scholarly journals Determinants of Interval Matrices

2018 ◽  
Vol 33 ◽  
pp. 99-112
Author(s):  
Jaroslav Horáček ◽  
Milan Hladík ◽  
Josef Matějka
Keyword(s):  
Computational Time ◽  
Interval Matrix ◽  
Real Interval ◽  
Interval Matrices ◽  
Np Hard Problem ◽  
Cramer’S Rule ◽  
Class Of Matrices ◽  
Exact Bounds ◽  
Symmetric Positive Definite Matrices ◽  
Np Hardness

In this paper we shed more light on determinants of real interval matrices. Computing the exact bounds on a determinant of an interval matrix is an NP-hard problem. Therefore, attention is first paid to approximations. NP-hardness of both relative and absolute approximation is proved. Next, methods computing verified enclosures of interval determinants and their possible combination with preconditioning are discussed. A new method based on Cramer's rule was designed. It returns similar results to the state-of-the-art method, however, it is less consuming regarding computational time. Other methods transferable from real matrices (e.g., the Gerschgorin circles, Hadamard's inequality) are discussed. New results about classes of interval matrices with polynomially computable tasks related to determinant are proved (symmetric positive definite matrices, class of matrices with identity midpoint matrix, tridiagonal H-matrices). The mentioned methods were compared for random general and symmetric matrices.

Computing the spectral decomposition of interval matrices and a study on interval matrix powers

2021 ◽  
Vol 403 ◽  
pp. 126174
Author(s):  
David Hartman ◽  
Milan Hladík ◽  
David Říha
Keyword(s):  
Spectral Decomposition ◽  
Interval Matrix ◽  
Interval Matrices

scholarly journals A Sparse Optimization Approach for Energy-Efficient Timetabling in Metro Railway Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-19 ◽  
Author(s):  
Ziyan Luo ◽  
Xiaoyu Li ◽  
Naihua Xiu
Keyword(s):  
Optimization Model ◽  
Energy Efficient ◽  
Convex Relaxation ◽  
Sparse Optimization ◽  
Computational Time ◽  
Optimization Approach ◽  
Railway Systems ◽  
Alternating Direction ◽  
Np Hard Problem ◽  
Regenerative Energy

In this paper, we propose a sparse optimization approach to maximize the utilization of regenerative energy produced by braking trains for energy-efficient timetabling in metro railway systems. By introducing the cardinality function and the square of the Euclidean norm function as the objective function, the resulting sparse optimization model can characterize the utilization of the regenerative energy appropriately. A two-stage alternating direction method of multipliers is designed to efficiently solve the convex relaxation counterpart of the original NP-hard problem and then to produce an energy-efficient timetable of trains. The resulting approach is applied to Beijing Metro Yizhuang Line with different instances of service for case study. Comparison with the existing two-step linear program approach is also conducted which illustrates the effectiveness of our proposed sparse optimization model in terms of the energy saving rate and the efficiency of our numerical optimization algorithm in terms of computational time.

scholarly journals Generalization of real interval matrices to other fields

2019 ◽  
Vol 35 ◽  
pp. 285-296
Author(s):  
Elena Rubei
Keyword(s):  
Full Rank ◽  
Maximal Rank ◽  
Interval Matrix ◽  
Real Interval ◽  
Rational Matrix ◽  
Rank One ◽  
Rational Rank ◽  
Q Matrix ◽  
Rational Interval ◽  
Rank Of A Matrix

An interval matrix is a matrix whose entries are intervals in $\R$. This concept, which has been broadly studied, is generalized to other fields. Precisely, a rational interval matrix is defined to be a matrix whose entries are intervals in $\Q$. It is proved that a (real) interval $p \times q$ matrix with the endpoints of all its entries in $\Q$ contains a rank-one matrix if and only if it contains a rational rank-one matrix, and contains a matrix with rank smaller than $\min\{p,q\}$ if and only if it contains a rational matrix with rank smaller than $\min\{p,q\}$; from these results and from the analogous criterions for (real) inerval matrices, a criterion to see when a rational interval matrix contains a rank-one matrix and a criterion to see when it is full-rank, that is, all the matrices it contains are full-rank, are deduced immediately. Moreover, given a field $K$ and a matrix $\al$ whose entries are subsets of $K$, a criterion to find the maximal rank of a matrix contained in $\al$ is described.

