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Journal articles on the topic 'Multidimensional Vector Packing Problem'

Author: Grafiati

Published: 6 September 2023

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1

Alves, Cláudio, José Valério de Carvalho, François Clautiaux, and Jürgen Rietz. "Multidimensional dual-feasible functions and fast lower bounds for the vector packing problem." European Journal of Operational Research 233, no. 1 (February 2014): 43–63. http://dx.doi.org/10.1016/j.ejor.2013.08.011.

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2

Chekanin, Vladislav A., and Alexander V. Chekanin. "Multilevel Linked Data Structure for the Multidimensional Orthogonal Packing Problem." Applied Mechanics and Materials 598 (July 2014): 387–91. http://dx.doi.org/10.4028/www.scientific.net/amm.598.387.

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The actual NP-completed orthogonal bin packing problem is considered in the article. In practice a solution of a large number of different practical problems, including problems in logistics and scheduling comes down to the bin packing problem. A decision of an any packing problem is represented as a placement string which contains a sequence of objects selected to pack. The article proposes a new multilevel linked data structure that improves the effectiveness of decoding of the placement string and as a consequence, increases the speed of packing generation. The new data structure is applicable for all multidimensional orthogonal bin packing problems.

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3

Chekanin, Alexander V., and Vladislav A. Chekanin. "Effective Data Structure for the Multidimensional Orthogonal Bin Packing Problems." Advanced Materials Research 962-965 (June 2014): 2868–71. http://dx.doi.org/10.4028/www.scientific.net/amr.962-965.2868.

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The actual in industry multidimensional orthogonal packing problem is considered in the article. Solution of a large number of different practical optimization problems, including resources saving problem, optimization problems in logistics, scheduling and planning comes down to the orthogonal packing problem which is NP-hard in strong sense. One of the indicators characterizing the efficiency of packing constructing algorithm is the efficiency of the used data structure. In the article a multilevel linked data structure that increases the speed of constructing of a packing is proposed. The carried out computational experiments show the high efficiency of the new data structure. Multilevel linked data structure is applicable for multidimensional orthogonal bin packing problems any kind.

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4

Chekanin, Alexander V., and Vladislav A. Chekanin. "Improved Packing Representation Model for the Orthogonal Packing Problem." Applied Mechanics and Materials 390 (August 2013): 591–95. http://dx.doi.org/10.4028/www.scientific.net/amm.390.591.

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The multidimensional NP-hard orthogonal bin packing problem is considered in the article. Usually the problem is solved using heuristic algorithms of discrete optimization which optimize a selection sequence of objects to be packed in containers. The quality and speed of getting the resulting packing for a given sequence of placing objects is determined by the used packing representation model. In the article presented a new packing representation model for constructing the orthogonal packing. The proposed model of potential containers describes all residual free spaces of containers in packing. The developed model is investigated on well-known standard benchmarks of three-dimensional orthogonal bin packing problem. The model can be used in development of applied software for the optimal allocation of orthogonal resources.

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Chekanin, Vladislav A., and Alexander V. Chekanin. "Improved Data Structure for the Orthogonal Packing Problem." Advanced Materials Research 945-949 (June 2014): 3143–46. http://dx.doi.org/10.4028/www.scientific.net/amr.945-949.3143.

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In paper is considered the actual in industry and engineering orthogonal multidimensional packing problem. This problem is NP-hard in strong sense therefore an important role is played effectiveness of the used packing representation model. To increase the speed of placement of a given set of orthogonal objects into containers is offered a new data structure – multilevel linked data structure. The carried out computational experiments demonstrate high time efficiency of the proposed data structure compared to the ordered simple linked list.

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6

Hu, Qian, Lijun Wei, and Andrew Lim. "The two-dimensional vector packing problem with general costs." Omega 74 (January 2018): 59–69. http://dx.doi.org/10.1016/j.omega.2017.01.006.

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7

Caprara, Alberto, and Paolo Toth. "Lower bounds and algorithms for the 2-dimensional vector packing problem." Discrete Applied Mathematics 111, no. 3 (August 2001): 231–62. http://dx.doi.org/10.1016/s0166-218x(00)00267-5.

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8

Hu, Qian, Andrew Lim, and Wenbin Zhu. "The two-dimensional vector packing problem with piecewise linear cost function." Omega 50 (January 2015): 43–53. http://dx.doi.org/10.1016/j.omega.2014.07.004.

