Mathematical programming via the least-squares method

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Evald Übi
Keyword(s):  
Least Squares ◽  
Dual Problem ◽  
Least Squares Method ◽  
Linear Inequalities ◽  
Initial Solution ◽  
Test Problems ◽  
Minimal Norm ◽  
Stable Algorithm ◽  
System Of Linear Inequalities ◽  
Linear And Nonlinear Programming

AbstractThe least-squares method is used to obtain a stable algorithm for a system of linear inequalities as well as linear and nonlinear programming. For these problems the solution with minimal norm for a system of linear inequalities is found by solving the non-negative least-squares (NNLS) problem. Approximate and exact solutions of these problems are discussed. Attention is mainly paid to finding the initial solution to an LP problem. For this purpose an NNLS problem is formulated, enabling finding the initial solution to the primal or dual problem, which may turn out to be optimal. The presented methods are primarily suitable for ill-conditioned and degenerate problems, as well as for LP problems for which the initial solution is not known. The algorithms are illustrated using some test problems.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

scholarly journals Improved Least-squares Error Estimates for Scalar Hyperbolic Problems

2001 ◽  
Vol 1 (2) ◽  
pp. 115-124 ◽  
Author(s):  
P.B. Bochev ◽  
J. Choi
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Least Squares ◽  
Error Estimates ◽  
Dual Problem ◽  
Least Squares Method ◽  
Hyperbolic Problems ◽  
Elliptic Regularity ◽  
Projection Property ◽  
Variational Framework

AbstractWe consider an L_2-norm least-squares principle for a scalar hyperbolic problem. A proper variational framework for the associated finite element method is developed and studied. Analysis of the discretization error based on the least-squares projection property shows a gap of one. This number cannot be improved with a standard duality argument because the least-squares dual does not possess full elliptic regularity. Using a perturbed dual problem we are able to show that the actual gap of the least-squares method in the constant convection case is not worse than 2/3.

scholarly journals A general iterative solver for unbalanced inconsistent transportation problems

2016 ◽  
Vol 37 (1) ◽  
pp. 7-13
Author(s):  
Doina Carp ◽  
Constantin Popa ◽  
Cristina Serban
Keyword(s):  
Least Squares ◽  
Transport Model ◽  
Linear Inequalities ◽  
Practical Reasons ◽  
Iterative Solver ◽  
Minimal Norm ◽  
Minimal Norm Solution ◽  
Classical Formulation ◽  
Inconsistent Systems ◽  
General Projection

The transportation problem, as a particular case of a linear programme, has probably the highest relative frequency with which appears in applications. At least in its classical formulation, it involves demands and supplies. When, for practical reasons, the total demand cannot satisfy the total supply, the problem becomes unbalanced and inconsistent, and must be reformulated as e.g. finding a least squares solution of an inconsistent system of linear inequalities. A general iterative solver for this class of problems has been proposed by S. P. Han in his 1980 original paper. The drawback of Han’s algorithm consists in the fact that it uses in each iteration the computation of the Moore-Penrose pseudoinverse numerical solution of a subsystem of the initial one, which for bigger dimensions can cause serious computational troubles. In order to overcome these difficulties we propose in this paper a general projection-based minimal norm solution approximant to be used within Han-type algorithms for approximating least squares solutions of inconsistent systems of linear inequalities. Numerical experiments and comparisons on some inconsistent transport model problems are presented.

scholarly journals Approximate Solutions for a Class of Nonlinear Fredholm and Volterra Integro-Differential Equations Using the Polynomial Least Squares Method

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2692
Author(s):  
Bogdan Căruntu ◽  
Mădălina Sofia Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Analytical Solutions ◽  
General Class ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Test Problems ◽  
Numerical Examples ◽  
Approximate Analytical Solutions ◽  
Very High

We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very high, a fact that is illustrated by the numerical examples presented. The comparison with previous approximations computed for the included test problems emphasizes the method’s simplicity and accuracy.

scholarly journals Modified Han algorithm for inconsistent linear inequalities

2015 ◽  
Vol 31 (1) ◽  
pp. 45-52
Author(s):  
DOINA CARP ◽  
◽  
CONSTANTIN POPA ◽  
CRISTINA SERBAN ◽  
◽  
...  
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Numerical Experiments ◽  
Linear Inequalities ◽  
Convergence Properties ◽  
Minimal Norm ◽  
Direct Solver ◽  
Least Squares Solution ◽  
Inconsistent Systems ◽  
Systems Of Linear Inequalities

In this paper we present a modified version of S. P. Han iterative method for solving inconsistent systems of linear inequalities. Our method uses an iterative Kaczmarz-type solver to approximate the minimal norm least squares solution of the problems involved in each iteration of Han’s algorithm. We prove some convergence properties for the sequence of approximations generated in this way and present numerical experiments and comparisons with Han’s and other direct solver based methods for inconsistent linear inequalities.

scholarly journals The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2324
Author(s):  
Bogdan Căruntu ◽  
Mădălina Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Analytical Solutions ◽  
Efficient Method ◽  
General Class ◽  
Least Squares Method ◽  
Test Problems ◽  
Approximate Analytical Solutions

We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the simplicity and accuracy of the method.

‘Least squares’ method—theorem or law?

1980 ◽  
Vol 59 (9) ◽  
pp. 8
Author(s):  
D.E. Turnbull
Keyword(s):  
Least Squares ◽  
Least Squares Method
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