Exact Solver Research Articles

A customized multi-neighborhood search algorithm using the tabu list for a sustainable closed-loop supply chain network under uncertainty

Environmental pollution and social inequalities have prompted the agricultural industry to implement sustainable supply chain management practices. However, supply chain operations and planning in developing countries such as Iran are often not economically, environmentally, and socially sustainable. To address this challenge, this paper presents a practical optimization model for sustainable closed-loop supply chain (SCLSC) management in the agricultural industry of Iran, with a focus on the olive crop. Based on the triple bottom line concept, which considers economic, environmental, and social sustainability, the proposed model makes location, allocation, and inventory decisions under uncertainty by developing a scenario-based robust optimization model. To solve this complex network design problem, we propose a metaheuristic algorithm with a multi-neighborhood procedure that efficiently handles the complexity of the problem. Specifically, we develop a customized Simulated Annealing (SA) algorithm using a tabu list to improve the initial solution found by a constructive heuristic algorithm. Our extensive analysis and comparison of our metaheuristic algorithm against the exact solver and two other powerful metaheuristic algorithms demonstrate the applicability of our SCLSC model for the agricultural industry in Iran and the high performance of the proposed metaheuristic algorithm for solving large-scale networks.

FluxNet: A physics-informed learning-based Riemann solver for transcritical flows with non-ideal thermodynamics

Traditional Riemann solvers fall into two broad categories: exact solvers, which require multiple iterations to achieve high accuracy, and approximate linearized solvers, which achieve fast runtime at the expense of reduced accuracy. Here we explore learning-based Riemann solvers, termed FluxNets, as a new approach that balances accuracy and computational efficiency especially for transcritical and supercritical flow problems involving non-ideal thermodynamics. Various designs and training strategies are applied to fully connected neural networks. By comparing data-driven versus physics-informed loss functions, as well as neural networks of varying size, the results show that order-of-magnitude reductions in error compared to the Roe solver can be achieved with compact architectures. Numerical validation on 1D and 2D test cases reveals that a physics-informed FluxNet promotes smoothness, numerical stability, and generalizability, and is capable of producing entropy-stable results free of non-physical expansion shocks as compared to the Roe solver. Parallelization can be leveraged to accelerate inference such that the gains in accuracy are achieved at one quarter the runtime of exact solvers. The trade-off in accuracy versus efficiency may be justified in the case of non-ideal flows where even minor errors can result in spurious oscillations and instabilities.

A machine learning optimization approach for last-mile delivery and third-party logistics

Third-party logistics is now an essential component of efficient delivery systems, enabling companies to purchase carrier services instead of an expensive fleet of vehicles. However, carrier contracts have to be booked in advance without exact knowledge of what orders will be available for dispatch. The model describing this problem is the variable cost and size bin packing problem with stochastic items. Since it cannot be solved for realistic instances by means of exact solvers, in this paper, we present a new heuristic algorithm able to do so based on machine learning techniques. Several numerical experiments show that the proposed heuristics achieve good performance in a short computational time, thus enabling its real-world usage. Moreover, the comparison against a new and efficient version of progressive hedging proves that the proposed heuristic achieves better results. Finally, we present managerial insights for a case study on parcel delivery in Turin, Italy.

An athlete–referee dual learning system for real-time optimization with large-scale complex constraints

Constrained optimization (CO) has made a profound impact in solving many real-world problems. Due to the high computation burden in exact solvers, data-driven CO based on machine learning techniques is recently receiving extensive research interests for its capability to solve CO problems in real time. The existing data-driven CO approaches only serve for optimization problems with rather simple constraints that can be directly incorporated into model training. However, constraints that are computationally infeasible or burdensome to evaluate are commonly experienced in realistic optimization applications, especially in the engineering sector. This paper proposes an athlete–referee dual learning system (ARDLS) for end-to-end CO with large-scale complex constraints, where an athlete model is trained as the main optimizer while a referee model is trained as a probabilistic constraint classifier to guide the athlete training. A risk-based constrained loss function is designed to fine-tune the athlete model for constraint satisfaction. A case study on electric power system emergency control application is conducted to validate the proposed ARDLS, where the testing results demonstrate the excellent capability of ARDLS to improve the likelihood of satisfying large-scale complex constraints in CO.

