An identity for multivariate Bernstein polynomials

We prove an identity for multivariate Bernstein polynomials on simplices, which may be considered a pointwise orthogonality relation. Its integrated version provides a new representation for the polynomial dual basis of Bernstein polynomials. An identity for the reproducing kernel is used to define quasi-interpolants of arbitrary order.     

An efficient algorithm for decomposing multivariate polynomials and its applications to cryptography

In this paper, we present an efficient and general algorithm for decomposing multivariate polynomials of the same arbitrary degree. This problem, also known as the Functional Decomposition Problem (FDP), is classical in computer algebra. It is the first general method addressing the decomposition of multivariate polynomials (any degree, any number of polynomials). As a byproduct, our approach can be also used to recover an ideal I from its kth power I^k. The complexity of the algorithm depends on the ratio between the number of variables (n) and the number of polynomials (u). For example, polynomials of degree four can be decomposed in , when this ratio is smaller than . This work was initially motivated by a cryptographic application, namely the cryptanalysis of 2R⁻ schemes. From a cryptographic point of view, the new algorithm is so efficient that the principle of two-round schemes, including 2R⁻ schemes, becomes useless. Besides, we believe that our algorithm is of independent interest.     

A new algorithm for sparse interpolation of multivariate polynomials

To reconstruct a black box multivariate sparse polynomial from its floating point evaluations, the existing algorithms need to know upper bounds for both the number of terms in the polynomial and the partial degree in each of the variables. Here we present a new technique, based on Rutishauser’s qd$qd$-algorithm, in which we overcome both drawbacks.     

针对选址问题的一种遗传算法改进探究

选址问题是现代地理信息资源配置的重要研究领域之一,通用性强、鲁棒性高的遗传算法可以较好地解决这类问题。常用方法是使用二进制编码的遗传算法对栅格数据地图进行选址。为克服二进制编码的标准遗传算法在解决选址问题过程中易陷入早熟的缺点,在研究了使用不同算子、引入观测概念这两大类解决标准遗传算法陷入早熟问题的方法后,针对选址问题的特点,选择了引入多样性测度与应用小生境技术对遗传算法进行改进,并深入探究了引入多样性测度与应用小生境技术后,遗传算法解决选址问题的过程中准确性、在线性能函数、离线性能函数的改善;接着提出了进一步改进小生境技术的方法,使得遗传群体中的每一个个体都参与遗传操作,并且避免了两个相同的个体参与交叉操作的情况。最后通过地图选址实验,将改进的小生境遗传算法与多样性测度结合,成功提高了遗传算法的性能。     

求解旅行商问题的改进蚁群算法研究

针对蚁群算法收敛速度慢的问题,对蚁群算法信息素更新规则进行研究,提出一个基于迭代思想的信息素更新规则。对信息残留因子进行实验,确定在新的信息素更新规则下信息素挥发系数的最佳合理值。最后针对eil51问题和dantzig42问题两个例子的仿真实验对比基本蚁群算法。实验结果表明,改进的蚁群算法在收敛性和求得最优解方面都明显优于基本蚁群算法和其它人工智能算法。     

An improved exact algorithm for the domatic number problem

The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time n^2.695 (up to polynomial factors) and in polynomial space. This result improves the previous bound of n^2.8805, which is due to Björklund and Husfeldt. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Δ(G) by a randomized polynomial-space algorithm, whose running time is better than the previous bound due to Riege and Rothe whenever Δ(G)?5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.     

An improved fireworks algorithm for the capacitated vehicle routing problem

The capacitated vehicle routing problem (CVRP), which aims at minimizing travel costs, is a wellknown NP-hard combinatorial optimization. Owing to its hardness, many heuristic search algorithms have been proposed to tackle this problem. This paper explores a recently proposed heuristic algorithm named the fireworks algorithm (FWA), which is a swarm intelligence algorithm. We adopt FWA for the combinatorial CVRP problem with several modifications of the original FWA: it employs a new method to generate "sparks" according to the selection rule, and it uses a new method to determine the explosion amplitude for each firework. The proposed algorithm is compared with several heuristic search methods on some classical benchmark CVRP instances. The experimental results show a promising performance of the proposed method. We also discuss the strengths and weaknesses of our algorithm in contrast to traditional algorithms.     

