Algorithm For Large-scale Problems Research Articles

A novel memory-based simulated annealing algorithm to solve multi-line facility layout problem

In this paper, a memory-based simulated annealing algorithm called the Dual Memory Simulated Annealing Algorithm (DMSA) is presented to solve multi-line facility layout problems. The objective is to minimize the total material handling cost. Two memory buffers and a restart mechanism are considered. Two benchmark problems were selected from the literature review papers and solved using the standard simulated annealing (SA) algorithm and the DMSA. The obtained results show that solutions provided by the DMSA algorithm are cost-effective compared to the standard SA algorithm and the other algorithms used for solving these test cases. Moreover, to further evaluate the performance of the DMSA algorithm in large scale problems, eleven test cases were selected from the benchmark library of the quadratic assignment problem (QAP). According to the results, the performance of the algorithm in finding solutions to complex problems is exemplary.

Energy-Efficient General PoI-Visiting by UAV with a Practical Flight Energy Model

Unmanned aerial vehicles (UAVs) are being widely exploited for various applications, <i>e.g</i>., traversing to collect data from ground sensors, patrolling to monitor key facilities, moving to aid mobile edge computing. We summarize these UAV applications and formulate a problem, namely the <i>general waypoint-based PoI-visiting problem</i>. Since energy is critical due to the limited onboard storage capacity, we aim at minimizing flight energy consumption. In our problem, we pay special attention to the energy consumption for turning and switching operations on flight planning, which are usually ignored in the literature but play an important role in practical UAV flights according to our real-world measurement experiments. We propose specially designed graph parts to model the turning and switching cost and thus transfer the problem into a classic graph problem, <i>i.e</i>., general traveling salesman problem, which can be efficiently solved. Theoretical analysis shows that such problem transformation has the graph redefinition approximation ratio upper bound, <inline-formula><tex-math notation="LaTeX">$max\lbrace \Theta /\delta ,2\rbrace$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\Theta$</tex-math></inline-formula> is related to the designed graph parts and <inline-formula><tex-math notation="LaTeX">$\delta$</tex-math></inline-formula> is a constant. Finally, we evaluate our proposed algorithm by simulations. The results show that it costs less than 107&#x0025; of the optimal minimum energy consumption for small scale problems and costs only 50&#x0025; as much energy as a naive algorithm for large scale problems.

An optimization model for evacuating people with disability in extreme disaster conditions: A case study

The increasing severity and frequency of disasters have posed major challenges for people. Amongst, the risks of fatalities and injuries of people with disabilities (PWDs) have significantly increased. The Sendai Framework for Disaster Risk Reduction (SFDRR) initiated a movement to create a "disability-accessible and inclusive environment" which highlighted the problems PWDs faced during disasters. One of the most important issues is providing evacuation and accommodation according to the special needs of PWDs. In this study, a MILP model is proposed to pick up PWDs from different locations and transfer them to shelters. Throughout this research, diverse disabilities, heterogonous vehicles, compatibility types of disabilities and vehicles, multi-depot and adept and amateur operators were considered to help evacuate PWDs. Additionally, 27 problems are solved to examine the efficacy of (μ+1) EA algorithm in large scale problems. Subsequently, a real case study with 500 nodes including pick up, shelters, and depot nodes are analyzed. The computational results illustrate that by adding small-sized (car) and medium-sized (Van) vehicles to the current fleet, the time for tours traveled significantly reduces. Finally, a sensitivity analysis has been conducted to prepare some managerial implications for crisis managers during the occurrence of disasters to help PWDs during evacuation.

