Direct search

The large databases CA, Betlstein, and Gmelin do not provide methods for directly searching spectroscopic data. Detailed retrieval of spectroscopic information is provided in databases that contain one or more types of spectra of chemical compounds. Section 5.18 gives an ovei view of the contents of larger databases including IR, NMR, and mass spectra. 

Wind direction search option (if no, specify desired angle)... 

The retrosynthetic analysis of fumagillol, the alcohol from which the antibiotic fumagillin is derived, has been outlined in Section 2.3. The experimentally demonstrated synthesis of fumagillol was derived by T-goal directed search to apply the Diels-Alder transform. 

Luss, R. and lakola, T.H., 1973. Optimisation by direct search and systematic reduction of the size of search region, American Institute of Chemical Engineers Journal, 19, 760. 

Some approaches to the directed search of drugs on the basis of nicotinic acid 99KFZ(4)6. 

In view of this a direct search for limit cycles on the topological basis was found to be extremely difficult, as we shall see later. 

The direct search for a global optimum may not uncover some of the Pareto-optimal solutions close to the overall optimum, which might be good trade-off solutions of interest to the decision maker. 

The rotational constants, although difficult to establish with the accuracy needed for a direct search on the telescope, should be precise enough to identify the deuterated isomers in the laboratory. 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

There is a variety of general purpose unconstrained optimization methods that can be used to estimate unknown parameters. These methods are broadly classified into two categories direct search methods and gradient methods (Edgar and Himmelblau, 1988 Gill et al. 1981 Kowalik and Osborne, 1968 Sargent, 1980 Reklaitis, 1983 Scales, 1985). 

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. 

Direct search methods use only function evaluations. They search for the minimum of an objective function without calculating derivatives analytically or numerically. Direct methods are based upon heuristic rules which make no a priori assumptions about the objective function. They tend to have much poorer convergence rates than gradient methods when applied to smooth functions. Several authors claim that direct search methods are not as efficient and robust as the indirect or gradient search methods (Bard, 1974 Edgar and Himmelblau, 1988 Scales, 1986). However, in many instances direct search methods have proved to be robust and reliable particularly for systems that exhibit local minima or have complex nonlinear constraints (Wang and Luus, 1978). 

One of the most reliable direct search methods is the LJ optimization procedure (Luus and Jaakola, 1973). This procedure uses random search points and systematic contraction of the search region. The method is easy to program and handles the problem of multiple optima with high reliability (Wang and Luus, 1977, 1978). A important advantage of the method is its ability to handle multiple nonlinear constraints. 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

In this section we first present an efficient step-size policy for differential equation systems and we present two approaches to increase the region of convergence of the Gauss-Newton method. One through the use of the Information Index and the other by using a two-step procedure that involves direct search optimization. 

A simple procedure to overcome the problem of the small region of convergence is to use a two-step procedure whereby direct search optimization is used to initially to bring the parameters in the vicinity of the optimum, followed by the Gauss-Newton method to obtain the best parameter values and estimates of the uncertainty in the parameters (Kalogerakis and Luus, 1982). 

For example let us consider the estimation of the two kinetic parameters in the Bodenstein-Linder model for the homogeneous gas phase reaction of NO with 02 (first presented in Section 6.5.1). In Figure 8.4 we see that the use of direct search (LJ optimization) can increase the overall size of the region of convergence by at least two orders of magnitude. 

The set of points over which the minimum of

gradient method for the location of the minimum of the stability function, we advocate the use of direct search. The rationale behind this choice is that first we avoid any local minima and second the computational requirements for a direct search over the interpolated and the given experimental data are rather negligible. Hence, the minimization of Equation 14.24 should be performed subject to the following constraint... 

Luus, R., and T.H.I. Jaakola, "Optimization by Direct Search and Systematic Reduction of the Search Region", AlChEJ, 19, 760 (1973). 

Professor Kalogerakis acknowledges the support of the Technical University of Crete in completing this book Professor Luus for his encouragement and help with direct search procedures and all his colleagues at the University of Calgary for the many discussions and help he received over the years. 

Hooke, R., andT.A. Jeeves, 1961. Direct Search solution of numerical and statistical problems, J. Assoc. Comp. Mach., 8 (2), 212-229. 

Deterministic methods. Deterministic methods follow a predetermined search pattern and do not involve any guessed or random steps. Deterministic methods can be further classified into direct and indirect search methods. Direct search methods do not require derivatives (gradients) of the function. Indirect methods use derivatives, even though the derivatives might be obtained numerically rather than analytically. 

Various search strategies can be used to locate the optimum. Indirect search strategies do not use information on gradients, whereas direct search strategies require this information. These methods always seek to improve the objective function in each step in a search. On the other hand, stochastic search methods, such as simulated annealing and genetic algorithms, allow some deterioration... 

Solution To determine the location of the azeotrope for a specified pressure, the liquid composition has to be varied and a bubble-point calculation performed at each liquid composition until a composition is identified, whereby X = y,-. Alternatively, the vapor composition could be varied and a dew-point calculation performed at each vapor composition. Either way, this requires iteration. Figure 4.5 shows the x—y diagram for the 2-propanol-water system. This was obtained by carrying out a bubble-point calculation at different values of the liquid composition. The point where the x—y plot crosses the diagonal line gives the azeotropic composition. A more direct search for the azeotropic composition can be carried out for such a binary system in a spreadsheet by varying T and x simultaneously and by solving the objective function (see Section 3.9) ... 

The direct search methods8 use many of the basic ideas developed so far. They suppose that if a step in a given direction is good a larger one in the same direction will be better. Conversely, if a step results in a worse response, in the future a smaller step should be made in the opposite direction. The method follows. 

All the algebraic and geometric methods for optimization presented so far work when either there is no experimental error or it is smaller than the usual absolute differences obtained when the objective functions for two neighboring points are subtracted. When this is not the case, the direct search and gradient methods can cause one to go in circles, and the geometric method may cause the region containing the maximum to be eliminated from further consideration. 

Hooke, R., Jeeves, T.A. Direct Search Solution of Numerical and Statistical Problems, Journal of the Association for Computing Machinery, Apr. 1961, p. 212. 

For a 10-compressor, 2-path problem Bickel et al. (B7) gave a solution time of 353 CPU seconds on a CDC 6600 computer. As a comparison, the same problem was later solved by Chen and Fan using a direct search procedure (F2). The solution was obtained in less than 3 minutes on an IBM 370/158 computer. 

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