Objective functional

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... 

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

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

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

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... 

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. 

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... 

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... 

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. )... 

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

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. 

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

FIND EQUILIBRIUM OBJECTIVE FUNCTION F AND UNNORMALIZEO COMPOSITIONS... 

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

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

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

Bubble-point temperature or dew-point temperatures are calculated iteratively by applying the Newton-Raphson iteration to the objective functions given by Equations (7-23) or (7-24) respectively. 

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

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

Secondly, the linearized inverse problem is, as well as known, ill-posed because it involves the solution of a Fredholm integral equation of the first kind. The solution must be regularized to yield a stable and physically plausible solution. In this apphcation, the classical smoothness constraint on the solution [8], does not allow to recover the discontinuities of the original object function. In our case, we have considered notches at the smface of the half-space conductive media. So, notche shapes involve abrupt contours. This strong local correlation between pixels in each layer of the half conductive media suggests to represent the contrast function (the object function) by a piecewise continuous function. According to previous works that we have aheady presented [14], we 2584... 

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

Unconstrained optimization methods [W. II. Press, et. ah, Numerical Recipes The An of Scieniific Compulime.. Cambridge University Press, 1 9H6. Chapter 101 can use values of only the objective function, or of first derivatives of the objective function. second derivatives of the objective function, etc. llyperChem uses first derivative information and, in the Block Diagonal Newton-Raphson case, second derivatives for one atom at a time. TlyperChem does not use optimizers that compute the full set of second derivatives (th e Hessian ) because it is im practical to store the Hessian for mac-romoleciiles with thousands of atoms. A future release may make explicit-Hessian meth oils 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. 

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

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. 

We note that if the crack opening is zero on F,, i.e. [%] = 0, the value of the objective functional Js u) is zero. We also assume that near F, the punch does not interact with the shell. It turns out that in this case the solution X = (IF, w) of problem (2.188) is infinitely differentiable in a neighbourhood of points of the crack. This property is local, so that a zero opening of the crack near the fixed point guarantees infinite differentiability of the solution in some neighbourhood of this point. Here it is undoubtedly necessary to require appropriate regularity of the curvatures % and the external forces u. The aim of the following discussion is to justify this fact. At this point the external force u is taken to be fixed. 

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. 

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