A Double Traveling Salesman Problem With Three-Dimensional Loading Constraints for Bulky Item Delivery

This article considers bulky item delivery problems in which multiple items are retrieved and loaded onto a vehicle from different warehouses and then delivered. This problem is described as a double traveling salesman problem with three-dimensional container loading constraints with multiple stacks. The double TSP with multiple stacks is used to determining the shortest route performing pickups and deliveries in two separated networks (one for pickups and one for deliveries) using only one container. Repacking is not allowed after loading the items into the container. An integer linear programming model is proposed to solve this problem, a standard genetic algorithm and an improved genetic algorithm is designed. In the improved genetic algorithm, a Lin-Kernighan algorithm is used to improve the delivery route, a k-means clustering algorithm, and a heuristic packing scheme improvement rules work together to improve the loading route. The results show that the improved genetic algorithm is superior to the standard genetic algorithm in large scale problems.

Convergence of Newton-MR under Inexact Hessian Information

Recently, there has been a surge of interest in designing variants of the classical Newton-CG in which the Hessian of a (strongly) convex function is replaced by suitable approximations. This is mainly motivated by large-scale finite-sum minimization problems that arise in many machine learning applications. Going beyond convexity, inexact Hessian information has also been recently considered in the context of algorithms such as trust-region or (adaptive) cubic regularization for general nonconvex problems. Here, we do that for Newton-MR, which extends the application range of the classical Newton-CG beyond convexity to invex problems. Unlike the convergence analysis of Newton-CG, which relies on spectrum preserving Hessian approximations in the sense of Löwner partial order, our work here draws from matrix perturbation theory to estimate the distance between the range spaces underlying the exact and approximate Hessian matrices. Numerical experiments demonstrate a great degree of resilience to such Hessian approximations, amounting to a highly efficient algorithm in large-scale problems.

Integrated routing optimization for post-disaster rapid-detailed need assessment

The post-disaster need assessment for the affected area is of significance for the effective distribution of relief supplies. This paper innovatively concentrates on the integrated routing optimization problem for the rapid need assessment stage and the detailed need assessment stage. A bi-objective model with consideration of the effects of site familiarity and route familiarity is proposed. The proposed nonlinear model is equivalently transformed into a mixed-integer linear programming. A tabu search algorithm is designed to solve the proposed model, and its solving accuracy is validated by comparing with the solution of CPLEX based on a small-scale problem. To demonstrate the efficiency of the proposed model and algorithm for large-scale problems, the case of Ya'an earthquake in China is investigated. The experimental results reveal that comparing to the separate routing optimization and the sequential routing optimization, the integrated routing optimization can achieve a better coordination between the decisions of two stages.

An Efficient v-Minimum Absolute Deviation Distribution Regression Machine

Support Vector Regression (SVR) and its variants are widely used regression algorithms, and they have demonstrated high generalization ability. This research proposes a new SVR-based regressor: v-minimum absolute deviation distribution regression (v-MADR) machine. Instead of merely minimizing structural risk, as with v-SVR, v-MADR aims to achieve better generalization performance by minimizing both the absolute regression deviation mean and the absolute regression deviation variance, which takes into account the positive and negative values of the regression deviation of sample points. For optimization, we propose a dual coordinate descent (DCD) algorithm for small sample problems, and we also propose an averaged stochastic gradient descent (ASGD) algorithm for large-scale problems. Furthermore, we study the statistical property of v-MADR that leads to a bound on the expectation of error. The experimental results on both artificial and real datasets indicate that our v-MADR has significant improvement in generalization performance with less training time compared to the widely used v-SVR, LS-SVR, e-TSVR, and linear e-SVR. Finally, we open source the code of v-MADR at https://github.com/AsunaYY/v-MADR for wider dissemination.

