Local LaGrange multiplier

Here (r) is a local Lagrange multiplier, and the e,- values are global Lagrange multipliers that ensure that Eqs. [32] and [33], respectively, are satisfied at the solution point. The signs in front of the Lagrange multipliers in q. [34] are arbitrary and have been set negative for convenience. Assuming that the functions < ) (r) vanish at the boundary, then the < >ft(r), which satisfy the constraints and make T,[ < >jt(r) ] a minimum, are the solutions of the equation... 

The constrained equations of motion in cartesian eoordinates can be solved by the SHAKE or (the essentially equivalent) RATTLE method (see [8]) which requires the solution of a non-linear system of equations in the Lagrange multiplier funetion A. The equivalent formulation in local coordinates ean still be integrated by using the explicit Verlet method. 

Let x be a local minimum or maximum for the problem (8.15), and assume that the constraint gradients Vhj(x ),j — 1,m, are linearly independent. Then there exists a vector of Lagrange multipliers A = (Af,..., A ) such that (x A ) satisfies the first-order necessary conditions (8.17)-(8.18). 

At each iteration, NLP algorithms form new estimates not only of the decision variables x but also of the Lagrange multipliers A and u. If, at these estimates, all constraints are satisfied and the KTC are satisfied to within specified tolerances, the algorithm stops. At a local optimum, the optimal multiplier values provide useful sensitivity information. In the NLP (8.25)-(8.26), let V (b, c) be the optimal value of the objective/at a local minimum, viewed as a function of the right-hand sides of the constraints b and c. Then, under additional conditions (see Luenberger, 1984, Chapter 10)... 

Let it be a local optimum of problem (3.3), the functions f(x),h(x),g(x) be twice continuously differentiable, and the second-order constraint qualification holds at x. If there exist Lagrange multipliers A, fi satisfying the KKT first-order necessary conditions ... 

If jf = (0,0) is a local minimum, then it satisfies the first-order necessary KKT conditions. From the gradient with respect to x1( the Lagrange multiplier Ji can be obtained ... 

The Lagrange multiplier p. determined by normalization, is the chemical potential [232], such that pt = dE/dN when the indicated derivative is defined. This derivation requires the locality hypothesis, that a Frechet derivative of Fs p exists as a local function (r). 

We now impose not only a given overall strength but also a given envelope. The average local intensity is thus also specified. This requires the introduction of the set of N constraints (3.13), each of which is assigned the Lagrange multiplier yf. The distribution of maximal entropy subject to the three constraints (1)—(3)is... 

For fixed normalization the Lagrange multiplier terms in 8Ts vanish. If these constants are undetermined, it might appear that they could be replaced by a single global constant pt. If so, this would result in the formula [22] 8Ts = J d3r p, — v(r) 8p(r). Then the density functional derivative would be a local function vr(v) such that STj/Sp = Vj-(r) = ix — v(r). This is the Thomas-Fermi equation, so that the locality hypothesis for vT implies an exact Thomas-Fermi theory for noninteracting electrons. 

Yet another technique is to minimize the Kohn-Sham energy as a functional of the density matrix, Eq. (65). In the first method of this type, due to Li et al. [47], the ground state of the system was only a local minimum of the energy functional. To surmount this difficulty, one may choose to explicitly impose the idempotency constraint on the density matrix, Eq. (71), with a Lagrange multiplier. For exam-... 

The localized orbitals (being some other orthonormal basis set in the space spanned by the canonical orbitals) satisfy the Fock equation (8.19) with the off-diagonal Lagrange multipliers. 

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