSAHMCP instances (several times each) from the Australian Post (AP) dataset using the proposed GARVND, as well as by the Genetic Algorithm, Variable Neighborhood Search, Simulated Annealing, Tabu Search algorithms, and a hybrid algorithm consisting of Grey Wolf Optimizer and Simulated Annealing, in addition to the Gurobi exact solver. Extensive computational results and statistical analyses showed that GARVND yields favorable solutions, outperforming all the compared algorithms by producing smaller gaps with global optimal objective values. In addition, to illustrate the practical value of the CSAHMCP and the GARVND method, we performed a cost-flow sensitivity analysis for the AP Central Sydney network and came up with the best expansion scenarios to be adopted by the management.

Large-volume LEO satellite imaging data networked transmission scheduling problem: Model and algorithm

With the current miniaturization and cost reduction trend, many small imaging satellites are deployed in low Earth orbit (LEO). Efficient scheduling of transmission links is crucial for handling the large volume of imaging data in the time-varying inter-satellite broadband communication networks. This ensures optimal utilization of inter-satellite link resources and gives rise to a novel problem known as the Large-volume LEO satellite Imaging Data Networked Transmission Scheduling Problem (LLSIDNTSP). This problem requires the integrity of the imaging data received on the ground station. A Specific Time-Evolving Graph (STEG) model is introduced to formulate this problem, while a highly effective Sequential Two-Phased Heuristic Algorithm (STPHA) is proposed to solve it. STPHA determines an optimized Contact Plan (CP) with a Contact-Fast Construction Algorithm (CFCA) in the first phase. With this CP as input, it produces an optimal transmission schedule with Linear Programming in the second phase. The CFCA incorporates a novel heuristic called Mission Residual Volume Factor(MRVF), which guides the selection of a proper inter-satellite and satellite-to-ground link at each construction step. Extensive simulation results demonstrate the effectiveness of the proposed STPHA. In particular, it can achieve the optimal solution for small-sized scenarios with one-hundredth of the computing time used by the exact solver CPLEX. Also, it is consistently better than other heuristic algorithms adapted from the literature for large-sized scenarios. Additional experimental analysis is conducted to showcase the effectiveness of the innovative components of STPHA.

A genetic algorithm for the Resource-Constrained Project Scheduling Problem with Alternative Subgraphs using a boolean satisfiability solver

This study evaluates a new solution approach for the Resource-Constrained Project Scheduling with Alternative Subgraphs (RCPSP-AS) in case that complex relations (i.e. nested and linked alternatives) are considered. In the RCPSP-AS, the project activity structure is extended with alternative activity sequences. This implies that only a subset of all activities should be scheduled, which corresponds with a set of activities in the project network that model an alternative execution mode for a work package. Since only the selected activities should be scheduled, the RCPSP-AS comes down to a traditional RCPSP problem when the selection subproblem is solved. It is known that the RCPSP and, hence, its extension to the RCPSP-AS is NP-hard. Since similar scheduling and selection subproblems have already been successfully solved by satisfiability (SAT) solvers in the existing literature, we aim to test the performance of a GA-SAT approach that is derived from the literature and adjusted to be able to deal with the problem-specific constraints of the RCPSP-AS. Computational results on small- and large-scale instances (both artificial and empirical) show that the algorithm can compete with existing metaheuristic algorithms from the literature. Also, the performance is compared with an exact mathematical solver and learning behaviour is observed and analysed. This research again validates the broad applicability of SAT solvers as well as the need to search for better and more suited algorithms for the RCPSP-AS and its extensions.

A variable neighborhood search approach for solving a real-world hierarchical multi-echelon vehicle routing problem involving HCT vehicles

This paper studies the Hierarchical Multi-Switch Multi-Echelon VRP (HMSME-VRP), a newly introduced VRP variant based on a real-world case involving High Capacity Vehicles (HCV). The problem originates from the policies of a distribution company in the Nordic countries where HCVs of up to 34.5 m and up to 76 tons are allowed. The HMSME-VRP offer a new way to model distribution problems to cover large geographical areas without substantial costs in infrastructure. Furthermore, it adds complexity to the standard VRP and, as such, remains NP-hard and difficult to solve to optimality. Indeed, it has been demonstrated that only very small instances can be solved to optimality by a commercial solver. Thus, in order to handle instances of real-world size, we propose two General Variable Neighborhood Search (GVNS) procedures, the second of which is adaptive, utilizing an intelligent reordering mechanism. In order to evaluate the proposed procedures, 48 benchmark instances of various sizes and characteristics are generated and made publicly available, comprising of clustered, random, and semi-clustered customers. The computational results show that both GVNS procedures outperform the exact solver. Additionally, the adaptive version outperforms the conventional version based on both average and best solutions. Furthermore, we present a statistical analysis to verify the superiority of the adaptive version.