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9

Billaut, Jean-Charles, Federico Della Croce, and Andrea Grosso. "A single machine scheduling problem with two-dimensional vector packing constraints." European Journal of Operational Research 243, no. 1 (May 2015): 75–81. http://dx.doi.org/10.1016/j.ejor.2014.11.036.

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10

Morales, Fernando. "Application and assessment of divide-and-conquer-based heuristic algorithms for some integer optimization problems." Yugoslav Journal of Operations Research, no. 00 (2022): 30. http://dx.doi.org/10.2298/yjor2111015030m.

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In this paper three heuristic algorithms using the Divide-and-Conquer paradigm are developed and assessed for three integer optimizations problems: Multidimensional Knapsack Problem (d-KP), Bin Packing Problem (BPP) and Travelling Salesman Problem (TSP). For each case, the algorithm is introduced, together with the design of numerical experiments, in order to empirically establish its performance from both points of view: its computational time and its numerical accuracy.

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11

Bougeret, Marin, Guillerme Duvillié, and Rodolphe Giroudeau. "Approximability and exact resolution of the multidimensional binary vector assignment problem." Journal of Combinatorial Optimization 36, no. 3 (March 29, 2018): 1059–73. http://dx.doi.org/10.1007/s10878-018-0276-8.

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12

Song, Hai Na, Xiao Qing Zhang, and Zhong Tang He. "Multidimensional Resource Scheduling in Cloud Data Centers." Advanced Materials Research 1008-1009 (August 2014): 1513–16. http://dx.doi.org/10.4028/www.scientific.net/amr.1008-1009.1513.

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Cloud computing environment is regarded as a kind of multi-tenant computing mode. With virtulization as a support technology, cloud computing realizes the integration of multiple workloads in one server through the package and seperation of virtual machines. Aiming at the contradiction between the heterogeneous applications and uniform shared resource pool, using the idea of bin packing, the multidimensional resource scheduling problem is analyzed in this paper. We carry out some example analysis in one-dimensional resource scheduling, two-dimensional resource schduling and three-dimensional resource scheduling. The results shows that the resource utilization of cloud data centers will be improved greatly when the resource sheduling is conducted after reorganizing rationally the heterogeneous demands.

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13

Hwang, Hark-Chin, and Jang-Ha Kang. "Hardness of Approximation for Two-Dimensional Vector Packing Problem with Large Items." Journal of Korean Institute of Industrial Engineers 38, no. 1 (March 1, 2012): 1–6. http://dx.doi.org/10.7232/jkiie.2012.38.1.001.

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14

Dahmani, Nadia, François Clautiaux, Saoussen Krichen, and El-Ghazali Talbi. "Iterative approaches for solving a multi-objective 2-dimensional vector packing problem." Computers & Industrial Engineering 66, no. 1 (September 2013): 158–70. http://dx.doi.org/10.1016/j.cie.2013.05.016.

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15

Wei, Lijun, Minghui Lai, Andrew Lim, and Qian Hu. "A branch-and-price algorithm for the two-dimensional vector packing problem." European Journal of Operational Research 281, no. 1 (February 2020): 25–35. http://dx.doi.org/10.1016/j.ejor.2019.08.024.

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16

Spieksma, Frits C. R. "A branch-and-bound algorithm for the two-dimensional vector packing problem." Computers & Operations Research 21, no. 1 (January 1994): 19–25. http://dx.doi.org/10.1016/0305-0548(94)90059-0.

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17

Gabay, Michaël, and Sofia Zaourar. "Vector bin packing with heterogeneous bins: application to the machine reassignment problem." Annals of Operations Research 242, no. 1 (September 7, 2015): 161–94. http://dx.doi.org/10.1007/s10479-015-1973-7.

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18

Stakić, Đorđe, Miodrag Živković, and Ana Anokić. "A Reduced Variable Neighborhood Search Approach to the Heterogeneous Vector Bin Packing Problem." Information Technology and Control 50, no. 4 (December 16, 2021): 808–26. http://dx.doi.org/10.5755/j01.itc.50.4.29009.

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The two-dimensional heterogeneous vector bin packing problem (2DHet-VBPP) consists of packing the set of items into the set of various type bins, respecting their two resource limits. The problem is to minimize the total cost of all bins. The problem, known to be NP-hard, can be formulated as a pure integer linear program, but optimal solutions can be obtained by the CPLEX Optimizer engine only for small instances. This paper proposes a metaheuristic approach to the 2DHet-VBPP, based on Reduced variable neighborhood search (RVNS). All RVNS elements are adapted to the considered problem and many procedures are designed to improve efficiency of the method. As the Two-dimensional Homogeneous-VBPP (2DHom-VBPP) is more often treated, we considered also a special version of the RVNS algorithm to solve the 2DHom-VBPP. The results obtained and compared to both CPLEX results and results on benchmark instances from literature, justify the use of the RVNS algorithm to solve large instances of these optimization problems.