求解高维优化问题的改进正弦余弦算法*

提出一种改进的正弦余弦算法(简记为ISCA)。受粒子群优化(PSO)算法的启发,引入惯性权重以提高正弦余弦算法的收敛精度和加快收敛速度。此外,采取反向学习策略产生初始个体以提高种群的多样性和解的质量。采用8个高维基准测试函数进行仿真实验：在相同的最大适应度函数评价次数下,ISCA总体性能上均优于基本SCA算法和HGWO算法;当维数较高(D=1000)时,ISCA所用计算量远小于HDEOO。实验结果表明ISCA在收敛精度和收敛速度指标上均优于对比算法。     

An improved particle swarm optimization algorithm for flowshop scheduling problem

The flowshop scheduling problem has been widely studied and many techniques have been applied to it, but few algorithms based on particle swarm optimization (PSO) have been proposed to solve it. In this paper, an improved PSO algorithm (IPSO) based on the “alldifferent” constraint is proposed to solve the flow shop scheduling problem with the objective of minimizing makespan. It combines the particle swarm optimization algorithm with genetic operators together effectively. When a particle is going to stagnate, the mutation operator is used to search its neighborhood. The proposed algorithm is tested on different scale benchmarks and compared with the recently proposed efficient algorithms. The results show that the proposed IPSO algorithm is more effective and better than the other compared algorithms. It can be used to solve large scale flow shop scheduling problem effectively.     

An improved hybrid algorithm for the set covering problem

The state-of-the-art ant colony optimization (ACO) algorithm to solve large scale set covering problems (SCP) starts by solving the Lagrangian dual (LD) problem of the SCP to obtain quasi-optimal dual values. These values are then exploited by the ACO algorithm in the form of heuristic estimates. This article starts by discussing the complexity of this approach where a number of new parameters are introduced to escape local optimums and normalize the heuristic values. To avoid these complexities, we propose a new hybrid algorithm that starts by solving the linear programming (LP) relaxation of the SCP. This solution is used to eliminate unnecessary columns, and to estimate the heuristic information. To generate solutions, we use a Max–Min Ant System (MMAS) algorithm that employs a novel mechanism to update the pheromone trail limits to maintain a predetermined exploration rate. Computational experiments on different sets of benchmark instances prove that our proposed algorithm can be considered the new state-of-the-art meta-heuristic to solve the SCP.     

An improved firefly algorithm for solving dynamic multidimensional knapsack problems

There is a wide range of publications reported in the literature, considering optimization problems where the entire problem related data remains stationary throughout optimization. However, most of the real-life problems have indeed a dynamic nature arising from the uncertainty of future events. Optimization in dynamic environments is a relatively new and hot research area and has attracted notable attention of the researchers in the past decade. Firefly Algorithm (FA), Genetic Algorithm (GA) and Differential Evolution (DE) have been widely used for static optimization problems, but the applications of those algorithms in dynamic environments are relatively lacking. In the present study, an effective FA introducing diversity with partial random restarts and with an adaptive move procedure is developed and proposed for solving dynamic multidimensional knapsack problems. To the best of our knowledge this paper constitutes the first study on the performance of FA on a dynamic combinatorial problem. In order to evaluate the performance of the proposed algorithm the same problem is also modeled and solved by GA, DE and original FA. Based on the computational results and convergence capabilities we concluded that improved FA is a very powerful algorithm for solving the multidimensional knapsack problems for both static and dynamic environments.     

An improved hybrid firefly algorithm for capacitated vehicle routing problem

Firefly algorithm (FA) is a new meta-heuristic which is successfully applied to solve several optimization problems. However, it suffers from a drawback of easily getting stuck at local optima. This paper proposes a new hybrid FA, called CVRP-FA, to solve capacitated vehicle routing problem. In CVRP-FA, FA is integrated with two types of local search and genetic operators to enhance the solution’s quality and accelerate the convergence. The experiments are conducted over 82 benchmark instances. The results demonstrate that CVRP-FA has fast convergence rate and high computational accuracy. It significantly outperforms the other state-of-the-art FA variants in majority of the tested instances.     

An improved differential evolution algorithm for the task assignment problem

An improved differential evolution algorithm (IDE) is proposed to solve task assignment problem. The IDE is an improved version of differential evolution algorithm (DE), and it modifies two important parameters of DE algorithm: scale factor and crossover rate. Specially, scale factor is adaptively adjusted According to the objective function values of all candidate solutions, and crossover rate is dynamically adjusted with the increasement of iterations. The adaptive scale factor and dynamical crossover rate are combined to increase the diversity of candidate solutions, and to enhance the exploration capacity of solution space of the proposed algorithm. In addition, a usual penalty function method is adopted to trade-off the objective and the constraints. Experimental results demonstrate that the optimal solutions obtained by the IDE algorithm are all better than those obtained by the other two DE algorithms on solving some task assignment problems.     

