Augmented Lagrange function

You might ask vhy we use the augmented Lagrange function (12.23) rather just opting directly for the follo ving Lagrange function ... 

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... 

The problem at hand has only two constraints, namely. Equations (2.27) and (2.28). With the application of the Lagrange Multiplier Rule, this problem is equivalent to the minimization of the following augmented objective functional ... 

The above difficulty is surmounted by introducing an undetermined function, A(t), called Lagrange multiplier, in the augmented objective functional defined... 

Step 3 The augmented objective functional is M = J + pK where is a Lagrange multiplier. From the Lagrange Multiplier Theorem, assuming that... 

The state equation, G = 0, constitutes a partial differential equation constraint. Applying the Lagrange Multiplier Rule, the equivalent problem is to find the control function D c) that minimizes the augmented objective functional... 

It can be considered as the ordinary Lagrange function augmented by the quadratic term. 

Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method)... 

The point where the constraint is satisfied, (x0,yo), may or may not belong to the data set (xj,yj) i=l,...,N. The above constrained minimization problem can be transformed into an unconstrained one by introducing the Lagrange multiplier, to and augmenting the least squares objective function to form the La-grangian,... 

The problem of minimizing Equation 14.24 subject to the constraint given by Equation 14.26 or 14.28 is transformed into an unconstrained one by introducing the Lagrange multiplier, to, and augmenting the LS objective function, SLS(k), to yield... 

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

A more enlightening example is that in which not only is the constraint of normalization of the probability distribution imposed, but it is assumed that we know the average value of some quantity, for example, E) = EiPi Ei). In this case, we again consider an augmented function, which involves two Lagrange multipliers, one tied to each constraint, and given by... 

One application of partial derivatives is in the search for minimum and maximum values of a function. An extremum (minimum or maximum) of a function in a region is found either at a boundary of the region or at a point where all of the partial derivatives vanish. A constrained maximum or minimum is found by the method of Lagrange, in which a particular augmented function is maximized or minimized. 

To proceed, we need to apply the Lagrange Multiplier Rule, the details of which will be provided later in Chapter 4. According to this rule, the above constrained problem is equivalent to the problem of finding the control T t) that maximizes the following augmented functional ... 

In the present problem defined by Equation (3.4)-(3.6), we adjoin the state equation constraint to / using a Lagrange multiplier A and obtain the augmented functional... 

Using the Lagrange Multiplier Rule, the augmented functional is... 

This rule is based on the Lagrange Multiplier Theorem. Consider the augmented functional... 

Based on the Lagrange Multiplier Rule (see Section 4.3.3.2, p. 103), the above problem is equivalent to minimizing the augmented functional... 

We seek to minimize this energy, but for this case we need a constraint during the minimization, namely for the normalization of the wave function. (If we are seeking many wave functions, we then need to have constraints for orthonormality—each eigenfunction must be normalized and orthogonal to all the other wave functions.) The augmented functional with the Lagrange multiplier constraint is... 

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