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19

Ma, Jun Yan, Xiao Ping Liao, Juan Lu, and Hong Yao. "Digital Modeling and Movement Resolution of Rubber Band Analogy for 2D Packing Problem." Applied Mechanics and Materials 220-223 (November 2012): 2466–70. http://dx.doi.org/10.4028/www.scientific.net/amm.220-223.2466.

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Packing problem is how to arrange the components in available spaces to make the layout compact. This paper adopts a digital modeling algorithm to establish a novel rubber band convex hull model to solve this problem. A ray scanning method analogy QuickHull algorithm is presented to get extreme points of rubber band convex hull. A plural vector expression approach is adopted to movement resolution,which calculate the resultant vector to translate, rotate and slide the subbody to make the volume decrease. An experiment proved this digital modeling algorithm effective.

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20

Lobaty, A. A., and A. Y. Bumai. "Features of construction of evaluation algorithms multidimensional random processes." «System analysis and applied information science», no. 1 (March 27, 2020): 24–32. http://dx.doi.org/10.21122/2309-4923-2020-1-24-32.

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The problem of evaluating the information which is present in random signals from various sources-meters is considered. It is assumed that the random process which is evaluated and the meter output according mathematical description of the problem are vector random processes. In this case, the dimension of the vector of the measurement can be larger than the dimension of the vector of the process being evaluated. The analysis of analytical methods and algorithms of the estimation that based on the determination of the main probabilistic characteristics of a random process by both the a priori and the a posteriori methods with various optimality criteria is carried out. Based on the analysis, the problem of complexing of the meters of the random process is considered according the proposed criterion for the maximum of posterior verisimilitude, combining the criterion of maximum verisimilitude and the criterion of maximum the a posteriori probability, general methodology complexing is developed. Proposed example of the complexing shows the efficiency of the proposed method. This approach to the construction of the algorithms of the evaluation for multidimensional random processes allows to increase the accuracy of estimation, since it takes into account additional information and its complex processing.

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21

Korobov, V. I. "Time Optimality for Systems with Multidimensional Control and Vector Moment Min-Problem." Journal of Dynamical and Control Systems 26, no. 3 (November 7, 2019): 525–50. http://dx.doi.org/10.1007/s10883-019-09465-2.

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22

Recupero, Vincenzo. "Multidimensional play operators with arbitrary BV inputs." Mathematical Modelling of Natural Phenomena 15 (2020): 13. http://dx.doi.org/10.1051/mmnp/2019042.

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In this paper we provide an integral variational formulation for a vector play operator where the inputs are allowed to be arbitrary functions with (pointwise) bounded variation, not necessarily left or right continuous. We prove that this problem admits a unique solution, and we show that in the left continuous and right continuous cases it reduces to the well known existing formulations.

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23

Dahmani, Nadia, François Clautiaux, Saoussen Krichen, and El-Ghazali Talbi. "Self-adaptive metaheuristics for solving a multi-objective 2-dimensional vector packing problem." Applied Soft Computing 16 (March 2014): 124–36. http://dx.doi.org/10.1016/j.asoc.2013.12.006.

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24

Newton, Paul K., and Takashi Sakajo. "Point vortex equilibria and optimal packings of circles on a sphere." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 467, no. 2129 (December 2, 2010): 1468–90. http://dx.doi.org/10.1098/rspa.2010.0368.

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We answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of N =3–12 and 24, the only values of N for which the optimal packing problem (also known as the Tammes problem) has rigorously known solutions. We also address the cases N =13–23 where optimal packing solutions have been computed, but not proven analytically. As in Jamaloodeen & Newton (Jamaloodeen & Newton 2006 Proc. R. Soc. Lond. Ser. A 462 , 3277–3299. ( doi:10.1098/rspa.2006.1731 )), a logarithmic, or point vortex equilibrium is determined by formulating the problem as the one in linear algebra, , where A is a N ( N −1)/2× N non-normal configuration matrix obtained by requiring that all interparticle distances remain constant. If A has a kernel, the strength vector is then determined as a right-singular vector associated with the zero singular value, or a vector that lies in the nullspace of A where the kernel is multi-dimensional. First we determine if the known optimal packing solution for a given value of N has a configuration matrix A with a non-empty nullspace. The answer is yes for N =3–9, 11–14, 16 and no for N =10, 15, 17–24. We then determine a basis set for the nullspace of A associated with the optimally packed state, answer the question of whether N -equal strength particles, , form an equilibrium for this configuration, and describe what is special about the icosahedral configuration from this point of view. We also find new equilibria by implementing two versions of a random walk algorithm. First, we cluster sub-groups of particles into patterns during the packing process, and ‘grow’ a packed state using a version of the ‘yin-yang’ algorithm of Longuet-Higgins (Longuet-Higgins 2008 Proc. R. Soc. A (doi:10.1098/rspa.2008.0219)). We also implement a version of our ‘Brownian ratchet’ algorithm to find new equilibria near the optimally packed state for N =10, 15, 17–24.

