Subroutines, optimization

Computer simulation programs for process design optimization have been developed for the PUREX process utilizing these relationships (22). A subroutine has also been developed which describes the behavior of fission products (23). 

In this section assume a mathematical model is possible for the problem to be solved. The model may be encoded in a subroutine and be known only imphcitly, or the equations may be known explicitly. A general form for such an optimization problem is... 

The next example treats isothermal and adiabatic PFRs. Newton s method is used to determine the throughput, and Runge-Kutta integration is used in the Reactor subroutine. (The analytical solution could have been used for the isothermal case as it was for the CSTR.) The optimization technique remains the random one. 

The BLOCK DATA subroutine provides information needed for the optimization to labeled COMMON. This includes the number of variables (NV) to be optimized, their initial values (X), their maximum values (XMAX), their minimum values (XMIN), initial increments to use in varying X values (DELTAX) and an indication of how accurate the optimized variables should be (DEIMIN). The parameters NTRACE and MATRIX are output options available from STEPIT. Once STEPIT has been modified for use with CSMP, it can be used without further modification. All information required for an optimization problem is provided by means of the BLOCK DATA subroutine. Although we find this approach satisfactory for batch calculations, individuals with an interactive computer system may wish to modify STEPIT so that this information can be introduced more conveniently. 

In this chapter we described and illustrated only a few unidimensional search methods. Refer to Luenberger (1984), Bazarra et al. (1993), or Nash and Sofer (1996) for many others. Naturally, you can ask which unidimensional search method is best to use, most robust, most efficient, and so on. Unfortunately, the various algorithms are problem-dependent even if used alone, and if used as subroutines in optimization codes, also depend on how well they mesh with the particular code. Most codes simply take one or a few steps in the search direction, or in more than one direction, with no requirement for accuracy—only that fix) be reduced by a sufficient amount. 

To gain the maximum benefit from the use of a flowsheet program, the operator/designer must be adequately trained. A suitable program will have 20-30 standard units available, numerous equation-solving procedures, control facilities and probably optimization facilities. The unit-equipment subroutine must adequately represent the process equipment, recycle streams need to be specified, and suitable solution convergence is required. For the effective use of CAD packages, it... 

Technically, COSMO-RS meets all requirements for a thermodynamic model in a process simulation. It is able to evaluate the activity coefficients of the components at a given mixture composition vector, x, and temperature, T. As shown in Appendix C of [Cl 7], even the analytic derivatives of the activity coefficients with respect to temperature and composition, which Eire required in many process simulation programs for most efficient process optimization, can be evaluated within the COSMO-RS framework. Within the COSMOt/ierra program these analytic derivatives Eire available at negligible additionEd expense. COSMOt/ierra can Eilso be csdled as a subroutine, Euid hence a simulator program can request the activity coefficients and derivatives whenever it needs such input. 

The kinetic parameters obtained from this optical optimization are used as starting values for the FIBSAS optimization subroutine SIMPLEX. The procedure described above was applied to all trials (Runs 1-5), whereby some of the parameters obtained for the different trial runs still showed significant variation. A set of parameters valid for all runs was obtained from the linear regression (8) ... 

Equations 8, 10 and 11 are now set up in a linear programming framework. The optimization subroutine is called to determine Axj while optimizing the objective function. Caution is taken to keep Axj within pre-set limits so as not to cause numerical instability. If all variables are within bounds at this point, a converged solution is obtained. Otherwise, a re-linearization and re-optimization are made and the calculation process is repeated. 

This would certainly have resulted in an operational library in the shortest possible time but at considerable sacrifice in efficiency. All of the VPLIB subroutines were therefore written entirely in Z80A/MVP-9500 assembly language and this produced modules which contained, on average, one third of the assembler instructions produced by F80 for the same operation coded in FORTRAN. In addition to these straightforward savings a considerable amount of hand optimization was possible on the assembler level subroutines. 

In general, one of the first steps in optimizing codes for these architectures is implementation of standard basic linear algebra subroutines (BLAS). These routines—continuously being improved, expanded, and adapted optimally to more machines—perform operations such as dot products (xTy) and... 

Algorithm 566 FORTRAN Subroutines for Testing Unconstrained Optimization Software. 

This completes the derivation of the derivatives needed for the gradient of the energy functional. The compact matrix forms of these results can be manipulated using matrix algebra and are readily implemented using optimized computer subroutine libraries. 

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