The Lagrange Multiplier Method

By combining the Lagrange multiplier method with the highly efficient delocalized internal coordinates, a very powerfiil algoritlun for constrained optimization has been developed [ ]. Given that delocalized internal coordinates are potentially linear combinations of all possible primitive stretches, bends and torsions in the system, cf Z-matrix coordinates which are individual primitives, it would seem very difficult to impose any constraints at all however, as... 

To find the minimum crossing point we shall use the Lagrange multiplier method. That is, we consider... 

EXAMPLE 8.3 APPLICATION OF THE LAGRANGE MULTIPLIER METHOD WITH NONLINEAR INEQUALITY CONSTRAINTS... 

Solve the following problem via the Lagrange multiplier method ... 

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

Figure 6.1 Search for the minimum of the Gibbs function in a two-component space (nn and ni2 are mole numbers) with the mass conservation constraints Bn = q. The search direction is the projection of the gradient onto the constraint subspace. Minimum is attained when the gradient is orthogonal to the constraint direction, which is the geometrical expression of the Lagrange multiplier methods.

This direct method yields the same results as the Lagrange multiplier method for finding the kernel introduced in Ref. [16] for the square-rectangle case, and used for kernels in 2D and 3D in Refs. [7-12],... 

Constraints in optimization problems often exist in such a fashion that they cannot be eliminated explicitly—for example, nonlinear algebraic constraints involving transcendental functions such as exp(x). The Lagrange multiplier method can be used to eliminate constraints explicitly in multivariable optimization problems. Lagrange multipliers are also useful for studying the parametric sensitivity of the solution subject to the constraints. 

The Lagrange multiplier method, applied to this minimisation problem, allows to recover the usual Kohn-Sham equations, modulo an unitary transform within the space of occupied orbitals, as follows. One Lagrange multiplier for each orthonormalisation constraint is introduced, such that ... 

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

In Eq. (16), yci stands for the number of segments of a chain in conformation c located in layer i. The two equations express the obvious conditions that each lattice layer must be occupied and that the total number of chains is constant. The Lagrange multiplier method is used to calculate the minimum free energy subject to the above constraints. By introducing the multipliers at, for each of the constraints given by Eq. (16), and (i for the constraint expressed by Eq. (17), one can write... 

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) ... 

B. van de Graaf and J. M. A. Baas, /. Comput. Chem., 5 314 (1984). Empirical Force Field Calculations. 23. The Lagrange Multiplier Method for Manipulating Geometries. Implementation and Applications in Molecular Mechanics. 

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

Clearly, Eq. 8.8-9 gives the same equilibrium requirement as before (see Eq. 8.8-4). whereas Eq. 8.8-10 ensures that the stoichiometric constraints are satisfied in solving the problem. Thus the Lagrange multiplier method yields the same results as the direct substitution or brute-force approach. Although the Lagrange multiplier method appears awkward when applied to the very simple problem here, its real utility is for complicated problems in which the number of constraints is large or the constraints are nonlinear in the independent variables, so that direct substitution is very difficult or impossible. 

This simple example could, of course, have been solved by simply substituting the constraint equation into the original function, to give a function of just one of the variables. However, in many cases this is not possible. The Lagrange multiplier method provides a powerful approach which is widely applicable to problems involving constraints such as in constraint dynamics (Section 7.5) and in quantum mechanics. 

Thus we seek the conditional minimum. We will find it using the Lagrange multipliers method (see Appendix N available at booksite.elsevier.com/978-0-444-59436-5, p. el21). In this method, the equations of the constraints multiphed by the Lagrange multipliers are added (or subtracted, does not matter) to the original function that is to be minimized. Then we minimize the function as if the constraints did not exist. 

A conditional minimum can be found by using the Lagrange multipliers method, as described in Appendix N available at booksite.elsevier.com/978-0-444-59436-5. 

Well, let us see whether the Lagrange multipliers method will give the same result. 

In practical calculations making use of the Kohn-Sham method, the Kohn-Sham equation is used. This equation is a one-electron SCF equation applying the Slater determinant to the wavefunction of the Hartree method, similarly to the Hartree-Fock method. Therefore, in the same manner as the Hartree-Fock equation, this equation is derived to determine the lowest energy by means of the Lagrange multiplier method, subject to the normalization of the wavefunction (Parr and Yang 1994). As a consequence, it gives a similar Fock operator for the nonlinear equation. 

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