Smart Routes: A System for Development and Comparison of Algorithms for Solving Vehicle Routing Problems with Realistic Constraints

The problem of route optimization with realistic constraints is becoming extremely relevant in the face of global urban population growth. While we are aware of approaches that theoretically provide an exact optimal solution, their application becomes challenging as the problem size increases because of exponential complexity. We investigate the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) and compare solutions obtaining by exact solver SCIP [1] with heuristic algorithms such as LKH, 2-OPT, 3-OPT [2], the OR-Tools framework [3], and the deep learning model JAMPR [4]. We demonstrate that for problem of size 50 deep learning and classical heuristic solutions became close to SCIP exact solution but requires less time. Additionally for problems with size 100, SCIP exact methods ∼ 13 times slower that neural and classical heuristics with the same route cost and on ∼ 50% worse for the first feasible solution on the same time. To conduct experiments, we developed the Smart Routes platform for solving route optimization problems, which includes exact, heuristic, and deep learning models, and facilitates convenient integration of custom algorithms and datasets.

Exact ideal magnetohydrodynamic Riemann solutions considering the strength of intermediate shocks

Exact magnetohydrodynamic (MHD) Riemann solutions are the basis of constructing numerical schemes and benchmarks for verifying the schemes. However, non-strict hyperbolicity and nonconvexity of MHD equations contribute to the appearance of intermediate shocks, causing low efficiency of existing exact solvers and high dependence on iterative initials. Utilizing the magnetic critical Mach number proposed in this paper, all possible intermediate shocks are analyzed, parameterized, and categorized. Moreover, the possible wave structures on one side of contact discontinuity are revealed to have 25 cases, and initial conditions are classified into three categories according to the coplanar properties. Based on our findings, a new exact MHD Riemann solver is built. The robustness has been significantly improved after avoiding considerable judgments and the dependence on iterative initials. The analysis of the exact MHD Riemann solution is carried out by the characteristic properties of MHD shocks in the parameterization, and it is found that a solution space exists with the highest dimension of two dimensions under the given initial condition. It is proposed to adopt the intensities of 2 → 3 intermediate shocks as the free parameters of solution space, which can completely express the degree of solution space freedom. Finally, two examples that possess the solution space are used as verifications. The physical properties of MHD equations show that the dominant factor for the solution space is the unique characteristic property of 2 → 3 intermediate shock: the existence of an additional free parameter with tangential symmetry simultaneously.

A VNS-based approach for solving the Manhattan metric straddle carrier routing problem with buffer areas

This paper presents a metaheuristic approach for solving an optimization problem that arises at container terminals where straddle carriers (SCs) transport containers between the stacking areas and the seaside. In such container terminals, operational efficiency depends mainly on SC routing. SCs routes should consider the order in which containers are unloaded and loaded at the quay cranes (QCs), taking into account the limited capacity of the buffer area of each QC where containers are temporarily stored after being handled by a QC or an SC. Besides the precedence relations (i.e., container sequences) and buffer capacities, the solution framework considers safety constraints. Efficient routing of SCs directly contributes to minimizing the idle time of QCs, thereby improving their overall productivity and minimizing the turnaround time of vessels, which is the objective of the problem. Specifically, we present two different variants of the Variable Neighborhood Search (VNS) algorithm. Each variant is initialized in both a greedy and a random manner. These algorithms address the problem by incorporating four LS operators commonly utilized in vehicle routing problems. We perform a comparative analysis of the results of these four approaches against each other and against solutions generated by an exact solver. Our numerical experiments show that the proposed algorithms perform better than the used solver, especially for bigger instances. A comparison with the results from the literature is also given and shows that the proposed VNS-based approach provides competitive results.

An integrated bus transit service for demand-responsive urban public transport

Recent mobility reports have highlighted the need of integrating Information and Communication Technologies (ICT) as an essential element in the development of intermodal and on-demand services to enhance transport flexibility. In this context, this paper investigates the improvements that could be obtained by integrating Dial-a-Ride into a bus transit service. We model the problem as a dynamic Dial-a-Ride problem with both fixed and dynamic requests. We extend an integer linear programming (ILP) model for its application to the dynamic case. To reduce computational effort, the problem is decomposed at the single vehicle level, where each best route is computed using an exact solver and new dynamic requests are assigned using a greedy insertion heuristic. We have developed the optimization model as a GIS-based tool and applied it to a realistic case of a bus transit service in the city of Benevento (Italy). The results show the benefits that could be obtained with the proposed integrated service when also compared to fixed routes and micro-mobility.

