Multivariable optimization

The advantage of the GA variable selection approach over the univariate approach discussed earlier is that it is a true search for an optimal multivariate regression solution. One disadvantage of the GA method is that one must enter several parameters before it... 

Kalivas, J.H. and Green, R.L., Pareto optimal multivariate calibration for spectroscopic data, Appl. Spectra sc., 55, 1645-1652, 2001. 

There are a niunber of different experimental design techniques that can be used for medium optimization. Four simple methods that have been used successfully in titer improvement programs are discussed below. These should provide the basis for initial medium-improvement studies that can be carried out in the average laboratory. Other techniques requiring a deeper knowledge of statistics, including simplex optimization, multivariate analysis, and principle-component analysis, have been reviewed (5,6). 

Chapter 5 considers optimal regulator control problems. The Kalman linear quadratic regulator (LQR) problem is developed, and this optimal multivariable proportional controller is shown to be easily computable using the Riccati matrix differential equation. The regulator problem with unmeasurable... 

Optimal Multivariate Interpolation 399 The entries of the symmetric Gram matrix B of size N + 2) are given... 

Thus loops, utility paths, and stream splits offer the degrees of freedom for manipulating the network cost. The problem is one of multivariable nonlinear optimization. The constraints are only those of feasible heat transfer positive temperature difference and nonnegative heat duty for each exchanger. Furthermore, if stream splits exist, then positive bremch flow rates are additional constraints. 

Moreover, multivariate optimization, the simultaneous optimization of several properties, will increasingly come into focus. A drug should have high selectivity in binding to different receptors and minimal toxicity, good solubility and penetration, and so on. A hair color should have a brilliant shine, be absorbed well, not be washed out, not damage the hair, not be toxic, and be stable under sunlight, etc. 

The analogous procedure for a multivariate problem is to obtain many experimental equations like Eqs. (3-55) and to extract the best slopes from them by regression. Optimal solution for n unknowns requires that the slope vector be obtained from p equations, where p is larger than n, preferably much larger. When there are more than the minimum number of equations from which the slope vector is to be extracted, we say that the equation set is an overdetermined set. Clearly, n equations can be selected from among the p available equations, but this is precisely what we do not wish to do because we must subjectively discard some of the experimental data that may have been gained at considerable expense in time and money. 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequahty constraints are present and all equahty constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objec tive func tion. Veiy few reahstic problems in process optimization are unconstrained. However, it is desirable to have efficient unconstrained optimization techniques available since these techniques must be applied in real time and iterative calculations cost computer time. The two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

In multivariable optimization, the difficulty of deahng with multi-... 

The second class of multivariable optimization techniques in principle requires the use of partial derivatives, although finite difference formulas can be substituted for derivatives such techniques are called indirect methods and include the following classes ... 

Nonlinear Programming The most general case for optimization occurs when both the objective function and constraints are nonlinear, a case referred to as nonlinear programming. While the idea behind the search methods used for unconstrained multivariable problems are applicable, the presence of constraints complicates the solution procedure. 

In a multivariable optimal regulator system, the plant state equations are... 

This is a multivariable robust eontrol problem that ealeulates the optimal Hrx, eontroller. The MATLAB eommand hinf opt undertakes a number of iterations by varying a parameter 7 until a best solution, within a given toleranee, is aehieved. 

Chu, C.C. (1985) Hoo-Optimization and Robust Multivariable Control, PhD Thesis, University of Minnesota, Minneapolis, MN. 

At the beginning of this chapter, several points were made about general metrics principles that are particularly applicable within the context of equipment and operability. It is worth to revisit them for just a moment to say that good process metrics for these categories are especially dependent on an understanding of the overall process, and optimization of a process should be done from a multivariate perspective. Metrics in these categories should be seen as having considerable dependencies on each other and on the materials and chemicals used in the process. 

In the introduction to Part A we discussed the arch of knowledge [1] (see Fig. 28.1), which represents the cycle of acquiring new knowledge by experimentation and the processing of the data obtained from the experiments. Part A focused mainly on the first step of the arch a proper design of the experiment based on the hypothesis to be tested, evaluation and optimization of the experiments, with the accent on univariate techniques. In Part B we concentrate on the second and third steps of the arch, the transformation of data and results into information and the combination of information into knowledge, with the emphasis on multivariate techniques. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

---