DYNAMIC POINT LABELING IS STRONGLY PSPACE-COMPLETE

2014 ◽  
Vol 24 (04) ◽  
pp. 373-395 ◽  
Author(s):  
KEVIN BUCHIN ◽  
DIRK H. P. GERRITS
Keyword(s):  
Hard Problem ◽  
Np Hard ◽  
Interactive Maps ◽  
Np Hard Problem ◽  
Automated Cartography ◽  
Np Hardness

An important but strongly NP-hard problem in automated cartography is how to best place textual labels for point features on a static map. We examine the complexity of various generalizations of this problem for dynamic and/or interactive maps. Specifically, we show that it is strongly PSPACE-complete to decide whether there is a smooth dynamic labeling (function from time to static labelings) when the points move, when points are added and removed, or when the user pans, rotates, and/or zooms their view of the points. In doing so we develop a framework from which a wide variety of labeling hardness results can be obtained, including (next to the PSPACE-hardness results) both known and new results on the NP-hardness of static labeling.

scholarly journals Solving the Traveling Salesman Problem Based on The Genetic Reactive Bone Route Algorithm whit Ant Colony System

2016 ◽  
Vol 4 (2) ◽  
pp. 65 ◽  
Author(s):  
Majid Yousefikhoshbakht ◽  
Nasrin Malekzadeh ◽  
Mohammad Sedighpour
Keyword(s):  
Traveling Salesman ◽  
Ant Colony ◽  
Metaheuristic Algorithms ◽  
Computational Time ◽  
Ant Colony System ◽  
Hard Problem ◽  
Np Hard Problem ◽  
The Traveling Salesman Problem ◽  
Route Algorithm

<p>The TSP is considered one of the most well-known combinatorial optimization tasks and researchers have paid so much attention to the TSP for many years. In this problem, a salesman starts to move from an arbitrary place called depot and after visits all of the nodes, finally comes back to the depot. The objective is to minimize the total distance traveled by the salesman.  Because this problem is a non-deterministic polynomial (NP-hard) problem in nature, a hybrid meta-heuristic algorithm called REACSGA is used for solving the TSP. In REACSGA, a reactive bone route algorithm that uses the ant colony system (ACS) for generating initial diversified solutions and the genetic algorithm (GA) as an improved procedure are applied. Since the performance of the Metaheuristic algorithms is significantly influenced by their parameters, Taguchi Method is used to set the parameters of the proposed algorithm. The proposed algorithm is tested on several standard instances involving 24 to 318 nodes from the literature. The computational result shows that the results of the proposed algorithm are competitive with other metaheuristic algorithms for solving the TSP in terms of better quality of solution and computational time respectively. In addition, the proposed REACSGA is significantly efficient and finds closely the best known solutions for most of the instances in which thirteen best known solutions are also found.</p>

scholarly journals Estimation of all-terminal network reliability using press forward GA with stochastic selection method

2021 ◽  
Vol 12 (8) ◽  
pp. 2875-2893
Author(s):  
Anuradha, Et. al.
Keyword(s):  
Genetic Algorithm ◽  
Evaluation Method ◽  
Network Reliability ◽  
Reliability Evaluation ◽  
Network Size ◽  
Computational Time ◽  
Close Analysis ◽  
The Press ◽  
Np Hard Problem ◽  
Stochastic Selection

Network reliability is an important measure of how well the network meets its design aim and mathematically is the probability that a network will perform communication efficiently for at least a given period of time. The reliability analysis is used for determining the probability of correct functioning of a system in terrain conditions. Estimating the all-terminal network reliability is an NP-hard problem, the press forward Genetic Algorithm is also an alternative to solve the problem of all-terminal network reliability. This paper presents a reliability evaluation method by using a feed-forward genetic algorithm with a random selection of nodes and links i.e., estimate the network reliability such that the flow is not less than a given threshold. To verify the efficiency of an algorithm, various sets of runs are applied to identify the most reliable network and the algorithm is tested on the network size of 2,4,6,8, --------256 nodes with link N(N-1)/2. The results of the algorithm are obtained for all-terminal reliability problem based on close analysis of the complexity of the network. Our assumption for reliability evaluation of node 1 to node N, with link reliability of 50%. Another result of the proposed GA approach is the significant reduction in computational time.

scholarly journals Vertex Weighting-Based Tabu Search for p-Center Problem

2020 ◽  
Author(s):  
Qingyun Zhang ◽  
Zhipeng Lü ◽  
Zhouxing Su ◽  
Chumin Li ◽  
Yuan Fang ◽  
...  
Keyword(s):  
Tabu Search ◽  
Search Strategy ◽  
Set Covering ◽  
Computational Time ◽  
Center Problem ◽  
Local Optima ◽  
Maximum Cost ◽  
Benchmark Instances ◽  
Np Hard Problem ◽  
Distinguishing Features

The p-center problem consists of choosing p centers from a set of candidates to minimize the maximum cost between any client and its assigned facility. In this paper, we transform the p-center problem into a series of set covering subproblems, and propose a vertex weighting-based tabu search (VWTS) algorithm to solve them. The proposed VWTS algorithm integrates distinguishing features such as a vertex weighting technique and a tabu search strategy to help the search to jump out of the local optima. Computational experiments on 138 most commonly used benchmark instances show that VWTS is highly competitive comparing to the state-of-the-art methods in spite of its simplicity. As a well-known NP-hard problem which has already been studied for over half a century, it is a challenging task to break the records on these classic datasets. Yet VWTS improves the best known results for 14 out of 54 large instances, and matches the optimal results for all remaining 84 ones. In addition, the computational time taken by VWTS is much shorter than other algorithms in the literature.

