Lagrange multipliers, applications

Lagrange multipliers, applications of, 631 for optimum conditions, 402-403 Land, cost of, 176 Lang factors, 184... 

Application of Lagrange Multipliers in Analytical Chemistry, Anal. Chem. 57, 1985, 956-957. 

The best quality to be found may be a temperature, a temperature program or profile, a concentration, a conversion, a yield of preferred product, kind of reactor, size of reactor, daily production, profit or cost — a maximum or minimum of some of these factors. Examples of some of these cases are in this group of problems. When mathematical equations can be formulated, peaks or valleys are found by elementary mathematics or graphically. With several independent variables quite sophisticated mathematical procedures are available to find optima. Here a case of two variables occurs in problem P4.12.ll that is solved graphically. The application of Lagrange Multipliers for finding constrained optima is made in problem P4.ll.19. 

EXAMPLE 8.3 APPLICATION OF THE LAGRANGE MULTIPLIER METHOD WITH NONLINEAR INEQUALITY CONSTRAINTS... 

Execution times for the overall ammonia plant model, of which the C02 capture system is a small part, are on the order of 30 s for the parameter estimation case, and less than a minute for an Optimize case. The model consists of over 65,000 variables, 60,000 equations, and over 300,000 nonzero Jacobian elements (partial derivates of the equation residuals with respect to variables). This problem size is moderate for RTO applications since problems over four times as large have been deployed on many occasions. Residuals are solved to quite tight tolerances, with the tolerance for the worst scaled residual set at approximately 1.0 x 10 9 or less. A scaled residual is the residual equation imbalance times its Lagrange multiplier, a measure of its importance. Tight tolerances are required to assure that all equations (residuals) are solved well, even when they involve, for instance, very small but important numbers such as electrolyte molar balances. 

Maximum eld of first-order consecutive reactions in CSTR by application of Lagrange multipliers... 

Hurley proposed a simple, sufficient condition for the applicability of the Hellmann-Feynman theorem. " If within a variational framework, the family of trial functions is invariant to changes in parameter a, then the Hellmann-Feynman theorem is satisfied by the optimum trial function. In variational approaches involving Lagrange multipliers, for example, in the Hartree-Fock and multiconfigurational self-consistent field methods, Hurley s condition is fulfilled. ... 

B. van de Graaf and J. M. A. Baas, /. Comput. Chem., 5 314 (1984). Empirical Force Field Calculations. 23. The Lagrange Multiplier Method for Manipulating Geometries. Implementation and Applications in Molecular Mechanics. 

An important aspect of CC3 and other approximate triples methods is that due to the approximations made in the triples equations, it is possible to avoid the storage of the triples parts of the cluster amplitudes and Lagrange multipliers. This is essential since otherwise the storage and not the CPU requirements would limit the applicability of these methods. 

To prove this assertion, it is first useful to consider the mathematical technique of Lagrange multipliers, a method used to find the extreme (maximum or minimum) value of a function subject to constraints. Rather than develop the method in complete generality, we merely introduce it by application to the problem just considered equilibrium in a single-phase, multiple-chemical reaction system. 

The rest of this section provides the details of how the functional form given in Eq. (62) is derived and with the practical aspect of how to determine the values of the Lagrange multipliers Xr, r = 0, 1,. . ., M. The reader interested in applications can proceed directly to Sec. IV.B. 

A practical route to an initial esiimate of the Lagrange multipliers X/ s is via the Burg algorithm (104,105). This generates a spectrum of maximum entropy provided the times tr are equally spaced, tr = rht. In many applications (see Sec. IV.B), there are specific reasons to preferring unequally spaced time intervals. One option is to use the... 

This simple example could, of course, have been solved by simply substituting the constraint equation into the original function, to give a function of just one of the variables. However, in many cases this is not possible. The Lagrange multiplier method provides a powerful approach which is widely applicable to problems involving constraints such as in constraint dynamics (Section 7.5) and in quantum mechanics. 

Finally note that we could combine the three equations (10.4.43)-(10.4.45) to eliminate the two Lagrange multipliers. But doing so produces the stoichiometric equation (10.4.31) that relates the equilibrium constant to the mole fractions. In other words, the stoichiometric and nonstoicdiiometric developments are merely two different formulations of the same equations, though in particular applications one approacdi or the other may be easier to use. 

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 ... 

Before considering the applications of the Lagrange Multiplier Theorem, it is worthwhile to generalize the theorem to handle several equahty constraints and functions. 

In order to minimize F, the calculus of variations was used with the aid of a Lagrange multiplier. Then, application of the conservation of mass, Onsager reciprocity relation for the mobilities, and neglecting terms beyond the quadratic derivatives in Eq. (A.4) (De Fontaine, 1967), one obtains... 

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