Convergence of first‐order finite volume method based on exact Riemann solver for the complete compressible Euler equations

AbstractRecently developed concept of dissipative measure‐valued solution for compressible flows is a suitable tool to describe oscillations and singularities possibly developed in solutions of multidimensional Euler equations. In this paper we study the convergence of the first‐order finite volume method based on the exact Riemann solver for the complete compressible Euler equations. Specifically, we derive entropy inequality and prove the consistency of numerical method. Passing to the limit, we show the weak and strong convergence of numerical solutions and identify their limit. The numerical results presented for the spiral and the Richtmyer‐Meshkov problem are consistent with our theoretical analysis.

Solution approaches for the bi-objective Skiving Stock Problem

The Skiving Stock Problem (SSP) aims to determine an optimal plan for producing as many large objects as possible by combining small items. The skiving process may need different considerations depending on the production environment and the product characteristics. In this study, we address bi-objective 1D-SSP with two conflicting objectives. One common objective is to minimize the trim loss remaining after skiving, as removing the excess width is an extra procedure. When welding is an element of the skiving process, increasing the number of items for each product indicates compromised quality. Therefore, minimizing the number of small items for each product becomes a primary objective in such cases. To solve this bi-objective version of the NP-hard problem, we implement a Lexicographic Method (LM) in which the importance of the objectives imposes their preference orders. We propose two methodologies within the LM framework. The first methodology integrates Column Generation (CG) and Branch & Bound (B&B) to search for an exact solution. Given the excessive computational time an exact solver may require for tight or large-sized problems, we propose a heuristic method integrating the Dragonfly Algorithm (DA) and a Constructive Heuristic (CH). Real-world application results validate the exact solver and demonstrate comparable results for the heuristic solver in terms of solution quality and computational time. The efficiency of the solution methodologies for a preemptive multi-objective SSP aims to support decision-makers with make-or-buy decisions.

When architecture meets AI: A deep reinforcement learning approach for system of systems design

How to design System of Systems has been widely concerned in recent years, especially in military applications. This problem is also known as SoS architecting, which can be boiled down to two subproblems: selecting a number of systems from a set of candidates and specifying the tasks to be completed for each selected system. Essentially, such a problem can be reduced to a combinatorial optimization problem. Traditional exact solvers such as branch-bound algorithm are not efficient enough to deal with large scale cases. Heuristic algorithms are more scalable, but if input changes, these algorithms have to restart the searching process. Re-searching process may take a long time and interfere with the mission achievement of SoS in highly dynamic scenarios, e.g., in the Mosaic Warfare. In this paper, we combine artificial intelligence with SoS architecting and propose a deep reinforcement learning approach DRL-SoSDP for SoS design. Deep neural networks and actor–critic algorithms are used to find the optimal solution with constraints. Evaluation results show that the proposed approach is superior to heuristic algorithms in both solution quality and computation time, especially in large scale cases. DRL-SoSDP can find great solutions in a near real-time manner, showing great potential for cases that require an instant reply. DRL-SoSDP also shows good generalization ability and can find better results than heuristic algorithms even when the scale of SoS is much larger than that in training data.

A positivity‐preserving Lagrangian discontinuous Galerkin scheme with exact Riemann solver for gas‐water compressible flows

AbstractIn this study, a new cell‐centered Lagrangian discontinuous Galerkin (DG) scheme is presented to simulate gas‐water compressible flows. The two‐phase flows are governed by the compressible Euler equation with ideal and stiffened gas equations of state. We integrate the Lagrangian DG scheme with the exact gas‐water Riemann solver for calculating the numerical fluxes so that the inherent stiff features of gas‐water compressible flows are well addressed. Furthermore, in order to guarantee the positivity of the density and internal energy during the high Mach number calculation, the positivity‐preserving limiter and strong stability preserving temporal integral are incorporated to ensure the numerical stability. Six numerical examples are tested to show the accuracy, robustness and positivity‐preserving property of the present scheme. It can be found that the present scheme can handle challenging numerical cases involving large density ratio (up to 1000) and strong shock. The results obtained with the typical HLLC flux are also given for comparison and the present scheme with the exact Riemann solver shows higher accuracy, particularly in the presence of large density ratio at the gas‐water interface.