A multi-objective two-echelon vehicle routing problem with multiple delivery options

The classical two-echelon vehicle routing problem (2E-VRP) has commodities transported from depots to intermediate facilities and then delivered to customers from these facilities. In this study, we consider another type of 2E-VRP, in which vehicles from both echelons can be used for home delivery. We also present a sustainable model, which considers customers' preferred delivery locations, economic and environmental costs. Also, we present a meta-heuristic algorithm to solve real-world size instances in a timely manner. The computational results demonstrate that the proposed solution method can effectively solve small instances with a minimal gap when compared to an exact solver, and it can also handle large instances in a timely manner.

Aptenodytes forsteri optimization algorithm for low-carbon logistics network under demand uncertainty.

As China's "double carbon" goal continues to advance, logistics as a key area of carbon emissions and low-carbon logistics center site selection are key links in the process. However, existing studies on logistics center location often ignore the impact of demand uncertainty, which leads to a waste of resources in the planning and construction processes. We take logistics cost and carbon emission as the objectives, and the multi-objective site selection model established based on stochastic programming theory takes demand uncertainty as a stochastic constraint. We transform the stochastic constraint model into a 0-1 mixed integer multi-objective planning model by utilizing the idea of equivalence transformation. The Aptenodytes Forsteri Optimization (AFO) algorithm is combined with the Ideal Point Method to solve the model, and the algorithm is compared with the Particle Swarm Optimization (PSO), Differential Evolutionary (DE), Tabu Search (TS), Sparrow Search (SS) algorithms, and the exact solver Linear Interactive and General Optimizer (LINGO). The examples verify the validity of the models and algorithms, with an average reduction of 6.2% and 3.6% in logistics costs and carbon emissions in the case of demand determination, and at the confidence level of 0.9 under demand uncertainty, both logistics costs and carbon emissions are decreased to varying degrees. This study provides a new research idea for the low-carbon logistics location problem under demand uncertainty, which helps to promote the transformation of the logistics industry to low-carbon and high-efficiency.

A cooperative evolutionary algorithm with simulated annealing for integrated scheduling of distributed flexible job shops and distribution

Production and distribution are two essential parts in supply chains. An integration of production and distribution has received amount of attention from both academia and industry. This article investigates an integrated scheduling problem of distributed flexible job shops and distribution. First, a mixed integer programming model is developed to minimize maximum completion time. Second, a cooperative evolutionary algorithm with simulated annealing methods is proposed in consideration of the problem's characteristics. A three-link integer string is designed to represent three parts, i.e., factory assignment, machine assignment and operation sequence, in a solution. Three populations are constructed to search the above three parts via genetic operations and simulated annealing methods, respectively. A cooperation strategy is designed to generate a schedule. Then, a heuristic rule is devised to determine the delivery scheme for distribution. Finally, experiments are carried out via extending a set of benchmark instances. The proposed algorithm is compared to four state-of-the-art algorithms and an exact solver CPLEX. The results and discussions verify that the developed model and method have strong competitiveness for solving the considered problems.

Automatic Cable Harness Layout Routing in a Customizable 3D Environment

Designing cable harnesses can be time-consuming and complex due to many design and manufacturing aspects and rules. Automating the design process can help to fulfil these rules, speed up the process, and optimize the design. To accommodate this, we formulate a harness routing optimization problem to minimize cable lengths, maximize bundling by rewarding shared paths, and optimize the cables’ spatial location with respect to case-specific information of the routing environment, e.g., zones to avoid. A deterministic and computationally effective cable harness routing algorithm has been developed to solve the routing problem and is used to generate a set of cable harness topology candidates and approximate the Pareto front. Our approach was tested against a stochastic and an exact solver and our routing algorithm generated objective function values better than the stochastic approach and close to the exact solver. Our algorithm was able to find solutions, some of them being proven to be near-optimal, for three industrial-sized 3D cases within reasonable time (in magnitude of seconds to minutes) and the computation times were comparable to those of the stochastic approach.