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25

DA SILVA, FABRICIO ALVES BARBOSA, and ISAAC D. SCHERSON. "EFFICIENT PARALLEL JOB SCHEDULING USING GANG SERVICE." International Journal of Foundations of Computer Science 12, no. 03 (June 2001): 265–84. http://dx.doi.org/10.1142/s0129054101000485.

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Gang scheduling has been widely used as a practical solution to the dynamic parallel job scheduling problem. To overcome some of the limitations of traditional Gang scheduling algorithms, Concurrent Gang is proposed as a class of scheduling policies which allows the flexible and simultaneous scheduling of multiple parallel jobs. It hence improves the space sharing characteristics of Gang scheduling while preserving all other advantages. To provide a sound analysis of Concurrent Gang performance, a novel methodology based on the traditional concept of competitive ratio is also introduced. Dubbed dynamic competitive ratio, the new method is used to compare dynamic bin packing algorithms used in this paper. These packing algorithms apply to the Concurrent Gang scheduling of a workload generated by a statistical model. Moreover, dynamic competitive ratio is the figure of merit used to evaluate and compare packing strategies for job scheduling under multiple constraints. It will be shown that for the unidimensional case there is a small difference between the performance of best fit and first fit; first fit can hence be used without significant system degradation. For the multidimensional case, when memory is also considered, we concluded that the packing algorithm must try to balance the resource utilization in all dimensions simulataneously, instead of given priority to only one dimension of the problem.

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26

Mambo, Lewis N. K. "From Multidimensional Ornstein - Uhlenbeck Process to Bayesian Vector Autoregressive Process." Journal of Mathematics Research 15, no. 1 (February 1, 2023): 32. http://dx.doi.org/10.5539/jmr.v15n1p32.

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The main purpose of  this paper is to make the connexion between stochastic analysis, the Bayesian Statistics, and time series analysis for policy analysis. This approach solves the problem of mathematical modelling - the presence of uncertainties in the models and parameters -  that reduces the  policy analysis and forecasting   effectiveness. By using the multiple It\^o  integral, the multidimensional Ornstein - Uhlenbeck process can be written as a Vector Autoregressive with lag 1 (VAR(1)) that is the generalization of Vector Autoregressive process. The  limit of this approach is in fact it requires  the strong foundations of stochastic analysis, the Bayesian Statistics, and time series analysis.

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27

HWANG, HARK-CHIN. "A 4/3-APPROXIMATION ALGORITHM FOR CASSETTE PACKING IN STEEL INDUSTRY." Asia-Pacific Journal of Operational Research 24, no. 05 (October 2007): 715–24. http://dx.doi.org/10.1142/s0217595907001462.

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In packing steel products of coils into cassettes, we need to consider both the width and weight capacity of cassettes. Each coil has weight in (0, 1/3] and width in (1/6, 1/3] when scaling both the weight and width capacities to 1. With the objective of minimizing the number of cassettes to pack the coils, the problem is modeled by two-dimensional vector packing. To efficiently pack the coils having sizes specified by the ranges, we develop a 4/3-approximation algorithm.

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28

Dow, Eli M. "Decomposed multi-objective bin-packing for virtual machine consolidation." PeerJ Computer Science 2 (February 24, 2016): e47. http://dx.doi.org/10.7717/peerj-cs.47.

