Lagrange Multiplier Rule functions

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 ... 

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 ... 

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

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

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... 

In the course of study, students should master material that is both simple and complex. Much of this involves familiarity with the set of mathematical tools repeatedly used throughout this book. The Appendices provide ample reference to such a toolbox. These include matrix algebra, determinants, vector spaces, vector orthogonalization, secular equations, matrix diagonalization, point group theory, delta functions, finding conditional extrema (Lagrange multipliers, penalty function methods), Slater-Condon rules, as well as secondary quantization. 

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

Table 14.1 Wigner s 2n + 1 rule for the wave-function parameters and the 2n -l- 2 rule for the Lagrange multipliers...

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