Lagrangian functions

Introducing the Dirac bra and ket notation for operators and states of the electrons, while explicitly stating the nuclear coordinates, the operator t Q,Q, t) is then expanded in the states Xn(Q))- The Lagrangian functional becomes... 

To show that this constrained minimization is indeed equivalent to the steady-state formulation, let us adjoin the equality constraints to the objective function to form the Lagrangian function,... 

The Lagrangian function is defined as the difference between the kinetic and potential energies... 

The definition of a Lagrangian function can be generalized to a system with many particles and eventually also to a field that represents a continuously... 

The first-order necessary conditions (8.7) and (8.8) can be used to find an optimal solution. Assume x and A are unknown. The Lagrangian function for the problem in Example 8.1 is... 

Define Lagrange multipliers A, associated with the equalities and Uj for the inequalities, and form the Lagrangian function... 

Solution. Although the objective function of this problem is convex, the inequality constraint does not define a convex feasible region as shown in Figure E8.4. The geometric interpretation is to find the points in the feasible region closest to (1, 0). The Lagrangian function for this problem is... 

The second order sufficiency conditions show that the first two of these three Kuhn-Tucker points are local minima, and the third is not. The Hessian matrix of the Lagrangian function is... 

The augmented Lagrangian is a smooth exact penalty function. For simplicity, we describe it for problems having only equality constraints, but it is easily extended to problems that include inequalities. The augmented Lagrangian function is... 

Successive quadratic programming (SQP) methods solve a sequence of quadratic programming approximations to a nonlinear programming problem. Quadratic programs (QPs) have a quadratic objective function and linear constraints, and there exist efficient procedures for solving them see Section 8.3. As in SLP, the linear constraints are linearizations of the actual constraints about the selected point. The objective is a quadratic approximation to the Lagrangian function, and the algorithm is simply Newton s method applied to the KTC of the problem. 

To accommodate the constraint (b)9 a Lagrangian function L is formed by augmenting/with Equation (b), using a Lagrange multiplier (o... 

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... 

Step 1. For a given set of Lagrange multipliers and penalty parameter minimize the Lagrangian function L R) to obtain an improved estimate of the factorized 2-RDM at the energy minimum. 

Two major forms of the OCFE procedure are common and differ only in the trial functions used. One uses the Lagrangian functions and adds conditions to make the first derivatives continuous across the element boundaries, and the other uses Hermite polynomials, which automatically have continuous first derivatives between elements. Difficulties in the numerical integration of the resulting system of equations occur with the use of both types of trial functions, and personal preference must then dictate which is to be used. The final equations that need to be integrated after application of the OCFE method in the axial dimension to the reactor equations (radial collocation is performed using simple orthogonal collocation) can be expressed in the form... 

There are N spin orbitals, so there are N(N+ 1)/2 independent constraints (note a b) = (6 a) ), so we need that many undetermined multipliers in our lagrangian function, for which we present... 

The restricted sum will prove inconvenient, but it can be eliminated. By taking the complex conjugate of the constraints and the lagrangian function, equation 86,... 

This allows for a more compact notation to be employed in writing the variance in the lagrangian function... 

The variable L is known as the Lagrangian function or kinetic potential, defined as... 

It is well known that when (3.1) contains no inequality constraints the Lagrangian function... 

Independently of this work, Fletcher 53 started with (3.4) and sought to make y and Q continuous functions of x which would converge to the required values at the stationary point. Later 64 he generalized the approach to deal with inequality constraints, and showed that the Lagrangian function for (3.1) ... 

Very recently Sargent 88 has shown that, although the Lagrangian function is not an exact penalty function, it can indeed be used in a descent test to force global convergence. Moreover the test is satisfied without step-reduction in the final stages, so the convergence is superlinear. Thus the penalty parameter and its associated problems are eliminated. 

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