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In this paper, we describe a novel solution to the problem of virtual machine (VM) consolidation, otherwise known as VM-Packing, as applicable to Infrastructure-as-a-Service cloud data centers. Our solution relies on the observation that virtual machines are not infinitely variable in resource consumption. Generally, cloud compute providers offer them in fixed resource allocations. Effectively this makes all VMs of that allocation type (or instance type) generally interchangeable for the purposes of consolidation from a cloud compute provider viewpoint. The main contribution of this work is to demonstrate the advantages to our approach of deconstructing the VM consolidation problem into a two-step process of multidimensional bin packing. The first step is to determine the optimal, but abstract, solution composed of finite groups of equivalent VMs that should reside on each host. The second step selects concrete VMs from the managed compute pool to satisfy the optimal abstract solution while enforcing anti-colocation and preferential colocation of the virtual machines through VM contracts. We demonstrate our high-performance, deterministic packing solution generation, with over 7,500 VMs packed in under 2 min. We demonstrating comparable runtimes to other VM management solutions published in the literature allowing for favorable extrapolations of the prior work in the field in order to deal with larger VM management problem sizes our solution scales to.

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29

Trevisan, Luca, and Fatos Xhafa. "The Parallel Complexity of Positive Linear Programming." Parallel Processing Letters 08, no. 04 (December 1998): 527–33. http://dx.doi.org/10.1142/s0129626498000511.

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In this paper we study the parallel complexity of Positive Linear Programming (PLP), i.e. the special case of Linear Programming in packing/covering form where the input constraint matrix and constraint vector consist entirely of positive entries. We show that the problem of exactly solving PLP is P-complete.

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30

Ma, Jun Yan, Feng Ying Long, Xiao Ping Liao, Biao Chen, Juan Lu, Wei Xia, and Xiao Mo Yu. "Digital Modeling and Movement Resolution of Rubber Ballon Analogy for 3D Packing Problem." Advanced Materials Research 753-755 (August 2013): 1670–74. http://dx.doi.org/10.4028/www.scientific.net/amr.753-755.1670.

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Packing problem is how to arrange the components in available spaces to make the layout compact. This paper adopts a digital modeling algorithm to establish a novel rubber ballon convex hull model to solve this problem. A visible point puls maximum distance method analogy QuickHull algorithm is presented to get extreme points of rubber ballon convex hull. Movement resolution aim at force analyze and calculate the resultant vector to translate, rotate and slide the component to make the volume decrease in detail. An experiment proved this digital modeling algorithm effective.

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31

Liksonova, D. I., and A. V. Medvedev. "Problem of Modeling and Control of Multi-Dimensional Timeless Processes with Dependent Output Variables." Vestnik NSU. Series: Information Technologies 20, no. 2 (October 8, 2022): 37–49. http://dx.doi.org/10.25205/1818-7900-2022-20-2-37-49.

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In this paper, we consider the problems of modeling and controlling multidimensional inertialess systems with delay under nonparametric uncertainty. We are talking about the multidimensional processes which are in conditions where the form of parametric equations for various channels of the object is absent due to lack of a priori information. The main emphasis is placed on the case when the components of the vector of output variables are randomly connected in an unforeseen way. In the case of a stochastic dependence of the output variables, the mathematical description of the object is reduced to a system of implicit equations, the form of which is unknown. Therefore, the main task of modeling is to find the predicted values of output variables from known input variables. When managing a multidimensional object, an important feature is the definition of setting actions. The main thing here is that the setting influences of the control system should not be chosen arbitrarily from the corresponding areas, but depending on the definition of the previous ones. Nonparametric identification and control algorithms for multidimensional systems are proposed. Computational experiments that show the efficiency of using the proposed nonparametric identification and control algorithms are presented.

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32

Cerri, Andrea, and Patrizio Frosini. "ADVANCES IN MULTIDIMENSIONAL SIZE THEORY." Image Analysis & Stereology 29, no. 1 (May 3, 2011): 19. http://dx.doi.org/10.5566/ias.v29.p19-26.

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Size Theory was proposed in the early 90's as a geometrical/topological approach to the problem of Shape Comparison, a very lively research topic in the fields of Computer Vision and Pattern Recognition. The basic idea is to discriminate shapes by comparing shape properties that are provided by continuous functions valued in R, called measuring functions and defined on topological spaces associated to the objects to be studied. In this way, shapes can be compared by using a descriptor named size function, whose role is to capture the features described by measuring functions and represent them in a quantitative way. However, a common scenario in applications is to deal with multidimensional information. This observation has led to considering vector-valued measuring functions, and consequently the multidimensional extension of size functions, namely the k-dimensional size functions. In this work we survey some recent results about size functions in this multidimensional setting, with particular reference to the localization of their discontinuities.

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33

Wang, Ting, Qian Hu, and Andrew Lim. "An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs." European Journal of Operational Research 300, no. 1 (July 2022): 20–34. http://dx.doi.org/10.1016/j.ejor.2021.10.011.