An improved monkey algorithm for a 0-1 knapsack problem

The 0-1 knapsack problem is a classic combinational optimization problem. However, many exiting algorithms have low precision and easily fall into local optimal solutions to solve the 0-1 knapsack problem. In order to overcome these problems, this paper proposes a binary version of the monkey algorithm where the greedy algorithm is used to strengthen the local search ability, the somersault process is modified to avoid falling into local optimal solutions, and the cooperation process is adopted to speed up the convergence rate of the algorithm. To validate the efficiency of the proposed algorithm, experiments are carried out with various data instances of 0-1 knapsack problems and the results are compared with those of five metaheuristic algorithms.     

An improved model for vehicle routing problem with time constraint based on genetic algorithm

A vehicle routing problem (VRP) with time constraint is one of the important problems in distribution and transportation. Thus the generic VRP and its practical extensions are discussed in great detail in the literatures. In the VRP, the service of a customer must start and finish within a given time interval. The objective of this problem is to minimize the cost of servicing the set of customers without being tardy or exceeding the capacity or travel time of the vehicles. In this research we concentrated on developing a GA–TSP model by improving the genetic algorithm (GA) operators and the initial population. For the computational purpose, we developed a GUI (graphic user interface)-type computer program according to the proposed method. The computational results show that the proposed method is very effective on a set of standard test problems and it can be potentially useful in solving the VRPs.     

An improved LNS algorithm for real-time vehicle routing problem with time windows

This paper studies the dynamic vehicle routing problem with hard time windows (DVRPTW). The main study course of this problem was briefly reviewed. The solving strategy and algorithm of the problem are put forward. First of all, DVRPTW problem is decomposed into a series of static VRPTW. When and how to decompose the DVRP is the issue, that must be addressed. An event-trigger mechanism has been proposed and used to decompose the DVRPTW into a series of system delay-snapshots. The trigger event to be adopted is a new request arrival during the stable operation. And each snapshot is regarded as a static VRPTW. Whether each static VRPTW can quickly and efficiently be solved within a given time or a shorter time, i.e. the solving time is another issue for the DVRPTW. In the solving process, how to merge the latest requirement to the current solution is the third issue that must be solved. An improved large neighborhood search (LNS) algorithm is proposed to solve the static problem. Utilizing the remove-reinsert process of the LNS, the latest request nodes are regarded as a part of the removed nodes; these nodes can be inserted into the original or current solution in good time in the reinsertion process; meanwhile, its computing speed is high and effective for it does not need to resolve primal problem each time. Computational results of a large number of test problems, which cited from Solomon's static benchmarks and Lacker’s dynamic data set, show that our method is superior to other methods in most instances.     

校车路径问题的改进迭代局部搜索算法

针对考虑站点服务时间、学生最大乘车时间约束的校车路径问题(SBRP),提出一种改进迭代局部搜索(ILS)算法以提升求解质量。该算法使用大规模邻域搜索(LNS)算法作为扰动算子;在解的破坏过程中,设计一组解的破坏因子并赋以一定的选择概率,每隔若干次迭代后根据解的质量自适应更改破坏因子的选择概率,进而调整解的破坏程度。为提升ILS解的多样性,算法采用了基于偏差系数的邻域解接受准则。在国际基准测试案例上进行了测试,测试结果表明在ILS算法中使用自适应调整破坏程度的LNS扰动比常规扰动和其他破坏扰动的求解质量有大幅提升;与蚁群算法的比较结果进一步验证了改进算法的有效性。     

An improved multi-objective evolutionary algorithm for the vehicle routing problem with time windows

The vehicle routing problem with time windows is a complex combinatorial problem with many real-world applications in transportation and distribution logistics. Its main objective is to find the lowest distance set of routes to deliver goods, using a fleet of identical vehicles with restricted capacity, to customers with service time windows. However, there are other objectives, and having a range of solutions representing the trade-offs between objectives is crucial for many applications. Although previous research has used evolutionary methods for solving this problem, it has rarely concentrated on the optimization of more than one objective, and hardly ever explicitly considered the diversity of solutions. This paper proposes and analyzes a novel multi-objective evolutionary algorithm, which incorporates methods for measuring the similarity of solutions, to solve the multi-objective problem. The algorithm is applied to a standard benchmark problem set, showing that when the similarity measure is used appropriately, the diversity and quality of solutions is higher than when it is not used, and the algorithm achieves highly competitive results compared with previously published studies and those from a popular evolutionary multi-objective optimizer.