Numerical optimization

Many problems in chemical engineering are expressed mathematically as optimization problems, and involve finding the particular x that minimizes some cost function F x). Each component ofx may vary either continuously or discretely. In this chapter, we assume that eachvy varies continuously. In Chapter 7, we consider stochastic techniques that can be used with discretely-varying parameters. 

An optimization problem may be unconstrained, in which case each xj can take any real value, or it can be constrained, such that an allowable jc must satisfy some collection of equality and inequality constraints 

We consider first unconstrained problems, and then treat constraints. Here, the focus is upon methods that identify local minima i.e., points that are lower in cost function than their neighbors. The stochastic methods of Chapter 7 return (eventually) global minima therefore, the reader is referred to that discussion if identifying the global minimum is necessary. 

Numerical optimization problems arise in many contexts. To predict the geometry of a molecule, we find the conformation of its atoms with the lowest potential energy. In process design x contains parameters such as equipment sizes, flow rates, temperatures, etc., and the cost function is a measure of the economic cost of operating the process. We fit a mathematical model for a system by minimizing the sum of squared differences between the model predictions and experimental data. In optimal control, we choose tiie best set of control inputs to maintain a process at the desired set point. 

In addition to a discussion of the basic techniques for identifying local minima in continuous parameter space, the use of optimization routines in A AB is demonstrated. As these routines are part of an optional tiiatin tit, an alternative routine is provided that can be used without the toolkit. 

To make practical use of our ability to perform numerically converged DFT calculations, we also need methods that can help us effectively cope with situations where we want to search through a problem with many degrees of freedom. This is the topic of numerical optimization. Just like understanding reciprocal space, understanding the central concepts of optimization is vital to using DFT effectively. In the remainder of this chapter, we introduce these ideas. 

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Numerous optimization methods aim at approximating the Hessian (or its inverse) in various ways. 

El-Halwagi, M. M. (1995). Introduction to numerical optimization approaches to pollution prevention. In Waste Minimization Through Process Design, (A. P. Rossiter, ed.) pp. 199-208, McGraw Hill, New York. 

N. Hofmann, S. Olive, G. Laschet, F. Hediger, J. Wolf, P. R. Sahm. Numerical optimization of process control variables for the Bridgman casting process. Model Simula Mater Sci Eng 5 23, 1997. 

Numerical optimizations are available for methods lacking analytic gradients (first derivatives of the energy), but they are much, much slower. Similarly, frequencies may be computed numerically for methods without analytic second derivatives. 

The friction factor plot is available in many handbooks, so that given a value of Re, one can find the corresponding value of /. In the context of numerical optimization, however, using a graph is a cumbersome procedure. Because all of the constraints should be expressed as mathematical relations, we select the Blasius correlation for a smooth pipe (Bird et al., 1964) ... 

In carrying out analytical or numerical optimization you will find it preferable and more convenient to work with continuous functions of one or more variables than with functions containing discontinuities. Functions having continuous derivatives are also preferred. Case A in Figure 4.1 shows a discontinuous function. Is case B also discontinuous ... 

Nocedal, J. and S. J. Wright. Numerical Optimization. Springer Series in Operations Research, New York (1999). 

Secanell, M., Games, B., Suleman, A., and Djilali, N. Numerical optimization of proton exchange membrane fuel ceU cathodes. Electrochimica Acta 2007 52 2668-2682. 

The refinement is automated by defining a goodness of the fit (GOT) parameter and using numerical optimization routines to do the search in a computer. One of the most useful GOF s for direct comparison between experimental and theoretical intensities is the... 

Numerical optimization is quite a broad general concept for finding optima, given an objective Junction of your problem at hand. In general, an objective function has a number of input variables (multivariate) and a real-valued output. For the sake of simplicity, we wiU assume that the objective function should be minimized. 

Practically all virtual screening procedures rely at least in part on some numerical optimization, be it an optimization of overlap (as in many alignment programs) [81-90], the generation of energetically favorable conformations of a molecule (for example CONCORD ]91] and CORINA [92]), or the relaxation of a compound in complex with the protein (for example [93-97]). The particular virtual screening problem as a whole may be solved this way. Once a decent scoring function is defined, numerical methods... 

Machine Learning as a general technique is quite broad topic and its application in virtual screening could easily fill a chapter on its own [139]. Therefore, similar to the topic on numerical optimization, only the tip of the iceberg can be covered here. 

A numerical optimization procedure which keeps track of all these factors has heen published by Schweikert, Krieg and Noack (67). The algorithm used in this procedure is almost unique as far as we know, there is only one alternative published by Kimmich et al. (73), that achieves a similar degree of completeness. 

The examples above demonstrate several general properties of numerical optimization that are extremely important to appreciate. They include ... 

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