Lagrangian multiplier method

To maximize In W, subject to the above constraints, it is convenient to use the Lagrangian multiplier method ... 

With the restriction of Eq. (B-17) and using the Lagrangian multiplier method, we get for the free energy of adsorption AF2... 

An improved method was developed by Chirlian and Francl and called CHELP (CHarges from ELectrostatic Potentials). Their method, which uses a Lagrangian multiplier method for fitting the atomic charges, is fast and noniterative and avoids the initial guess required in the standard least-squares methods. In this approach, the best least-squares fit is obtained by minimizing y ... 

Woods et al. [159] again use the Lagrangian multipliers method, adding the constraint on the dipole moment. Attention is focussed on the numerical aspects of the method, in particular on the effect of number and location of the sampling points, and on the variance of the results with respect to a rotation of the position of these points. The paper extends the analysis to several molecules of medium size, opening the way to the problem of transferability of the PD-AC values obtained for small molecules. 

Besler, Merz and Kollman [163] examine PD-AC values obtained with the MNDO and AM 1 methods (using the procedure of deorthogonalization of the semiempirical wavefunctions) and compare them with STO-3G and 6-3IG values. The authors find that MNDO charges are superior to AMI charges in correlating with PD-AC values obtained from 6-3IG wavefunctions. The analysis is with 20 compounds, and the fitting exploits the Lagrangian multipliers method. 

The principle and procedure of the Lagrangian multiplier method was described in section 5.3.3. It has been shown that the numerical algorithm is based on solving a set of non-linear equations... 

Another important factor in determining chemical equilibria by the Lagrangian multiplier method is determination of the first approximation of the multipliers k = 1,2, M. This first approximation is required in order to enable solution of the set of equations (5.140) by one of the differentiation methods (e.g. Newton s method with reduction parameter) or, for the application of one of the methods in which differentiation is not performed. Let us assume that the first approximation... 

The remaining part of the calculation is identical with that of the Lagrangian multiplier method. 

A similar effect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to infinity. In practice, small positive values remove most of the spin contamination. 

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

By the method of Lagrangian multipliers (a and / in the following) it is found in all cases for arbitrary SNi that... 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

The solution is obtained by means of the Lagrange multipliers method. The Lagrangian for this problem is... 

Assuming uniform prior probabilities, we maximise S subject to these constraints. This is a standard variation problem solved by the use of Lagrangian multipliers. A numerical solution using standard variation methods gives i.p6j=. 05435, 0.07877, 0.11416, 0.16545, 0.23977, 0.34749 with an entropy of 1.61358 natural units. 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

In a convenient method, due to Hamilton (1964), the Lagrangian multipliers representing the constraint are algebraically eliminated from the least-squares expressions. The linear constraints are defined as... 

Constraints may be imposed on a set of simultaneous linear equations by the method of Lagrangian multipliers. Let the Lagrangian multipliers be — - Therefore, add to equation (A.28) the quantity... 

Since the x1 and x averages coincide by definition, to solve the posed problem it is necessary to find a minimum of dispersion (5.25) with conditions (5.22) satisfied. Let us use the method of uncertain Lagrangian multipliers and form an auxiliary expression ... 

On the basis of the Lagrangian method (Lagrangian multipliers) and the conservation of total number of particles and total energy of the system, show that the Maxwell-Boltzmann distribution can take the form... 

Only three of the four variables in Eq. (42) are independent. Under these conditions, optimization can be accomplished by use of the Lagrange multiplier method The necessary relationship for applying the constant Lagrangian multiplier A is given by Eq. (43) ... 

Such constrained extrema can be found by the Lagrange multipliers method One form the Lagrangian ... 

The problem of finding extrema of a function / (xi, X2,..., x, ) = / (x) subject to n constraints may be solved by using the method of Lagrangian multipliers. In the absence of such constraints the necessary condition for the existence of extrema may be stated as... 

In such a case the method of Lagrangian multipliers is feasible such a Lagrangian multiplier e is chosen so that the functional... 

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