Extensions of Lagrange Multipliers in Nonlinear Programming

1969 ◽  
Vol 17 (6) ◽  
pp. 1280-1297 ◽  
Author(s):  
F. J. Gould
Keyword(s):  
Nonlinear Programming ◽  
Lagrange Multipliers

scholarly journals PEMECAHAN MASALAH NONLINEAR PROGRAMMING DENGAN LAGRANGEAN METHOD MENGGUNAKAN SUBSTITUTION METHOD ATAU MICROSOFT EXCEL

2017 ◽  
Vol 4 (2) ◽  
Author(s):  
Rudy Santosa Sudirga
Keyword(s):  
Nonlinear Programming ◽  
Programming Problem ◽  
Lagrange Multipliers ◽  
Optimization Problems ◽  
Stationary Points ◽  
Mathematical Tool ◽  
Nonlinear Programming Problem ◽  
Microsoft Excel ◽  
Substitution Method ◽  
General Technique

<p><em>Lagrange Multipliers are a mathematical tool for constrained optimization of differentiable functions. The given procedure defines the Lagrangean Method for identifying the stationary points of optimization problems with equality constraints; L (ℷ, X) = f(X) – ℷg(X – C). This function is called the Lagrangean function and the parameters ℷ is the Lagrange Multipliers. The partial derivatives of this equation ∂L/∂ℷ = 0 and ∂L/∂X = 0 will give the optimal result for X.</em></p><p><em>However, in some cases, sometimes we have been facing some difficulties to solve nonlinear programming problem with this method, therefore we would like to introduce the <strong>Substitution Method</strong> or <strong>Microsoft Excel Method</strong>, which is simple, easier and faster to solve the nonlinear programming problem. Because of the mathematical nature of nonlinear programming models, and in management science we do not have a single general technique to solve all mathematical models that can arise in practice, therefore simpler approaches should be explored first. In some cases, a “common sense” solution may be reached through simple observations.  </em><em>  </em></p>

Uniqueness of multipliers in optimal control: the missing piece

2018 ◽  
Vol 36 (4) ◽  
pp. 1395-1411
Author(s):  
Jorge A Becerril ◽  
Karla L Cortez ◽  
Javier F Rosenblueth
Keyword(s):  
Optimal Control ◽  
Nonlinear Programming ◽  
Optimality Conditions ◽  
Lagrange Multipliers ◽  
Constraint Qualification ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Simple Condition ◽  
Necessary And Sufficient

Abstract In a well-known paper by Kyparisis it is proved that, in nonlinear programming, the uniqueness of Lagrange multipliers is equivalent to a strict version of the Mangasarian–Fromovitz constraint qualification which, in turn, implies the satisfaction of second-order necessary optimality conditions. This is no longer the case in optimal control where, as shown in a recent paper, the corresponding strict constraint qualification is only sufficient for the uniqueness of multipliers. In this paper we exhibit the missing piece: a new, simple condition, implied by the strict constraint qualification, which is necessary and sufficient for the uniqueness of multipliers in optimal control.

Applying the Method of Lagrange Multipliers to Derive an Estimator for Unsampled Soil Properties

2014 ◽  
Vol 11 (1) ◽  
pp. 15
Author(s):  
Set Foong Ng ◽  
Pei Eng Ch’ng ◽  
Yee Ming Chew ◽  
Kok Shien Ng
Keyword(s):  
Soil Properties ◽  
Lagrange Multipliers ◽  
Unbiased Estimator ◽  
Soil Property ◽  
Minimum Variance ◽  
Distance Method ◽  
True Value ◽  
A Site ◽  
Method Of Lagrange Multipliers ◽  
Inverse Distance

Soil properties are very crucial for civil engineers to differentiate one type of soil from another and to predict its mechanical behavior. However, it is not practical to measure soil properties at all the locations at a site. In this paper, an estimator is derived to estimate the unknown values for soil properties from locations where soil samples were not collected. The estimator is obtained by combining the concept of the ‘Inverse Distance Method’ into the technique of ‘Kriging’. The method of Lagrange Multipliers is applied in this paper. It is shown that the estimator derived in this paper is an unbiased estimator. The partiality of the estimator with respect to the true value is zero. Hence, the estimated value will be equal to the true value of the soil property. It is also shown that the variance between the estimator and the soil property is minimised. Hence, the distribution of this unbiased estimator with minimum variance spreads the least from the true value. With this characteristic of minimum variance unbiased estimator, a high accuracy estimation of soil property could be obtained.

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