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Akhmedov, Yunus, Shuhrat Asadov, and Bobir Azimov. "Two-sided estimation of linear approximation error second - order hypersurfaces." Journal of Physics: Conference Series 2388, no. 1 (December 1, 2022): 012124. http://dx.doi.org/10.1088/1742-6596/2388/1/012124.

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Abstract The following aspects of multidimensional space are considered in the article. The parametrizations of the hypersurface of the second order are considered, which allow visualizing them in the process of discretization. The development of the principle of vector modeling for displaying objects of multidimensional space on a two-dimensional plane made it possible to visually display the solution to the problem of linear approximation of a second-order hypersurface. The proposed method for discretizing a second-order hypersurface makes it possible to create a simple algorithm for its linear ash-approximation, which is the basis for solving the linear programming problem in automatic mode. The developed method of two-sided estimation of the error in the case of linear approximation of the second-order hypersurface allows us to present the problem of determining the bearing capacity of structures as a linear programming problem.

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35

Xie, Jianwen, Ruiqi Gao, Erik Nijkamp, Song-Chun Zhu, and Ying Nian Wu. "Representation Learning: A Statistical Perspective." Annual Review of Statistics and Its Application 7, no. 1 (March 9, 2020): 303–35. http://dx.doi.org/10.1146/annurev-statistics-031219-041131.

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Learning representations of data is an important problem in statistics and machine learning. While the origin of learning representations can be traced back to factor analysis and multidimensional scaling in statistics, it has become a central theme in deep learning with important applications in computer vision and computational neuroscience. In this article, we review recent advances in learning representations from a statistical perspective. In particular, we review the following two themes: ( a) unsupervised learning of vector representations and ( b) learning of both vector and matrix representations.

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36

Buldygin, V. V., and V. A. Koval. "Convergence to Zero and Boundedness of Operator-Normed Sums of Random Vectors with Application to Autoregression Processes." gmj 8, no. 2 (June 2001): 221–30. http://dx.doi.org/10.1515/gmj.2001.221.

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Abstract The problems of almost sure convergence to zero and almost sure boundedness of operator-normed sums of different sequences of random vectors are studied. The sequences of independent random vectors, orthogonal random vectors and the sequences of vector-valued martingale-differences are considered. General results are applied to the problem of asymptotic behaviour of multidimensional autoregression processes.

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37

Osypchuk, M. M. "On some perturbations of a stable process and solutions to the Cauchy problem for a class of pseudo-differential equations." Carpathian Mathematical Publications 7, no. 1 (July 3, 2015): 101–7. http://dx.doi.org/10.15330/cmp.7.1.101-107.

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A fundamental solution for some class of pseudo-differential equations is constructed by the method based on the theory of perturbations. We consider a symmetric $\alpha$-stable process in multidimensional Euclidean space. Its generator $\mathbf{A}$ is a pseudo-differential operator whose symbol is given by $-c|\lambda|^\alpha$, were the constants $\alpha\in(1,2)$ and $c>0$ are fixed. The vector-valued operator $\mathbf{B}$ has the symbol $2ic|\lambda|^{\alpha-2}\lambda$. We construct a fundamental solution of the equation $u_t=(\mathbf{A}+(a(\cdot),\mathbf{B}))u$ with a continuous bounded vector-valued function $a$.

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Piryatinska, Alexandra, and Boris Darkhovsky. "Retrospective Change-Points Detection for Multidimensional Time Series of Arbitrary Nature: Model-Free Technology Based on the ϵ-Complexity Theory." Entropy 23, no. 12 (December 2, 2021): 1626. http://dx.doi.org/10.3390/e23121626.

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We consider a retrospective change-point detection problem for multidimensional time series of arbitrary nature (in particular, panel data). Change-points are the moments at which the changes in generating mechanism occur. Our method is based on the new theory of ϵ-complexity of individual continuous vector functions and is model-free. We present simulation results confirming the effectiveness of the method.

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39

Luong, Thai Hien, and Thi Tam Dinh. "Support vector machines, presented for the problem of identifying two groups of points on the plane." Tạp chí Khoa học Đại học Văn Hiến 5, no. 2 (July 10, 2017): 106–14. http://dx.doi.org/10.58810/vhujs.5.2.2017.5211.

