Lagrange undetermined multipliers

There are two main methods for enforcing such constraints. One is the Penalty Function approach, the other the metlrod of Lagrange Undetermined Multipliers. 

By taking the minimum in Eq. (69) subject to the restrictions specified and using Lagrange undetermined multipliers (see, for example, Ref. 6), one finds a set of relationships satisfied by the defect chemical potentials. The results for the three basic types of intrinsic lattice disorder are as follows ... 

A more elegant and useful method was suggested by Lagrange. The fundamental difficulty is that there are fewer variables than the number of derivative conditions. As suggested by Lagrange, we can therefore introduce new constants Ai, A2,..., Ac ( Lagrange undetermined multipliers, one for each constraint) to define a new constrained function / given by... 

The important aspect of (13.70b) is that each pa=Pa(U, V, N) has maximal ( most probable ) character with respect to the natural control variables of S. The constrained maximization procedure to find this optimal distribution by the method of Lagrange undetermined multipliers [see Schrodinger (1949), Sidebar 13.4, for further details] is very similar to that described in Section 5.2. In particular, the pa must be maximal with respect to variations in each control variable, leading to the usual second-derivative curvature conditions such as... 

Write equations for minimization of total Gibbs free energy. This step employs the method of Lagrange undetermined multipliers for minimization under constraint for a discussion of this method, refer to mathematics handbooks. As for its application to minimization of total Gibbs free energy, see Perry and Chilton [7] and Smith and Van Ness [11]. 

Hi) Constrained Maximization Method of Lagrange Undetermined Multipliers The problem of constrained maximization may be posed in its most general form as follows ... 

Suppose you wish to find an extreme value of / (x) on a surface g (x) = c, with c a constant. Explain why the gradients of / and g must be parallel at the desired extrema. Suppose you wish to find an extreme value of / (x) on a surface for which (x) = Ci and g2 (x) = C2- Explain why the gradient of / must be contained in the plane defined by the gradients of gi and g2 at a desired extremum. Explain what this has to do with the Lagrange undetermined multiplier calculation. 

The constraint of constant total bond-order is introduced via the method of Lagrange undetermined multiplier, so that the functional that is minimized with respect to 5i is F = H + L, where L ii C being the undetermined multiplier. 

The one-to-one correspondence between electron density and effective potential, which is proven on the basis of the constrained search formulation, suggests that the effective potential can be determined directly from the electron density. Parr and coworkers developed a procedure for determining highly accurate exchange-correlation potentials from electron densities, which are calculated by high-level ab initio correlation wavefunction theories. This procedure is called the Zhao-Morrison-Parr (ZMP) method (Zhao et al. 1994). In this method, the effective potential is given by the Lagrange undetermined multiplier method with a potential. 

The mathematical problem of locating a minimum on the seam between two energy surfaces is a constrained optimization, i.e., the energy should be minimized subject to the constraint that the reactant and product energies are equal. This may be handled, for example, by the technique of Lagrange undetermined multipliers, and optimization of the Lagrange function may be done by NR techniques ... 

---