The Large-Scale Periodic Home Health Care Server Assignment Problem: A Region-Partition-Based Algorithm

This article addresses the periodic home health care (HHC) server assignment problem in HHC companies. This problem is a variant of the periodic vehicle routing problem with specific constraints for customers and servers (e.g., customers’ requirements for multiple medical skills and continuous care offered by a given number of servers and servers’ requirements for workload balance). Solving large-scale problems from practical applications of HHC companies within an acceptable computation time is challenging. Consequently, we develop an efficient region-partition-based algorithm to solve these large-scale problems. First, the algorithms assign customers and servers to many independent regions. Second, four different tabu search (TS) algorithms are designed to solve the optimization problem of each region. Finally, the algorithms iteratively adjust the assignment of customers and servers to regions and solve the problems of each region. The performances of different TS algorithms are discussed. The effectiveness of the proposed region-partition-based algorithm for large-scale problems is validated. Note to Practitioners —This article is motivated by our collaboration with an HHC company in Shanghai, China. The company delivers medical, paramedical, and social services to customers in their homes to help them improve their clinical and psychological conditions without hospital stays. In practice, customers require long-term and periodic service (e.g., weekly), which may last many months or even years. HHC companies have to assign a combination of service days and at least one server to each customer. Due to specific constraints in practical operations and the large scale of the practical problems (e.g., 1000–2000 customers) of the HHC in this article, the server assignment problem is challenging to the HHC company. The algorithm presented in this article takes into account the special constraints of the HHC and is able to solve the large-scale problem in a reasonable time span. The algorithm can be used to help decision-makers assign combinations of service days and servers to customers with the objective of balancing the workloads among servers.

The multi-route location-routing problem and zone price decision-making using a tabu and variable neighborhood search algorithm

Zone pricing is one of the most important pricing policies that are prevalently used by very companies. This paper considers integrating zone pricing and location-routing problems for profit maximization. Hence, a node-based model is proposed as mixed integer nonlinear programming to solve the problem. In order to handle the nonlinearity elements and change the model to a mixed integer linear programming, a piecewise linearization method is employed to approximate the problem. Moreover, we introduce a new heuristic variable neighborhood search (VNS) and tabu search algorithm for large-scale problems. The proposed algorithms consist of four intra-tour and seven inter-tour operators in its local search. Finally, some test instances with 20, 200, 400, and 600 nodes are solved and the results of them confirm the efficiency of the proposed algorithms and comparative superiority of the VNS algorithm over the tabu search algorithm.

Posterior Constraint Selection for Nonnegative Linear Programming

Posterior constraint optimal selection techniques (COSTs) are developed for nonnegative linear programming problems (NNLPs), and a geometric interpretation is provided. The posterior approach is used in both a dynamic and non-dynamic active-set framework. The computational performance of these methods is compared with the CPLEX standard linear programming algorithms, with two most-violated constraint approaches, and with previously developed COST algorithms for large-scale problems.

Customized algorithms for growing connected resistive networks

Abstract: We consider the problem of adding edges to connected resistive networks in order to optimally enhance their performance. The performance is captured by the H 2 norm of the closed-loop network and the l 1 regularization is introduced as a means to promote sparsity of the controller graph Laplacian. The resulting optimal control problem can be cast as a semidefinite program and standard interior point method solvers can be used to efficiently compute the optimal solution for small and medium size networks. In this paper, we develop two efficient customized algorithms for large-scale problems. Our customized algorithms are based on the proximal gradient method and the sequential quadratic approximation method. In the latter, the Newton direction is obtained using coordinate descent algorithm over the set of active variables. We provide comparison of these methods and show that both of them can be effectively employed to solve topology identification and optimal design problems for large-scale networks.

Scheduling of jobs with cross families in two stage manufacturing systems

This paper studies scheduling of n jobs in a two stage manufacturing system to minimize makespan when each stage has its own job families and families in each stage require sequence independent setups. In a manufacturing system that each stage has its own family; jobs belonging to different families in one stage may belong to the same family in another stage, and we call it cross families. To the best of our knowledge, there is no study on cross families in scheduling literature. Our problem is NP-hard for arbitrary number of families, and therefore we study the problem with fixed number of families. We first analyze some properties of the optimal schedule and show that Johnson sequence is optimal for jobs belonging to the same family on both machines. We develop an efficient branch and bound algorithm with complexity of O(nc), where c is a constant and a hybrid genetic algorithm for large scale problems, using the properties of the optimal schedule. Finally, we present computational experiment to demonstrate the effectiveness of our algorithms.

