Lagrange multipliers enforcing constraints

In eqn (5.17), PA(rA) and Oa are Lagrange multipliers enforcing constraints (5.12) and (5.15). All of this charge partitioning functional s parameters were theoretically derived, so its results can be regarded as non-empirical. 

K i) and i a( a)>0 and are Lagrange multipliers enforcing constraints (5.21) and (5.22) co is the unit sphere surface comprised of all possible endpoints for h, and integration over m means integration over all possible choices for h This spin partitioning functional has a unique minimum that is found by an iterative solution algorithm. ... 

Steps 1-3 are repeated until the maximum absolute error in the constraints falls below a target threshhold. Before the first iteration the Lagrange multipliers may be initialized to zero and the penalty parameter set to 0.1. The constraints are not fully enforced until convergence, and the energy in the primal program approaches the optimal value from below. 

Minimization of the functional (41) has to be performed under the orthonormality requirement in Eq. (4) for the NSOs, whereas the ONs conform to the N-representability conditions for D. Bounds on the ONs are enforced by setting rii = cos y, and varying y,- without constraints. The other two conditions may easily be taken into account by the method of Lagrange multipliers. 

These constraints can be enforced within the variational optimization of the energy function mentioned above by introducing a set of Lagrange multipliers jj, one for each constraint condition, and subsequently differentiating... 

To maximize the current and thus the round-trip rate, this integral must be minimized. However, there is a constraint H E) is a probability distribution and must remain normalized which can be enforced with a Lagrange multiplier ... 

We have assumed here that the variations are performed within the domain of normalized densities. Alternatively, the minimization can be performed using the Euler-Lagrange procedure. Then the densities are allowed to vary also outside the normalization domain. This we shall do by relaxing the normalization constraint of the wavefunctions and by using the definition (3) of the density also in the extended domain. The normalization constraint is enforced by means of a Lagrange multiplier (pi),... 

Above we have assumed that the minimization is carried out within the domain of normalized of densities. Alternatively, we can perform the minimization, using the Euler-Lagrange procedure. Then we use the extension of the functionals valid also outside the normalization domain and enforce the normalization constraint by a Lagrange multiplier.5 For the Levy-Lieb energy functional (70) this leads to... 

The truncation of the series expansion (4) at second order allows the solution to be found by solving a linear system of equations. The only complication is the need to enforce the constraint (2), which can be taken care of with the method of Lagrange multipliers. In this context, the Lagrange multiplier can be interpreted as the chemical potential, and the solution to the constrained problem is the charge distribution and chemical potential which minimizes the free energy... 

We cannot solve (10.3.31) merely by forming the total differential wrt the mole numbers and setting that differential to zero, because the dN,- are not independent instead, they are related through (10.3.32). In the stoichiometric development in 10.3.2, the constraint (10.3.32) was included in the problem through stoichiometric coefficients and an extent of reaction t,. Here we impose the constraint in a different way namely, we allow the N, in the equilibrium condition (10.3.31) to vary independently and enforce the constraints (10.3.32) via Lagrange multipliers (see Appendix I). 

How this mass parameter has to be chosen is extensively discussed in [3]. A critical point of view about the fictitious mass parameter and about arguments used for the justification of the CPMD approach is given in [16]. The dot in this Lagrangian indicates the time derivative thus it is apparent that the wavefunction fulfils the same task as the nuclear position variable. The potential is now a functional of the electronic energy plus the constraints which are enforced in order to satisfy quantum mechanics, i.e., the orbitals which are altered during time evolution are supposed to stay orthonormal see second term of (12). The additional constraint is introduced by the standard Lagrange multipliers approach, where the Aij are the Lagrange multipliers and by is the Kronecker delta ... 

This uses a data constraint for each pixel in the measured image, and requires the introduction of one Lagrange multiplier per pixel. The main problem with this is that it fails to recognize that the data are noisy. Consequently, exact agreement with the data should not be enforced. An alternative, the so-called constraint, requires only statistical agreement. In particular,... 

Se is tile set of equality constraints, Cm sX ) = 0 and Si is the set of inequality constraints, Cm<=Si x) > 0. If /LX is the vector of Lagrange multipliers that enforce the constraints, tiie constrained minimum satisfies... 

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

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