Optimization using derivatives

A finite difference formula is used to estimate the second derivatives of the coordinate vector with respect to time and S is now a function of all the intermediate coordinate sets. An optimal value of S can be found by a direct minimization, by multi-grid techniques, or by an annealing protocol [7]. We employed in the optimization analytical derivatives of S with respect to all the Xj-s. 

The different optimization equations derived in chapter 12 will then be used with these realistic chromatographic conditions in a simple optimization procedure. The conditions chosen are typical and might represent the average LC analysis. The values for (X) and (yp) are those estimated by Giddings [1] for a well-packed... 

There are some systems for which the default optimization procedure may not succeed on its own. A common problem with many difficult cases is that the force constants estimated by the optimization procedure differ substantially from the actual values. By default, a geometry optimization starts with an initial guess for the second derivative matrix derived from a simple valence force field. The approximate matrix is improved at each step of the optimization using the computed first derivatives. 

For this calculation we used the basis set s,2s,2p of Fraga and Ransil (35) which gives near HF limit quality for energy Ehfl = -1.13362957 (34)). The polarization functions were derived from the Is orbital only, like in He calculations. Their exponent was optimized using the maximum probability criterion (Copt = 1-1). Table 7 presents the obtained results. 

Parameter Estimation and Optimization Using Constrained Derivatives... 

The preparation of enantiomerically pure chemicals is also the theme of the next group of four procedures. The biopolymer polyhydroxybutyric acid, which is now produced on an industrial scale, serves as the starting material for the large scale synthesis of (R)-3-HYDROXYBUTANOIC ACID and (R)-METHYL 3-HYDROXYBUTANOATE. Esters of (-)-camphanic acid are useful derivatives for resolving and determining the enantiomeric purity of primary and secondary alcohols. An optimized preparation of (-)-(1S,4R)-CAMPHANOYL CHLORIDE is provided. The preparation of enantiomerically pure a-hydroxyketones from ethyl lactate is illustrated in the synthesis of (3HS)-[(tert)-BUTYL-DIPHENYLSILYL)OXY]-2-BUTANONE. One use of this chiral a-hydroxyketone is provided in the synthesis of (2S,3S)-3-ACETYL-8-... 

A discontinuity in a function may or may not cause difficulty in optimization. In case A in Figure 4.1, the maximum occurs reasonably far from the discontinuity which may or may not be encountered in the search for the optimum. In case B, if a method of optimization that does not use derivatives is employed, then the kink in /(x) is probably unimportant, but methods employing derivatives might fail, because the derivative becomes undefined at the discontinuity and has different signs on each side of it. Hence a search technique approaches the optimum, but then oscillates about it rather than converges to it. 

As mentioned earlier, nonlinear objective functions are sometimes nonsmooth due to the presence of functions like abs, min, max, or if-then-else statements, which can cause derivatives, or the function itself, to be discontinuous at some points. Unconstrained optimization methods that do not use derivatives are often able to solve nonsmooth NLP problems, whereas methods that use derivatives can fail. Methods employing derivatives can get stuck at a point of discontinuity, but the function-value-only methods are less affected. For smooth functions, however, methods that use derivatives are both more accurate and faster, and their advantage grows as the number of decision variables increases. Hence, we now turn our attention to unconstrained optimization methods that use only first partial derivatives of the objective function. 

From numerous tests involving optimization of nonlinear functions, methods that use derivatives have been demonstrated to be more efficient than those that do not. By replacing analytical derivatives with their finite difference substitutes, you can avoid having to code formulas for derivatives. Procedures that use second-order information are more accurate and require fewer iterations than those that use only first-order information(gradients), but keep in mind that usually the second-order information may be only approximate as it is based not on second derivatives themselves but their finite difference approximations. 

Martin and coworkers described an application of optimization to an existing tower separating propane and propylene. The lighter component (propylene) is more valuable than propane. For example, propylene and propane in the overhead product were both valued at 0.20/lb (a small amount of propane was allowable in the overhead), but propane in the bottoms was worth 0.12/lb and propylene 0.09/lb. The overhead stream had to be at least 95 percent propylene. Based on the data in Table E12.4A, we will determine the optimum reflux ratio for this column using derivations provided by McAvoy (personal communication, 1985). He employed correlations for column performance (operating equations) developed by Eduljee (1975). 

A biorefinery maximizes the value derived from the complex biomass feedstock by (a) optimal use and valorization of feedstock, (b) optimization and integration of processes for better efficiency, and (c) optimization of inputs (water, energy, etc.) and waste recycling/treatment. Integrated production of bioproducts, especially for bulk chemicals, biofuels, biolubricants and polymers, can improve their competitiveness and eco-efficiency. However, although a few examples of biorefineries already exist (Chapters 3 and 6), many improvements are still needed to enhance the process [5] ... 

The molecular structures used in the calculations were optimized using a molecular force field program. The force field parameters were derived from various sources The bond length was taken into accound according to a model of Dewar and Llano (7). The VALBOND method of Root et al. (8) was used for the calculation of bond angels. The dihedral angles were parametrized according to the PIMM method (9). 

Fig. 18 The solution of epothilone bound to tubulin by electron crystallography, a 2Fo-Fc map and model of epothilone A bound to tubulin at la (ltvk) [3], b Energy optimized model derived from ltvk through MAID protocol used as template for analysis of SAR and design of new analogs...

The optimization can be carried out by several methods of linear and nonlinear regression. The mathematical methods must be chosen with criteria to fit the calculation of the applied objective functions. The most widely applied methods of nonlinear regression can be separated into two categories methods with or without using partial derivatives of the objective function to the model parameters. The most widely employed nonderivative methods are zero order, such as the methods of direct search and the Simplex (Himmelblau, 1972). The most widely used derivative methods are first order, such as the method of indirect search, Gauss-Seidel or Newton, gradient method, and the Marquardt method. 

By its very nature, minimization of En is more expensive than the maximization of S, because it requires the construction of quantities corresponding to applications of the hamiltonian operator, but this may be achieved by adapting the efficient procedures already available in various CASSCF codes. It turns out, however, that the two sets of orbital representations tend to be rather similar, and so maximization of tends to be preferred. In either case, the actual optimization uses reliable Newton-Raphson-like procedures that utilize first and second derivatives. 

Thermoeconomic optimization using differentially derived prices, permitting the analysis of the system s local and global responses to well specified small changes in the state of the system, and leading to sensitivity analysis and optimization techniques. 

R. B. Schnabel and T.-T. Chow, SIAM j. Opt., 1, 293 (1991). Tensor Methods for Unconstrained Optimization Using Second Derivatives. 

---