scholarly journals Detecting Quantum Speedup of HHL Algorithm for Linear Programming Scenarios

2017 ◽  
Author(s):  
Volkan Erol ◽  
Mert Side
Keyword(s):  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Equation System ◽  
Linear Equation System ◽  
The Matrix ◽  
Speedup Ratio ◽  
Special Case ◽  
Quantum Speedup

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems, we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we look at quantum speedup ratios of HHL Algorithm for different scenarios of linear programming. In a first scenario we look quantum speedup ratio (S(N)) as a function of phase transition and the ratio (κ) between the greatest and smallest eigenvalues of the matrix in linear equation system. As a second scenario, we investigate the changes in S(N) as a function of κ and s, which is the coefficient for defining the matrix as s-sparse.

scholarly journals Applying Quantum Optimization Algorithms for Linear Programming

2017 ◽  
Author(s):  
Mert Side ◽  
Volkan Erol
Keyword(s):  
Resource Allocation ◽  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Quantum Optimization ◽  
Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

scholarly journals Applying Quantum Optimization Algorithms for Linear Programming

2017 ◽  
Author(s):  
Mert Side ◽  
Volkan Erol
Keyword(s):  
Resource Allocation ◽  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Quantum Optimization ◽  
Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

scholarly journals Applying Quantum Optimization Algorithms for Linear Programming

2017 ◽  
Author(s):  
Mert Side ◽  
Volkan Erol
Keyword(s):  
Resource Allocation ◽  
Linear Programming ◽  
Production Scheduling ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Computers ◽  
Quantum Optimization ◽  
Special Case

Quantum computers are machines that are designed to use quantum mechanics in order to improve upon classical computers by running quantum algorithms. One of the main applications of quantum computing is solving optimization problems. For addressing optimization problems we can use linear programming. Linear programming is a method to obtain the best possible outcome in a special case of mathematical programming. Application areas of this problem consist of resource allocation, production scheduling, parameter estimation, etc. In our study, we looked at the duality of resource allocation problems. First, we chose a real world optimization problem and looked at its solution with linear programming. Then, we restudied this problem with a quantum algorithm in order to understand whether if there is a speedup of the solution. The improvement in computation is analysed and some interesting results are reported.

scholarly journals Calculating Fuzzy Inverse Matrix Using Linear Programming Problem: An Improved Approach

2021 ◽  
pp. 983-996
Author(s):  
Fatemeh Babakordi ◽  
Nemat Allah Taghi-Nezhad
Keyword(s):  
Linear Programming ◽  
Programming Problem ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Equation System ◽  
Matrix Inverse ◽  
Fuzzy Matrix ◽  
Linear Equation System ◽  
The Matrix ◽  
Left And Right

Calculating the matrix inverse is a key point in solving linear equation system, which involves complex calculations, particularly  when the matrix elements are  (Left and Right) fuzzy numbers. In this paper, first, the method of Kaur and Kumar for calculating the matrix inverse is reviewed, and its disadvantages are discussed. Then, a new method is proposed to determine the inverse of  fuzzy matrix based on linear programming problem. It is demonstrated that the proposed method is capable of overcoming the shortcomings of the previous matrix inverse. Numerical examples are utilized to verify the performance and applicability of the proposed method.

A Fast Approach to Solve Matrix Games with Payoffs of Trapezoidal Fuzzy Numbers

2016 ◽  
Vol 33 (06) ◽  
pp. 1650047 ◽  
Author(s):  
Sanjiv Kumar ◽  
Ritika Chopra ◽  
Ratnesh R. Saxena
Keyword(s):  
Linear Programming ◽  
Linear Programming Problem ◽  
Fuzzy Numbers ◽  
Optimization Problems ◽  
Fuzzy Linear Programming ◽  
Matrix Game ◽  
Matrix Games ◽  
Trapezoidal Fuzzy Numbers ◽  
Fast Approach ◽  
The Matrix

The aim of this paper is to develop an effective method for solving matrix game with payoffs of trapezoidal fuzzy numbers (TrFNs). The method always assures that players’ gain-floor and loss-ceiling have a common TrFN-type fuzzy value and hereby any matrix game with payoffs of TrFNs has a TrFN-type fuzzy value. The matrix game is first converted to a fuzzy linear programming problem, which is converted to three different optimization problems, which are then solved to get the optimum value of the game. The proposed method has an edge over other method as this focuses only on matrix games with payoff element as symmetric trapezoidal fuzzy number, which might not always be the case. A numerical example is given to illustrate the method.