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SVM (Support Vector Machine) is a concept in statistics and computer science for a set of supervised learning methods related to each other for classification and regression analysis. SVM is a binary classification algorithm, Support vector machine (SVM) to build a hyperplane to classify the data set into two separate classes. A hyperplane is a function similar to the line equation, y = ax + b. In fact, if we need to classify a dataset with only two features, the hyperplane is now a straight line. In terms of ideas, SVM uses tricks to map the original dataset to more dimensional spaces. Once mapped to a multidimensional space, SVM will review and select the most suitable superlattice to classify that data set.

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Osadchiy, Serhii, Oleksandr Didyk, Oleksandr Serbul, and Viktor Kalich. "Method of Synthesis of an Optimal Multidimensional Filter for Extracting a Regular Signal Against a Background of Random Noise." National Interagency Scientific and Technical Collection of Works. Design, Production and Exploitation of Agricultural Machines, no. 51 (2021): 210–45. http://dx.doi.org/10.32515/2414-3820.2021.51.210-215.

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In terms of [1] we formulate the problem of synthesis of a complex system of allocation of a useful regular signal. Suppose that a multidimensional stationary filter best processes the input information about the same regular signal r(t), which is obtained by several "non-ideal" meters, and the result is transmitted to the evaluation system. In this article, a multi-channel measurement system, which is connected to the input of a multidimensional data transmission system, the dynamics of which is described by a system of linear differential equations. The input of the measuring system receives an n-dimensional vector of measured signals r (t), the components of which are deterministic functions. The vector r (t) belongs to the main group of "non-ideal" meters. Measurements are accompanied by interference, which is a random stationary process with zero mathematical expectation and a known fractional-rational matrix of spectral densities. An integral indicator of the quality of the system is the sum of the weighted integral quadratic error of the regular signal estimation and the variance of the random component of the error. Thus, a new algorithm for the synthesis of a complete optimal regular signal extraction system against the background of a multidimensional stationary random interference is obtained, which allows to find the structure and parameters of a multidimensional optimal filter taking into account the dynamics of the information transmission system.

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Gorelov, Y. N. "ABOUT ONE APPROACH TO MODELING OF OPTIMAL CONTROL OF MULTIDIMENSIONAL COMPOSITE LINEAR SYSTEMS." Vestnik of Samara University. Natural Science Series 19, no. 9.2 (June 6, 2017): 184–90. http://dx.doi.org/10.18287/2541-7525-2013-19-9.2-184-190.

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The approach to the synthesis of optimal control of multidimensional composite linear system based on solutions of optimal control problems for inclusion in its structure of partial linear systems with scalar controls is considered. Control tasks are reduced to the problem of moments in the L\ space for minimum functionals of the norm type, on minimum maximal Ноlder norms level (with indicators 2 and то) for the vector control composite system.

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Jin, Gang, Peng Zhang, Lei Liu, and Xiang Yu Meng. "A Virtual Machine Scheduling Strategy Based on Grouping Genetic Algorithm in Cloud Environment." Applied Mechanics and Materials 411-414 (September 2013): 203–6. http://dx.doi.org/10.4028/www.scientific.net/amm.411-414.203.

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With the emergence of cloud computing, data center is becoming resource pooling and the applications are becoming multi-tenant and heterogeneous which make the server resource load balancing problem become more and more important. In order to resolve contradiction between the heterogeneous applications and sharing resource pool, we think this problem as a multi-dimensional variable vector packing model. For the NP-hard feature, we design a virtual machine scheduling strategy and it resolve the data center resource balancing problem. The experiment indicated that the algorithm can reduce the number of open physical machine to get the most of it and it not affect the physical machine utilization rate and interior load balancing.

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Panov, Evgeny Yu. "On the Cauchy problem for scalar conservation laws in the class of Besicovitch almost periodic functions: Global well-posedness and decay property." Journal of Hyperbolic Differential Equations 13, no. 03 (September 2016): 633–59. http://dx.doi.org/10.1142/s0219891616500168.

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We study the Cauchy problem for a multidimensional scalar conservation law with merely continuous flux vector in the class of Besicovitch almost periodic functions. The existence and uniqueness of entropy solutions are established. We also uncover the necessary and sufficient condition for the decay of almost periodic entropy solutions as the time variable [Formula: see text]. Our results are then interpreted in the framework of conservation laws on the Bohr compact.

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Korzyuk, V. I. "THE GOURSAT PROBLEM FOR HYPERBOLIC LINEAR THIRD ORDER EQUATIONS." Mathematical Modelling and Analysis 6, no. 2 (December 15, 2001): 270–79. http://dx.doi.org/10.3846/13926292.2001.9637166.