Code and Data Repository for VRPSolverEasy: A Python Library for the Exact Solution of a Rich Vehicle Routing Problem

VRPSolverEasy is a Python package which provides a simple interface for VRPSolver, a state-of-the-art Branch-Cut-and-Price exact solver for vehicle routing problems (VRPs). The simplified interface is accessible for users without operations research background, meaning you do not need to know how to model your problem as an Integer Programming problem. As a price to pay for the simplicity, this interface is restricted to some standard VRP variants, which involve the following features and their combinations: capacitated vehicles customer time windows heterogeneous fleet multiple depots open routes optional customers with penalties parallel links to model transition time/cost trade-off incompatibilities between vehicles and customers customers with alternative locations and/or time windows To our knowledge, VRPSolver is the most efficient exact solver available for VRPs. Its particularity is to focus on finding and improving a lower bound on the optimal solution value of your instance. It is less efficient in finding feasible solutions but still can be better than available heuristic solvers for non-classic VRP variants.

Pyqcm: An open-source Python library for quantum cluster methods

Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

Codebase release 2.2 for Pyqcm

Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

Formal derivation of a new sediment transport model using a multiscale procedure: Numerical Validation

We study in this work structures formed under a unidirectional flow. The main problem that addresses sediment transport by aquatic flow. We present the formal derivation of a new sediment transport model by using a multiscale procedure in time and space. We consider two non-miscible layers of different materials with physical properties by including friction. Interactions at the fluid-sediment interface and those internally sediment are also taken into account. The derived model is the Saint–Venant–Exner type obtained by coupling a Saint-Venant system governing the hydrodynamic component and an Exner equation describing the morphodynamical component. The first part of this work takes into account two major contributions namely the formal derivation of new unidirectional Saint–Venant–Exner model inspired by the work in Fernández-Nieto et al. (2017) and the introduction of a new multiscale parameterization in space-time characterized by the combination of classical variability and rapid variability of solutions. However, we have limited ourselves to the case of a first order approximation which does not include viscosity terms. Further analysis of the second order approach leading to a viscous model can be deduced from the work in Fernández-Nieto et al. (2013). We also propose a numerical study of the derived model. Thus, we develop finite volume numerical schemes. The numerical approach employs a Riemann solver, and various options ranging from an exact solver to an approximate Riemann HLL-type and HLLC-type solvers. We did some numerical tests leading to consistent and effective results.

A hybrid deep-learning-metaheuristic framework for bi-level network design problems

This study proposes a hybrid deep-learning-metaheuristic framework with a bi-level architecture for road network design problems (NDPs). We train a graph neural network (GNN) to approximate the solution of the user equilibrium (UE) traffic assignment problem and use inferences made by the trained model to calculate fitness function evaluations of a genetic algorithm (GA) to approximate solutions for NDPs. Using three test networks, two NDP variants and an exact solver as benchmark, we show that on average, our proposed framework can provide solutions within 1.5 % gap of the best results in less than 0.5 % of the time used by the exact solution procedure. Our framework can be utilized within an expert system for infrastructure planning to determine the best infrastructure planning and management decisions under different scenarios. Given the flexibility of the framework, it can easily be adapted to many other decision problems that can be modeled as bi-level problems on graphs. Moreover, we foreseen interesting future research directions, thus we also put forward a brief research agenda for this topic. The key observation from our research that can shape future research is that the fitness function evaluation time using the inferences made by the GNN model was in the order of milliseconds, which points to an opportunity and a need for novel heuristics that 1) can cope well with noisy fitness function values provided by deep learning models, and 2) can use the significantly enlarged efficiency of the evaluation step to explore the search space effectively (rather than efficiently). This opens a new avenue for a modern class of metaheuristics that are crafted for use with AI-powered predictors.

Optimized Design of a Backbone Network for Public Transportation in Montevideo, Uruguay

This article presents the application of exact and metaheuristic approaches to the problem of designing the backbone network of a hierarchically public transportation system for Montevideo, Uruguay. This is a very relevant problem in nowadays smart cities, as it accounts for many social and environmental impacts and also affects the dynamics of the cities. The design of the proposed backbone network is conceived in combination with the bus network, with the main objective of improving the overall quality of service and reducing travel times. Three different variants of the problem are solved, considering different design premises. Exact solvers are proposed for simpler variants of the problem, which account for maximum resilience and bounded travel times. An evolutionary algorithm is proposed for a multiobjectie version of the problem that optimizes cost and quality of service. The main results indicate that the computed optimized designs provide reduced end-to-end travel times, which improve up to five times over the current system, and are economically viable to be implemented.