Numerical Investigation into the Performance of an OWC Device under Regular and Irregular Waves

A numerical investigation into the hydrodynamic efficiency of an oscillating water column (OWC) device for the production of energy from sea waves under the conditions of regular and irregular waves is proposed. The numerical simulations were carried out using a two-dimensional version of a recently published three-dimensional free-surface nonhydrostatic numerical model, which is based on a conservative form of the contravariant Navier–Stokes equations written for a moving co-ordinate system. The governing equations are spatially discretized by a finite volume shock-capturing scheme based on high-order wave-targeted essentially nonoscillatory reconstructions and an exact Riemann solver. Time discretization was performed by a predictor-corrector method that took into account the nonhydrostatic pressure component. The proposed numerical model allowed us to highlight the significant differences between the hydrodynamic efficiency obtained under irregular waves and those obtained under regular monochromatic waves and provides more realistic evaluations of the OWC device performances. The results of the above comparison showed a reduction in the hydrodynamic efficiency of the OWC from 0.78 to about 0.54 when passing from regular waves to the corresponding irregular ones. The model was applied to assess the potential energy production obtainable by a set of OWCs at the Cetraro harbor (southern Italy). The numerical results show that, by adopting the optimal dimensions of the OWC, the estimated mean annual energy production obtainable at the Cetraro harbor is equal to 1540.52 MWh, which corresponds to the energy production of about 10 wind turbines with a nominal power of 60 KW.

Hybrid divide-and-conquer approach for tree search algorithms

One of the challenges of quantum computers in the near- and mid- term is the limited number of qubits we can use for computations. Finding methods that achieve useful quantum improvements under size limitations is thus a key question in the field. In this vein, it was recently shown that a hybrid classical-quantum method can help provide polynomial speed-ups to classical divide-and-conquer algorithms, even when only given access to a quantum computer much smaller than the problem itself. In this work, we study the hybrid divide-and-conquer method in the context of tree search algorithms, and extend it by including quantum backtracking, which allows better results than previous Grover-based methods. Further, we provide general criteria for polynomial speed-ups in the tree search context, and provide a number of examples where polynomial speed ups, using arbitrarily smaller quantum computers, can be obtained. We provide conditions for speedups for the well known algorithm of DPLL, and we prove threshold-free speed-ups for the PPSZ algorithm (the core of the fastest exact Boolean satisfiability solver) for well-behaved classes of formulas. We also provide a simple example where speed-ups can be obtained in an algorithm-independent fashion, under certain well-studied complexity-theoretical assumptions. Finally, we briefly discuss the fundamental limitations of hybrid methods in providing speed-ups for larger problems.

QBoost for regression problems: solving partial differential equations

A hybrid algorithm based on machine learning and quantum ensemble learning is proposed to find an approximate solution to a partial differential equation with good precision and favorable scaling in the required number of qubits. The classical component consists in training several regressors (weak-learners), capable of solving a partial differential equation approximately using machine learning. The quantum component consists in adapting the QBoost algorithm to solve regression problems to build an ensemble of classical learners. We have successfully applied our framework to solve the 1D Burgers’ equation with viscosity, showing that the quantum ensemble method really improves the solutions produced by classical weak-learners. We also implemented the algorithm on the D-Wave Systems, confirming the good performance of the quantum solution compared to the simulated annealing and exact solver methods.

A deterministic bounding algorithm vs. a hybrid meta-heuristic to deal with a bilevel mixed-integer nonlinear optimization model for electricity dynamic pricing