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The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

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Hu, Qian, Wenbin Zhu, Hu Qin, and Andrew Lim. "A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function." European Journal of Operational Research 260, no. 1 (July 2017): 70–80. http://dx.doi.org/10.1016/j.ejor.2016.12.021.

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Luo, Qiang, and Yunqing Rao. "Improved Sliding Algorithm for Generating No-Fit Polygon in the 2D Irregular Packing Problem." Mathematics 10, no. 16 (August 15, 2022): 2941. http://dx.doi.org/10.3390/math10162941.

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This paper introduces an efficient and robust sliding algorithm for the creation of no-fit polygons. The improved algorithm can cope with complex cases and is given an implementation in detail. The proposed concept of a touching group can simplify the judging process when recognizing the potential translation vector for an orbital polygon. In addition, the generation of the no-fit polygon only involves three main steps based on the proposed concept. The proposed algorithm has a mechanism that searches other start positions to generate a complete no-fit polygon for handling complex cases. To improve the efficiency, many acceleration strategies have been proposed, such as point exclusion strategy and point inclusion test. The robust and efficient performance of the algorithm is tested by well-known benchmark instances and degenerate and complex cases, such as holes, interlocking concavities and jigsaw-type pieces. Experimental results show that the proposed algorithm can produce complete no-fit polygons for complex cases, and acceleration strategies can reduce the creation time of no-fit polygon on benchmark instances by more than sixteen percent on average.

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Srochko, Vladimir, and Elena Aksenyushkina. "Parametric Regularization of a Linear-quadratic Problem on a Set of Piecewise Linear Controls." Bulletin of Irkutsk State University. Series Mathematics 41 (2022): 57–68. http://dx.doi.org/10.26516/1997-7670.2022.41.57.

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A linear-quadratic problem with arbitrary matrices in the functional and multidimensional control with convex constraint is considered. Acceptable controls are piecewise linear vector functions within an uneven grid of possible corner points. The reduction of the optimal control problem into a finite-dimensional format is carried out using vector formalization of the linear spline construction and block matrices together with the corresponding operations. The possibility of influencing the functional in the original problem is provided by using parameters with quadratic forms. The choice of these parameters is focused on the regularization of the functional in the sense of providing it with the properties of convexity or concavity at the level of a finite-dimensional model. The conditions for the choice of parameters are in the nature of inequalities with respect to the extreme eigenvalues of the block matrices forming the objective function. The corresponding convex or concave optimization problems can be solved in a finite number of iterations.

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Olteanu, Octav. "On Markov Moment Problem and Related Results." Symmetry 13, no. 6 (June 1, 2021): 986. http://dx.doi.org/10.3390/sym13060986.

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We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued functions on a compact subset or an Lν1 space, where ν is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem.

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Eremeev, Anton V. "On Complexity of Optimal Recombination for Binary Representations of Solutions." Evolutionary Computation 16, no. 1 (March 2008): 127–47. http://dx.doi.org/10.1162/evco.2008.16.1.127.

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We consider the optimization problem of finding the best possible offspring as a result of a recombination operator in an evolutionary algorithm, given two parent solutions. The optimal recombination is studied in the case where a vector of binary variables is used as a solution encoding. By means of efficient reductions of the optimal recombination problems (ORPs) we show the polynomial solvability of the ORPs for the maximum weight set packing problem, the minimum weight set partition problem, and for linear Boolean programming problems with at most two variables per inequality, and some other problems. We also identify several NP-hard cases of optimal recombination: the Boolean linear programming problems with three variables per inequality, the knapsack, the set covering, the p-median, and some other problems.

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Dau, Hoan Manh, and Ning Xu. "Text Document Classification Using Support Vector Machine with Feature Selection Using Singular Value Decomposition." Advanced Materials Research 905 (April 2014): 528–32. http://dx.doi.org/10.4028/www.scientific.net/amr.905.528.

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Text document classification is content analysis task of the text document and then giving decision (or giving a prediction) whether this text document belongs to which group among given text document ones. There are many classification techniques such as decision method basing on Naive Bayer, decision tree, k-Nearest neighbor (KNN), neural network, Support Vector Machine (SVM) method. Among those techniques, SVM is considered the popular and powerful one, especially, it is suitable to huge and multidimensional data classification. Text document classification with characteristics of very huge dimensional numbers and selecting features before classifying impact the classification results. Support Vector Machine is a very effective method in this field. This article studies Support Vector Machine and applies it in the problem of text document classification. The study shows that Support Vector Machine method with choosing features by singular value decomposition (SVD) method is better than other methods and decision tree.

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