A Fast and Deterministic Approach to a Near Optimal Solution for the p-Median Problem

2012 ◽  
Vol 3 (3) ◽  
pp. 1-14 ◽  
Author(s):  
Xiang Li ◽  
Christophe Claramunt ◽  
Xihui Zhang ◽  
Yingping Huang
Keyword(s):  
Heuristic Method ◽  
Location Theory ◽  
Optimal Solution ◽  
Deterministic Algorithm ◽  
Initial Solution ◽  
Computational Time ◽  
Computationally Efficient ◽  
Median Problem ◽  
Research Issues ◽  
Np Hard Problem

Finding solutions for the p-median problem is one of the primary research issues in the field of location theory. Since the p-median problem has proved to be a NP-hard problem, several heuristic and approximation methods have been proposed to find near optimal solutions with acceptable computational time. This study introduces a computationally efficient and deterministic algorithm whose objective is to return a near optimal solution for the p-median problem. The merit of the proposed approach, called Relocation Median (RLM), lies in solving the p-median problem in superior computational time with a tiny percentage deviation from the optimal solution. This approach is especially relevant when the problem is enormous where, even when a heuristic method is applied, the computational time is quite high. RLM consists of two parts: The first part uses a greedy search method to find an initial solution; the second part sequentially substitutes medians in the initial solution with additional vertices to reduce the total travel cost. Experiments show that to solve the same p-median problem, the RLM approach can significantly shorten the computational time needed by a genetic algorithm based approach to obtain solutions with similar quality.

Application of an Information-Based Design for Assembly Theory to Assembly Workstation Design

1993 ◽  
Author(s):  
David O. Hunt ◽  
Robert H. Sturges
Keyword(s):  
Problem Formulation ◽  
Mixed Integer ◽  
Computational Time ◽  
Design For Assembly ◽  
Assembly Time ◽  
Aspect Ratios ◽  
Workstation Design ◽  
Orientation Time ◽  
Np Hard Problem ◽  
Assembly Tasks

Abstract An effective presentation of components at the workstation can have a significant impact in reducing assembly time. Our goal is to reduce the assembly time by optimization of the component presentation. The assembly factors recognized in Design for Assembly theory as relevant to both parts acquisition and assembly workstation layout are recognition, orientation, weight, and handling distance. This study considers a single manual assembler at an assembly station, with the components in rectangular bins of differing sizes and aspect ratios. Ninety degree rotations are allowed for minimizing potential handling distance. The assembly task is modelled with multiple assembly points representing the final location of the components. Components can be preoriented or random in the bins, with preorientation removing the recognition and orientation time penalties. The problem formulation employs Mixed Integer Non-Linear Programming (MINLP), and numerical evidence suggests an np-hard problem. Heuristic methods control computational time to practical levels for realistic assembly tasks. Our results show that numerical optimization of assembly workstation layout can reduce the expected level of difficulty over random or manual workstation design methods.

scholarly journals Minimization of Delay and Travel Time of Yard Trucks in Container Terminals Using an Improved GA with Guidance Search

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Z. X. Wang ◽  
Felix T. S. Chan ◽  
S. H. Chung ◽  
Ben Niu
Keyword(s):  
Travel Time ◽  
Large Scale ◽  
Container Terminal ◽  
Container Terminals ◽  
Computational Time ◽  
Small Scale ◽  
Storage Allocation ◽  
Total Delay ◽  
Truck Scheduling ◽  
Np Hard Problem

Yard truck scheduling and storage allocation problems (YTS-SAP) are two important issues that influence the efficiency of a container terminal. These two problems aim to determine the routing of trucks and proper storage locations for discharging containers from incoming vessels. This paper integrates YTS and SAP as a whole and tries to minimize the weighted summation of total delay and total yard trucks travel time. A genetic algorithm (GA) is proposed to deal with the problem. In the proposed GA, guidance mutation approach and exhaustive heuristic for local searching are used in order to force the GA to converge faster and be steadier. To test the performance of the proposed GA, both small scale and large scale cases are studied. The results of these cases are compared with CPLEX for the small scale cases. Since this problem is an NP-hard problem, which CPLEX cannot solve, a simple GA is studied for comparison in large scale cases. The comparison demonstrates that the proposed GA can obtain near optimal solutions in much shorter computational time for small scale cases. In addition, the proposed GA can obtain better results than other methods in reasonable time for large scale cases.

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