In the electricity retail market, the retailer company aims to determine the optimal time-of-use (ToU) prices to maximize profits resulting from buying energy in organized (long-term, day-ahead, balancing) markets and selling it to consumers. Therefore, the retailer should take into account the consumer’s demand response actions to minimize the electricity bill in face of time varying prices. In this paper, this problem is formulated as a bilevel mixed-integer nonlinear programming model in which the retailer is the leader and the consumer is the follower. The consumer’s problem encompasses the integrated optimization of all home energy resources, considering re-scheduling appliance operation, charging/discharging of electric vehicle and stationary batteries, local microgeneration and (buying and selling) exchanges with the grid. The accurate physical modelling of appliance operation to generate effective load scheduling solutions imposes a high computational burden. Two algorithms are proposed to address this problem: a deterministic bounding algorithm (DBA) using an optimal-value-function approach for bilevel optimization, and a hybrid meta-heuristic using a particle swarm optimization algorithm to tackle the upper-level problem that calls an exact mixed-integer linear programming solver to deal with the lower-level problem. In the framework of the DBA, three different techniques were implemented to deal with the nonlinearities arising from the products of integer and continuous variables (bilinear terms): 1) solving the (non-convex) subproblems of DBA using a mixed-integer nonlinear solver, 2) using the McCormick envelopes to approximate the bilinear terms by linear ones, and 3) expressing the integer variables by binary ones and linearizing the bilinear terms using an exact form. Computational experiments are presented and discussed for real data settings of the problem under study considering a computational budget to compare the different algorithms and techniques employed. The results showed that the DBA with the approximate linearization technique (2) leads to the best solutions.

Deep learning-enabled MCMC for probabilistic state estimation in district heating grids

Flexible district heating grids form an important part of future, low-carbon energy systems. We examine probabilistic state estimation in such grids, i.e., we aim to estimate the posterior probability distribution over all grid state variables such as pressures, temperatures, and mass flows conditional on measurements of a subset of these states. Since the posterior state distribution does not belong to a standard class of probability distributions, we use Markov Chain Monte Carlo (MCMC) sampling in the space of network heat exchanges and evaluate the samples in the grid state space to estimate the posterior. Converting the heat exchange samples into grid states by solving the non-linear grid equations makes this approach computationally burdensome. However, we propose to speed it up by employing a deep neural network that is trained to approximate the solution of the exact but slow non-linear solver. This novel approach is shown to deliver highly accurate posterior distributions both for classic tree-shaped as well as meshed heating grids, at significantly reduced computational costs that are acceptable for online control. Our state estimation approach thus enables tightening the safety margins for temperature and pressure control and thereby a more efficient grid operation.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

The Shallow Water equations model is commonly used to reproduce a wide variety of environmental flows. Its applications include river hydraulics and flood forecasting.Therefore, an algorithm to solve the Riemann problem for the Shallow Water equations can be a considerable asset for the development of numerical schemes devoted to the solution of such equations. Nevertheless, for real world applications, it becomes mandatory to consider the influence of the irregular shape of the cross section of natural channels in the solution.This work presents an energy based exact Riemann solver for the Shallow Water equations for channels with an arbitrarily shaped cross section with the presence of a non flat bottom and dry states.The procedure for solving the Riemann problem involves the solution of a system of non linear algebraic equations, for which a CPU time efficient solution strategy based on nested Newton-Raphson methods is presented.The proposed solver can be used to develop Finite Volume or Discontinuous Galerkin schemes, based on the Godunov method. For a non flat bottom, the Shallow Water equations are a non conservative system, once the bottom elevation is included in the state variables. Many available schemes take advantage of this fact to use approximate path-conservative Riemann solvers. In other cases, the term related to the non flat bottom is treated as a source term. The use of this exact solver avoids the necessity of considering the bottom topography as a source term and of reverting to approximate solvers. These strategies can in facts pose numerical problems in situations such as transonic rarefactions and flows approaching abrupt changes in bottom topography.The solver is tested by reproducing the solution of Riemann problems with a Godunov finite volume scheme. An analysis of the performance of the solver shows that it requires less computational time than path-conservative schemes and that it has better stability and convergence properties, especially in situations involving resonant waves, such as transonic flows across a bottom step.

Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria

The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.