Sensitivity Analysis and Optimization of Mechanical System Design

1988 ◽  
Author(s):  
W. J. Langner
Keyword(s):  
Linear Equation ◽  
Mechanical System ◽  
Numerical Integration ◽  
Equation System ◽  
Sensitivity Analyses ◽  
Design Parameters ◽  
Governing Equations ◽  
Integration Algorithm ◽  
Linear Equation System ◽  
The Matrix

Abstract The paper follows studies on simulation of three-dimensional mechanical dynamic systems with the help of sparse matrix and stiff integration numerical algorithms. For sensitivity analyses and the application of numerical optimization procedures it is substantial to calculate the effect of design parameters on the system behaviour by means of derivatives of state variables with respect to the design parameters. For static and quasi static analyses the computation of these derivatives from the governing equations leads to a linear equation system. The matrix of this set of linear equations shows to be the Jacobian matrix required in the numerical integration process solving the system of governing equations for the mechanical system. Thus the factorization of the matrix perfomed by the numerical integration algorithm can be reused solving the linear equation system for the state variable sensitivities. Some example demonstrate the simplicity of building the right hand sides of the linear equation system. Also it is demonstrated that the procedure proposed neatly fits into a modular concept for simulation model building and analysis.

scholarly journals MENENTUKAN DETERMINAN SUATU MATRIKS DENGAN METODE CHIO

2016 ◽  
Vol 6 (1:) ◽  
pp. 33-40
Author(s):  
Asep Saepudin
Keyword(s):  
Mathematical Model ◽  
Matrix Element ◽  
Matrix Theory ◽  
Human Life ◽  
Equation System ◽  
Inverse Matrix ◽  
Matrix Methods ◽  
Linear Equation System ◽  
The Matrix ◽  
Mathematical Sciences

Matrix theory is a branch of linear algebra that discussed in the mathematical sciences. Mathematical sciences play an important role in human life, it is necessary to solve problems that can not be solved directly. Thus, the problem can be transformed into the form of a mathematical model. One is the SPL (Linear Equation System). Various methods can be used to solve it. But for the SPL with a large variable can be solved by matrix methods, namely the inverse matrix. In the inverse matrix of the determinants involved. If the search value that ordo major determinant of the matrix (𝑛×𝑛), it would require an effective method. One is the method of Chio. Chio method can be applied to all square matrixas long as the element is 𝑎11 not equal to zero (𝑎11≠0). Chio method of calculating the determinant of the matrix by decomposing determinant will look into sub-determinant of degree two (2×2) using the matrix element row 1 and column 1 as pointof departure. The decomposition is performed using the following sized matrix:   Keywords: Matrix, Matrix Determinant, Chio method.

An approximation algorithm for solving standard quadratic optimization problems

2020 ◽  
Vol 39 (3) ◽  
pp. 4383-4392
Author(s):  
Lunshan Gao
Keyword(s):  
Linear Programming ◽  
Approximation Method ◽  
Heuristic Algorithms ◽  
Optimization Problems ◽  
Quadratic Optimization ◽  
Approximate Algorithm ◽  
Computational Time ◽  
Standard Quadratic Optimization ◽  
The Matrix ◽  
Possibility Distributions

Standard quadratic optimization problems (StQPs) are NP-hard in computational complexity theory when the matrix is indefinite. This paper describes an approximate algorithm of finding inner optimal values of StQPs. The approximate algorithm fuzzifies variable x ∈ Rn with normalized possibility distributions and simplifies the solving of StQPs. The approximation ratio is discussed and determined. Numerical results show that (1) the new algorithm achieves higher accuracy than the semidefinite programming method and linear programming approximation method; (2) the novel algorithm consumes less than one out of fourth computational time that is consumed by linear programming approximation method; (3) the computational time of the new algorithm does not correlate with the matrix densities whereas the computational times of the branch-and-bound and heuristic algorithms do.

scholarly journals Vertically Symmetric Alternating Sign Matrices and a Multivariate Laurent Polynomial Identity

10.37236/4436 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ilse Fischer ◽  
Lukas Riegler
Keyword(s):  
Linear Equation ◽  
Polynomial Identity ◽  
Computer Experiments ◽  
Equation System ◽  
Laurent Polynomial ◽  
Alternative Proof ◽  
Alternating Sign Matrices ◽  
Linear Equation System ◽  
The Matrix ◽  
Sign Matrix

In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers appearing in a conjecture concerning the number of vertically symmetric alternating sign matrices with respect to the position of the first 1 in the second row of the matrix establish the solution of a linear equation system similar to the one for the ordinary refined ASM numbers. In this paper we show how our attempt to prove this fact naturally leads to a more general conjectural multivariate Laurent polynomial identity. Remarkably, in contrast to the ordinary refined ASM numbers, we need to extend the combinatorial interpretation of the numbers to parameters which are not contained in the combinatorial admissible domain. Some partial results towards proving the conjectured multivariate Laurent polynomial identity and additional motivation why to study it are presented as well.

Interregional Equilibrium and Fuzzy Linear Programming: 2

1988 ◽  
Vol 20 (2) ◽  
pp. 219-230 ◽  
Author(s):  
Y Leung
Keyword(s):  
Linear Programming ◽  
Equilibrium Solution ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Fuzzy Linear Programming ◽  
Fuzzy Constraints ◽  
Programming Framework ◽  
Special Case

Extended on the approach in the first part of this two-part series of papers, multiobjective interregional equilibrium is analyzed within a multiobjective fuzzy linear programming framework. Interregional problems with precise objectives and precise constraints, with fuzzy objectives and precise constraints, and with fuzzy objectives and fuzzy constraints are individually examined. Properties of the equilibrium solution and of the associated dual optimal problem are investigated. The present framework comprises ordinary multiobjective interregional equilibrium problems as a special case. A variety of interregional optimization problems can be effectively solved through the use of fuzzy linear programming, as discussed in this two-part series of papers.

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