Derivative-free methods

The Simplex algorithm and that of Powell s are examples of derivative-free methods (Edgar and Himmelblau, 1988 Seber and Wild, 1989, Powell, 1965). In this chapter only two algorithms will be presented (1) the LJ optimization procedure and (2) the simplex method. The well known golden section and Fibonacci methods for minimizing a function along a line will not be presented. Kowalik and Osborne (1968) and Press et al. (1992) among others discuss these methods in detail. 

Solution methods for optimization problems that involve only continuous variables can be divided into two broad classes derivative-free methods (e.g., pattern search and stochastic search methods) and derivative-based methods (e.g., barrier function techniques and sequential quadratic programming). Because the optimization problems of concern in RTO are typically of reasonably large scale, must be solved on-line in relatively small amounts of time and derivative-free methods, and generally have much higher computational requirements than derivative-based methods, the solvers contained in most RTO systems use derivative-based techniques. Note that in these solvers the first derivatives are evaluated analytically and the second derivatives are approximated by various updating techniques (e.g., BFGS update). 

A number of methods for finding the root (roots) of equations of the implicit form of equation (2.3.3) or the explicit form of equation (2.3.4) have been developed. In general, they fall into derivative-free or derivative-based categories. Derivative-free methods are usually more stable, less sensitive to initial guesses, and converge less rapidly than derivative-based methods. 

A straightforward derivative-free method is that of halving the interval of uncertainty. Consider a monotonic function /(x), shown in Figure 2.10, which is continuous from x = a to x = 6 where the values /(o) and f b) have opposite signs. For the equation... 

The determination of the switching point can be done, by using Newton s method, the secant rule or by inverse interpolation. We prefer derivative-free methods, because the numerical computation of the time derivative of q involves an extra evaluation of the right hand side function due to... 

This evaluation may require an unnecessary additional computational effort. Another reason to use derivative-free methods is that in the DAE case derivatives for the algebraic variables are not available and may even not exist, see Sec. 5.1.2. Thus, we focus here only on the method of inverse interpolation. The secant rule is a special case of this approach. 

Minimisation Algorithms Production data integration must, in some way, involve forward simulation of the fluid flow model. In MLE or MAP, even with reparameterisation, an optimisation method must be used. There are two choices to use a derivative-free method, or to use derivatives. 

Derivative-Free Methods Derivative-free methods can simply call the simulator and use the results. A simple technique is simulated annealing which was investigated in [126] and [41]. Using a fast simulator such as a streamline method (fast by virtue of the IMPES approximation and the one-dimensional approximation along the streamlines) or a coarse grid simulator, this might be practical. 

The recent developments in streamline zonation use a simple iterative, derivative free, update of the permeability field along the streamlines. Gradient Methods If the aim is to find the MLE or MAP estimates then the method of choice must be a derivative method. Once the effort is made to modify the simulator so that derivatives can be calculated exactly (that is without using numerical differentiation) then the higher order convergence of a gradient method can easily outperform a derivative free method. 

Similarly, some INAA data contributed to the derivation of a reference value for Ba in SDO-i were biased high by an interference from Ru (Wandless 1993). The Ru is a fission product of U, whose concentration of 40 qg/g is relatively high in SDO-1. In this case, no appropriate reference sample was available for analysis to control the SDO-1 results the interference was identified through the disagreement between INAA data and data produced using XRF and ICP-AES methods on the same sample. A bias-free method again resulted when analysis of an atypical type led to detection of a rarely encountered but sizeable spectral overlap. Once identified, correction was straightforward. 

The gradient search methods require derivatives of the objective functions whereas the direct methods are derivative-free. The derivatives may be available analytically or otherwise they are approximated in some way. It is assumed that the objective function has continuous second derivatives, whether or not these are explicitly available. Gradient methods are still efficient if there are some discontinuities in the derivatives. On the other hand, direct search techniques, which use function values, are more efficient for highly discontinuous functions. 

An important group of methods relies on the inherent order of the data, typically time in kinetics or chromatography. These methods are often based on Evolving Factor Analysis and its derivatives. Another well known family of model-free methods is based on the Alternating Least-Squares algorithm that solely relies on restrictions such as positive spectra and concentrations. 

Aliphatic aldehydes typically provide only moderate yields in the Biginelli reaction unless special reaction conditions are employed, such as Lewis-acid catalysts or solvent-free methods, or the aldehydes are used in protected form [96]. The C4-unsubstituted DHPM can be prepared in a similar manner employing suitable formaldehyde synthons [96]. Of particular interest are reactions where the aldehyde component is derived from a carbohydrate. In such transformations, DHPMs having a sugar-like moiety in position 4 (C-nucleoside analogues) are obtained (see Section 4.7) [97-106]. Also of interest is the use of masked amino acids as building blocks [107, 108]. In a few cases, bisaldehydes have been used as synthons in Biginelli reactions [89, 109, 110]. 

By a similar but solvent-free method Plaquevent et al. produced the Michael adduct 30 from 2-pentyl-2-cyclopentenone in 91% yield and with 90% ee, by use of the quinine-derived catalyst 31 (Scheme 4.10) [16], When the quinidine-derived ammonium salt 32 was employed, 80% of the enantiomeric product ent-30 was ob-... 

Carbolithiation reactions of ketone a,-dianions, generated by the above amine-free method with several alkenes, such as styrenyl derivatives, vinyl sulfides and vinylsilanes, can lead to the generation of ketone a,5-dianions (Scheme 15)14. For example, when one equivalents of triphenylvinylsilane was treated with a ketone a,-dianion, in THF, at 0 C for 1 h and the resulting reaction mixture was quenched by 2.2 mol equivalents of trimethylchlorosilane, the corresponding bis-silylated enol silyl ether was obtained. Substituted styrenyl derivatives, such as 1,1-diphenylethylene and cinnamyl alcohol, also underwent a smooth carbolithiation to give the corresponding ketone a,5-dianions. Similar addition reactions of ketone a,f)-dianions to vinyl phenyl sulfide took place smoothly to give a,5-dianions with a sulfur attached in the 5-position. 

MP2/cc-pVTZ//MP2/cc-pvDZ)- (MP2/ccpVDZ)-H (CCSD(T)//cc-pVDZ//MP2/cc-pVDZ)-l-AG it. t-RTlnco ° C, derived from free energies at B3LYP/6-31 H-G //MP2/6-31-l-G 3 D, derived from free energies at BSLYP/aug-cc-pVTZ E, derived from method A -I- PM3-SM3. ... 

Free radicals react most efficiently with other free radicals, and the chemicals that have relatively stable free radicals, such as 2,2-diphenyl-1-pricylhydrazyl (DPPH), could be used, in principle, to quantitatively estimate the amount of free radicals (DPPH method). The peroxide formed on the surface that derived from free radical could be quantitatively analyzed by the determination of iodine liberated from KI solution, which could be used to calculate the amount of the original free radical from that peroxide was derived (iodine method). 

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