A new Bayesian approach to nonnegative matrix factorization: Uniqueness and model order selection

NMF is a blind source separation technique decomposing multivariate non-negative data sets into meaningful non-negative basis components and non-negative weights. There are still open problems to be solved: uniqueness and model order selection as well as developing efficient NMF algorithms for large scale problems. Addressing uniqueness issues, we propose a Bayesian optimality criterion (BOC) for NMF solutions which can be derived in the absence of prior knowledge. Furthermore, we present a new Variational Bayes NMF algorithm VBNMF which is a straight forward generalization of the canonical Lee–Seung method for the Euclidean NMF problem and demonstrate its ability to automatically detect the actual number of components in non-negative data.

Capital renewal optimisation for large-scale infrastructure networks: genetic algorithms versus advanced mathematical tools

Civil infrastructure assets require continuous renewal (repair, rehabilitation or replacement) actions to modernise the inventory and sustain its operability. Allocating limited renewal funds among numerous asset components, however, represents a complex optimisation problem. Earlier efforts using genetic algorithms (GAs) could optimise small size problems yet exhibiting steep degradation in solution quality as problem size increases. Even by applying sophisticated mechanisms such as ‘segmentation’ to improve the performance of GAs, large processing time hinders the practicality of the algorithm for large-scale problems. This article, therefore, aims at improving both processing speed and solution quality for very large-scale problems (up to 50,000 assets). The article develops optimisation models using an advanced modelling tool (GAMS/CPLEX), and compares its results with GAs on three different model formulations. Both approaches proved to be beneficial, yet the advanced mathematical approach showed superior performance.

Computing minimal hitting set based on immune genetic algorithm

The primary bottleneck of many model-based diagnosis approaches is computing minimal hitting sets (MHSs). Most existing algorithms for computing MHSs are deterministic. Such algorithms are sound and complete but with the complexity of exponential, since the problem of computing MHSs is NP-hard. To reduce the complexity, especially for large systems, we present a novel approach for multiple fault diagnosis, based on an immune genetic algorithm (IGA). This approach maps a hitting set problem onto an integer programming problem, followed by obtaining the lower bound on the size of the solution. The use of the bound in the immune operator guides the MHS search. An elite-set strategy and an appropriate termination criterion are proposed to improve the efficiency of the approach. Experimental results show that our approach solves MHSs faster than the traditional deterministic algorithm in large-scale problems. Compared with the incomplete algorithm, the speed and accuracy of our approach are also significantly improved.

A dwindling filter inexact projected Hessian algorithm for large scale nonlinear constrained optimization

In this paper, we propose a dwindling filter inexact projected Hessian algorithm for solving large scale nonlinear constrained optimization. For large-scale applications, inexact projected Hessian algorithm is needed to get search direction by solving one or more linear systems approximately using iterative linear algebra techniques. The envelope of the dwindling filter becomes thinner and thinner as the step size approaches zero so that the new filter has more flexibility for the acceptance of the trial step compared with traditional filter. Under mild conditions, global convergence and local superlinear convergence rate are obtained. The numerical experiments are reported to show the effectiveness of the proposed algorithm for large scale problems.

An observation‐space formulation of variational assimilation using a restricted preconditioned conjugate gradient algorithm

