Search problems

We begin by considering situations in which nothing is known about how the criterion of effectiveness depends on the operating variables. This means that to find the value of the criterion at any combination of operating variables we must take an experimental measurement. In this case, optimization involves using information from past experiments to direct the search for successively better values of the criterion. For this reason such situations are called search problems. 

There are several reasons why functions to be optimized, such as yield, profit, throughput, or cost, are often unknown in advance. Physical 

Search problems can be divided into two groups, depending on whether or not random experimental error is associated with each measurement. There are, indeed, significant problems that have no experimental error as when the function in question is given as an exact mathematical expression, but one too complicated to be optimized directly by calculus or by known methods of mathematical programming. Design problems are often of this latter nature. We shall discuss mainly the no-error case, since its principles are simple and can be used even in the presence of experimental error. 

Consider a function y, of a single independent variable x restricted to the interval (a, b) that is, a x b. We shall consider the greatest value of y that can occur in this interval to be the optimal value y. The value of x for which y is optimal will be denoted by x that is, 

Suppose that the function y is not known in advance and that the only way to obtain information about y is to measure it at different 

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Rous P J 1993 A global approach to the search problem in surface crystallography by low-energy electron diffraction Surf. Sc 296 358-73... 

To learn a more thorough approach to the solution of the substructure search problem. 

Mathematical theory of labeled colored graphs is exclusively used to formalize the structure and substructure search problem. There is almost a one-to-one correspondence between the terms used in graph theory and the ones used in chemical structure theory. Formally a graph G can be given by Eq. (1), where V is the set of graph vertices and H the set of edges. 

Tree representation of the conformation search problem for hexane. Unlike the tree in Figure 9.4 the path gth from the root node to any of the terminal nodes is constant. 

A Search Problem.—An example of an operations research problem that gives rise to an isoperimetric model is a search problem, first given by B. Koopman,40 that we only formulate here. Suppose that an object is distributed in a region of space with ... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

In this article, we will restrict ourselves to the case of finding the orientation and position of a known model in another unit cell, to help solve the molecular structure contained in this unit cell (Fig. 7.2). Traditionally, this six-dimensional (6D) search (three orientation angles and three translations are to be found) is divided into two separate and consecutive 3D search problems. 

In our two-dimensional space, these two search directions are perpendicular to one another. Saying this in more general mathematical terms, the two search directions are orthogonal. This is not a coincidence that occurs just for the specific example we have defined it is a general property of steepest descent methods provided that the line search problem defined by Eq. (3.18) is solved optimally. 

Part II deals with the subject of molecular structure-property relations. It addresses many of the search problems raised in the development of new products, and forms the intellectual core of the science behind product engineering. 

Theory must play a role in addressing the data storage and search problems associated with the increasingly large datasets generated by chemical imaging techniques. 

These methods have been enormously successful, yet the theoretical work developed for them so far is quite limited. The success of these methods is not trivial the huge number of sequences being searched through, the low concentrations of individual species, and the noise and biases inherent in the techniques would seem to make these experiments very difficult. Understanding why they work so well, and showing how they can perform better and for more complex molecular search problems, falls under the purview of theory. 

For virtual screening, the situation is somewhat different. Screening can be understood as a search process for finding the best lead structures available in a database. Search problems typically become easier to solve if the search... 

These, then, are the reasons why magnetic resonance methods, microwave or far-infrared laser, have had limited success with 2A diatomic radicals. Similar considerations apply to nonlinear polyatomic radicals in doublet states success in far-infrared laser magnetic resonance depends upon the magnitude of the spin-rotation coupling, and the size of the energy mismatch between the transition frequency and the laser frequency, since the mismatch has to be magnetically tuned. This becomes less of a limitation as more laser frequencies become available, except that one then needs to know in advance which laser frequency to choose. It becomes part of the search problem ... 

Another set of early studies came from the work of Judson and coworkers [35, 36], which emphasized using GAs for search problems on small molecules and peptides, especially cyclic peptides. A dihedral angle representation was used for the peptides with values encoded as binary strings, and the energy function used the standard CHARMM force field. Mutations were implemented as bit flips and crossovers were introduced by a cut-and-paste of the strings. The small size of the system enabled a detailed investigation of the various parameters and policies chosen. In Ref. [37], a comparison between a GA and a direct search minimization was performed and showed the advantages and weaknesses of each method. As many concepts are shared between search problems on small peptides and complete proteins, these studies have contributed to subsequent attempts on full proteins. 

In a search problem, almost nothing is known in advance about how the criterion of effectiveness depends upon the operating variables, the only way to learn being to perform experiments. Here the obstacle to using the calculus is the complete lack of a function that can be differentiated. The objective of the search is to get as close as possible to the optimum after only a limited number of experiments. Box and Wilson, with their paper The Experimental Attainment of Optimum Conditions published in 1951, were the first to interest engineers in search problems (B4). 

In our study of search problems we have seen that single variable systems can be optimized with ease two-variable systems, with some effort and multivariable systems, only with extreme difficulty if at all. As more variables enter a search problem, the number of experiments needed grows rapidly, and the unimodality assumption becomes less and less plausible. Thus our investigation of search problems leads directly to interaction problems, where the criterion of effectiveness depends on so many factors that it is impractical, or even impossible, to find the optimum by conventional methods. Successful techniques for solving interaction problems involve decomposing a big system into several smaller ones, as we have already done with our lines of search. 

As a literature-search problem, find out what is known about the development of a selected standard substance. One such problem could involve how a metallo-organic standard (say, one of Al, Ag, Cr, Cu, Mg, Fe, Ni, Ti or Si) has... 

Another method of systematic search is the use of databases of pregenerated conformations. Conformations are calculated once and the search problem is reduced to a rigid body docking procedure. This is the main concept of FLOG docking program. ... 

The Search Problem Must Be Clearly Defined. This means establishing the purpose, bounding the subject area, and setting limits on the time period to be covered. 

As medium-sized and smaller companies generally do not have patent departments, their patent searchers in general require more familiarity with patent fundamentals than those working for larger firms, where searches are organized by supervisors in consultation with patent attorneys. In smaller firms the searcher usually has to define the search problem and the scope of the search for himself. Being left more to his own devices, he must inform himself on matters that are frequently outside the province of the searcher in a larger company. 

Treatises. By definition, a treatise is a comprehensive survey of a large general field of knowledge, such as organic chemistry. Ideally the publication should be so comprehensive as to be complete. If so, every relevant reference from every periodical would be cited. One s searching problem would then be simple, at least if the indexing were adequately done. If the tieatise itself did not provide the necessary details, one need only turn to the appropriate reference, get the periodical, and read it. 

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