Fermi arcs from dynamical variational Monte Carlo

Variational Monte Carlo is a many-body numerical method that scales well with system size. It has been extended to study the Green function only recently by Charlebois and Imada [Phys. Rev. X 10, 041023 (2020)]. Here we generalize the approach to systems with open boundary conditions in the absence of translational invariance. Removing these constraints permits the application of embedding techniques like cluster perturbation theory (CPT). Using this approach we provide robust evidence of the existence of Fermi arcs in the one-band Hubbard model, an enduring problem in the physics of the pseudogap in cuprate high-temperature superconductors. We study the behavior of the Fermi surface and of the density of states as a function of hole doping for clusters of up to 64 sites, well beyond the reach of modern exact diagonalization solvers. We observe that the technique reliably captures the transition from a Mott insulator at half filling to a pseudogap, evidenced by the formation of Fermi arcs, and finally to a metallic state at large doping. The ability to treat large clusters with quantum cluster methods helps to minimize potential finite-size effects and enables the study of systems with long-range orders, which will help extend the reach of these already powerful methods and provide important insights on the nature of various strongly correlated many-electron systems, including the high-${\mathrm{T}}_{c}$ cuprate superconductors.

Two-Level Optimization to Reduce Waiting Time at Locks in Inland Waterway Transportation

Inland vessels often have to cross numerous locks before reaching their final destination, which leads to a significant delay and sometimes represents as much as half of the total travel time. The delay affects shipment costs and can affect other parts of the transport chain, adversely impacting this transportation mode’s growth. Therefore, this work presents a two-level solution to ensure a shorter waiting time at locks and improve inland waterway transport. On the one hand, the first level focuses on making infrastructural modifications by proposing an efficient Lock Automation Decision Making (Lock-ADM) method. The problem modeling consists of using a three-stage algorithm. Firstly, we calculate the optimal number of locks while minimizing the investment costs using the exact solver, CPLEX. Secondly, we measure the importance of locks in the network, and finally, we select the best locks to automate using the Genetic Algorithm (GA) metaheuristic. Based on real data, we achieved an average reduction of 33.7% in overall lock waiting time at a low cost. On the other hand, the second level proposes a Dynamic Lock Scheduling (Lock-DS) to efficiently manage vessels scheduling at locks by minimizing their waiting time and optimizing their speed. We achieve an average reduction of 69.9% in vessel waiting time and a reduction of 48.03% in total fuel consumption compared to existing scheduling methods. Automating the most important locks with Lock-ADM and managing their crossing with Lock-DS ensure shorter vessels’ waiting time and represent a significant first step towards the automation of inland navigation.

Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy

AbstractWe study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle‐point problems with a two‐times‐two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.

Neuro-Ising: Accelerating Large-Scale Traveling Salesman Problems via Graph Neural Network Guided Localized Ising Solvers

One of the most extensively studied combinatorial optimization problems is the Travelling Salesman Problem (TSP). Considerable research efforts in the past have resulted in exact solvers. However, the runtime of such hand-crafted solutions increases exponentially with problem size. Ising model based solvers have also gained prominence due to their abilities to find fast and approximate solutions for combinatorial optimization problems. However, such Ising based heuristics also suffer from scalability as the solution quality becomes increasingly sub-optimal with increase in problem size. In this work, we propose Neuro-Ising – a machine learning framework which uses Ising models to find clusters of near-optimal partial solutions of large scale TSPs and combines those solutions by employing a supervised data driven mechanism, which we model as a Graph Neural Network (GNN). The GNN is trained from solution instances obtained through exact solvers and hence, the proposed approach generalizes to unseen problems while avoiding the run-time complexity otherwise required, if the solution is built from scratch. Using standard computing resources, our proposed framework rapidly converges to near-optimal solutions for 15 TSPs (upto <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 5k$ </tex-math></inline-formula> cities) from the TSPLib benchmark suite. We report <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 10.66\times $ </tex-math></inline-formula> speedup over Tabu Search for 8 problems. Furthermore, compared to two state-of-the-art clustering-based TSP solvers, Neuro-Ising achieves <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 38 \times $ </tex-math></inline-formula> faster convergence along with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 8.9\%$ </tex-math></inline-formula> better quality of solution, on average.

Numerical studies of two-phase water hammer flows using Godunov methods

In this paper, an explicit finite volume Godunov-type scheme is applied for a two-phase homogeneous water hammer flow model. The developed finite volume method is based on an exact Riemann solver for a gas–liquid two-phase flow model. This paper investigates the case of a water hammer problem caused by a rapid closing valve located in upstream of the pipeline. The performance of the proposed numerical method is validated against experimental data from literature. The good agreement found between the numerical and experimental results confirms the accuracy and the robustness of both the numerical methods and the proposed exact Riemann solver.