AbstractWe consider parameter estimation problems involving a set of m physical observations, where an unknown vector of n parameters is defined as the solution of a nonlinear least‐squares problem. We assume that the problem is regularized by a quadratic penalty term. When solution techniques based on successive linearization are considered, as in the incremental four‐dimensional variational (4D‐Var) techniques for data assimilation, a sequence of linear systems with particular structure has to be solved. We exhibit a subspace of dimension m that contains the solution of these linear systems, and derive a variant of the conjugate gradient algorithm that is more efficient in terms of memory and computational costs than its standard form, when m is smaller than n. The new algorithm, which we call the Restricted Preconditioned Conjugate Gradient (RPCG), can be viewed as an alternative to the so‐called Physical‐space Statistical Analysis System (PSAS) algorithm, which is another approach to solve the linear problem. In addition, we show that the non‐monotone and somehow chaotic behaviour of PSAS algorithm when viewed in the model space, experimentally reported by some authors, can be fully suppressed in RPCG.Moreover, since preconditioning and re‐orthogonalization of residuals vectors are often used in practice to accelerate convergence in high‐dimension data assimilation, we show how to reformulate these techniques within subspaces of dimension m in RPCG. Numerical experiments are reported, on an idealized data assimilation system based on the heat equation, that clearly show the effectiveness of our algorithm for large‐scale problems. Copyright © 2009 Royal Meteorological Society

A Finite Algorithm for Almost Linear Complementarity Problems

We present an algorithm for solving a class of nonlinear complementarity problems called the almost linear complementarity problem (ALCP), which can be used to simulate free boundary problems. The algorithm makes use of a procedure for identifying an active index subset of an ALCP by bounding its solution with an interval vector. It is shown that an acceptable solution of the given ALCP can be obtained by solving at most n systems of equations. Numerical results are reported to illustrate the efficiency of the algorithm for large-scale problems.

Parallel Elastic Finite Element Analysis Using the Balancing Domain Decomposition

The iterative Domain Decomposition Method (DDM) is one of the most effective parallel algorithm for large scale problems due to its excellent parallelism and variety of researches have already been done. As the iterative DDM satisfies continuity among subdomains through the iterative calculations, it is indispensable to reduce the number of iterations with a preconditioning technique especially for speed-up for computing. Thus, we have chosen the Balancing Domain Decomposition (BDD) proposed by J. Mandel. The BDD is a Neumann-Neumann type preconditioner with a coarse grid correction, and then its convergence estimates are independent of the number of subdomains. In this research, we introduce the BDD algorithm and how to apply it for elastic problems, then we develop a parallel BDD based on the Hierarchical DDM (HDDM) and apply it to some numerical examples on parallel computers.

FINITE ELEMENT ALGORITHMS FOR DYNAMIC SIMULATIONS OF VISCOELASTIC COMPOSITE SHELL STRUCTURES USING CONJUGATED GRADIENT METHOD ON COARSE GRAINED AND MASSIVELY PARALLEL MACHINES

Recently much attention has been paid to high-performance computing and the development of parallel computational strategies and numerical algorithms for large-scale problems. In this present study, a finite element procedure for the dynamic analyses of anisotropic viscoelastic composite shell structures by using degenerated 3-D elements has been studied on vector and coarse grained and massively parallel machines. CRAY hardware performance monitors such as Flowtrace and Perftrace tools are used to obtain performance data for subroutine program modules and specified code segments. The performances of conjugated gradient method, the Cray sparse matrix solver and the Feable solver are evaluated. SIMD and MIMD parallel implementation of the finite element algorithm for dynamic simulation of viscoelastic composite structures on the CM-5 is also presented. The performance studies have been conducted in order to evaluate efficiency of the numerical algorithm on this architecture versus vector processing CRAY systems. Parametric studies on the CM-5 as well as the CRAY system and benchmarks for various problem sizes are shown. The second study is to evaluate how effectively the finite element procedures for viscoelastic composite structures can be solved in the Single Instruction Multiple Data (SIMD) parallel environment. CM-FORTRAN is used. A conjugate gradient method is employed for the solution of systems. In the third study, we propose to implement the finite element algorithm in a scalable distributed parallel environment using a generic message passing library such as PVM. The code is portable to a range of current and future parallel machines. We also introduced the domain decomposition scheme to reduce the communication time. The parallel scalability of the dynamic viscoelastic finite element algorithm in data parallel and scalable distributed parallel environments is also discussed. © 1997 by John Wiley & Sons, Ltd.