Objective functions

The object function we have to estimate is the relative conductivity fi = —— 

FIND NORM OF OBJECTIVE FUNCTION AND CHECK FOR DECREASE 260 FV ABS(F) 

Value of the objective function [(7-23) or (7-24)] at T + AT used for finite difference approximation of the derivative. 

FIND ENTHALPY OBJECTIVE FUNCTION G lAOIABATICI 255 CALL ENTH(N lOfKEEfOtX TNyPfHLfER) 

F Rachford-Rice objective function for liquid-liquid separa- 

FIND EQUILIBRIUM OBJECTIVE FUNCTION F AND UNNORMALIZEO COMPOSITIONS 

APPLY STEP-LIMITING PROCEDURE TO DECREASE OBJECTIVE FUNCTION 265 KD=l 

DFA Partial derivative of the Rachford-Rice objective function (7-13) with respect to the vapor-feed ratio. 

Here and below we emphasize the dependence of the objective functional on 5, because later we shall investigate the convergence of the solutions of problem (2.189) as 5 —> 0. 

In application of the Newton-Raphson iteration to these objective functions [Equations (7-23) through (7-26)], the near linear nature of the functions makes the use of step-limiting unnecessary. 

For bubble and dew-point calculations we have, respectively, the objective functions 

For liquid-liquid systems, the separations are isothermal and the objective function is one-dimensional, consisting of Equation (7-17). However, the composition dependence of the 

A step-limited Newton-Raphson iteration, applied to the Rachford-Rice objective function, is used to solve for A, the vapor to feed mole ratio, for an isothermal flash. For an adiabatic flash, an enthalpy balance is included in a two-dimensional Newton-Raphson iteration to yield both A and T. Details are given in Chapter 7. 

We shall investigate the problem of controlling the external forces with an objective functional describing the crack opening 

Change in extract-feed ratio from one iteration to the next. Partial derivative of Rachford-Rice objective function with respect to extract-feed ratio. 

For binary vapor-liquid equilibrium measurements, the parameters sought are those that minimize the objective function 

In the case of the adiabatic flash, application of a two-dimensional Newton-Raphson iteration to the objective functions represented by Equations (7-13) and (7-14), with Q/F = 0, is used to provide new estimates of a and T simultaneously. The derivatives with respect to a in the Jacobian matrix are found analytically while those with respect to T are found by finite-difference approximation 

The Newton-Raphson approach, being essentially a point-slope method, converges most rapidly for near linear objective functions. Thus it is helpful to note that tends to vary as 1/P and as exp(l/T). For bubble-point-temperature calculation, we can define an objective function 

The equation systems representing equilibrium separation calculations can be considered multidimensional, nonlinear objective functions 

Subroutine FUNDR. This subroutine calculates the required derivatives for REGRES by central difference, using EVAL to calculate the objective functions. 

Liquid-liquid equilibrium separation calculations are superficially similar to isothermal vapor-liquid flash calculations. They also use the objective function. Equation (7-13), in a step-limited Newton-Raphson iteration for a, which is here E/F. However, because of the very strong dependence of equilibrium ratios on phase compositions, a computation as described for isothermal flash processes can converge very slowly, especially near the plait point. (Sometimes 50 or more iterations are required. ) 

As indicated in Chapter 6, and discussed in detail by Anderson et al. (1978), optimum parameters, based on the maximum-likelihood principle, are those which minimize the objective function 

Unconstrained optimization methods [W. H. Press, et. al.. Numerical Recipes The Art of Scientific Computing, Cambridge University Press, 1986, Chapter 10] can use values of only the objective function, or of first derivatives of the objective function, second derivatives of the objective function, etc. HyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. HyperChem does not use optimizers that compute the full set of second derivatives (the Hessian) because it is impractical to store the Hessian for macromolecules with thousands of atoms. A future release may make explicit-Hessian methods available for smaller molecules but at this release only methods that store the first derivative information, or the second derivatives of a single atom, are used. 

Combinatorial. Combinatorial methods express the synthesis problem as a traditional optimization problem which can only be solved using powerful techniques that have been known for some time. These may use total network cost direcdy as an objective function but do not exploit the special characteristics of heat-exchange networks in obtaining a solution. Much of the early work in heat-exchange network synthesis was based on exhaustive search or combinatorial development of networks. This work has not proven useful because for only a typical ten-process-stream example problem the alternative sets of feasible matches are cal.55 x 10 without stream spHtting. 

Equations (7-8) and (7-9) are then used to calculate the compositions, which are normalized and used in the thermodynamic subroutines to find new equilibrium ratios,. These values are then used in the next Newton-Raphson iteration. The iterative process continues until the magnitude of the objective function 1g is less than a convergence criterion, e. If initial estimates of x, y, and a are not provided externally (for instance from previous calculations of the same separation under slightly different conditions), they are taken to be 

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