The optimisation algorithm

Some readers have complained that chapter 7 of ref. 190 is too lapidarlc and difficult to understand. As there are also too many errors in it (mine, not the typist s), and as I have changed the optimisation algorithm and widened the scope of optimisation, it seems necessary to rewrite that chapter completely. 

We revert to the original formulation of ref. 151 It was not realised at the time that the original algorithm is really a Leven-berg - Marquard one compare with for instance ref. 180. 

Let p be a vector whose components are the current values of those potential energy function parameters we want to optimise  

5p shall be a vector whose only non-zero component is a 

A general change in the current value of may now be written nopt 

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Catalyst preparation and activation conditions are included as parameters of importance in the optimisation algorithm. The preparation and activation procedures are very relevant aspects since minor variations in such conditions would cause major changes to the final phase of the solid and, consequently, to its catalytic properties. Typical preparation variables are promoter precursors, type of impregnation, calcination atmosphere, time and temperature, time and temperature for metal reduction and so forth. 

For each outer loop function and gradient evaluation 4 and 14 inner loop problems were solved respectively (a total of 124 inner loop problems). For the inner loop problems 12-14 iterations for Tasks 1 and 3 and 5-7 iterations for Tasks 2 and 4 were usually required. For this problem size and detail of dynamic and physical properties models the computation time of slightly over 5 hrs (using SPARC-1 Workstation) is acceptable. It is to note that the optimum number of plates and optimum recovery for Task 1 (Table 7.2) are very close to initial number of plates and recovery (Table 7.1). This is merely a coincidence. However, during function evaluation step the optimisation algorithm hit lower and upper bounds of the variables (shown in Table 7.1) a number of times. Note that the choices of variable bounds were done through physical reasoning as explained in detail in Chapter 6 and Mujtaba and Macchietto (1993). 

In theory one hidden layer neural network is sufficient to describe all input/output relations. More hidden layers can be introduced to reduce the number of neurons compared to the number of neurons in a single layer neural network. The same argument holds for the type of activation function and the choice of the optimisation algorithm. However, the emphasis of this work is not directed on the selection of the best neural network structure, activation function and training protocol, but to the application of neural networks as a means of non-linear function fit. 

A reduced FORTRAN code to fit the magnetic susceptibility is available on request from the author [17]. The optimisation algorithms selected there cover the simulated annealing method combined with the simplex method. 

A problem in ML approach is the estimation of experimental errors, seldom reported in original articles. Recommended values are AT= 1 K, AP= 133 Pa (ImmHg), zlc=0.005, 4y=0.015.Note that in the ML approach the equilibrium equations are considered as constraints of the optimisation algorithm. 

The objective of the optimisation is to minimise the batch time. Feed and reflux ratio profiles are considered as decision variables within the optimisation problem. Constraints to be taken into account are product specifications (acid value, carboxyl number, viscosity, water content), feed amount and limitation of the feed flow rate. The model DAE s are discretised and the resulting algebraic system is optimised with a NLP algorithm (e.g. a SQP solver). The objective function and the constraints can be defined as VBA macros and then be computed by CHEMCAD. Present work is concerned with the implementation of the optimisation algorithm. 

This ability to deal with general models means that much of the linear regression analysis cannot be performed exactly, since the underlying assumptirais are not valid any more. Nevertheless, most of the linear regression results hold if the number of data points is much larger than the number of parameters to be estimated. The optimisation algorithm can be written as... 

The order in which the parameters are varied generally depends on the distance to their final values, the so-called parameter shift. Parameters that will experience a large shift need to be refined first. Depending on the performance of the optimisation algorithm, damping factors may need to be placed to limit the parameter variation between the iterative optimisation cycles. 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

J. Schneider, I. Morgenstern, and J. M. Singer. Bouncing towards the optimum Improving the results of monte carlo optimisation algorithms. Phys. Rev. E,... 

The above findings have lead to the further exploration of the algorithms for the optimisation of drug extraction, the results of which have been presented in Section 7.2. 

As the first test case, we selected 20 aliphatic primary amines from a set of 493 commercially available ones by three different methods random reagent selection, entropy-optimised ProSAR selection and an occupancy-optimised method which purely maximises the occupancy of pharmacophore bins (56), i.e. ensures that as many bins as possible are covered by the reagent selection regardless of the pharmacophore distribution. As the greedy algorithm... 

Below, a pseudo-code is presented to perform a simplex optimisation (using the original algorithm). 

In this paper, after the description of the general features for the retrieval model (Sect. 2) and the optimisations implemented in both the forward model and Jacobian calculation for matching the run-time requirements (Sect. 3 and 4), we will focus on the validation tests that have been performed on the code (Sect.5) and on the accuracy and run-time performances of the retrieval algorithm (Sect. 6 and Sect. 7). 

Both the look-up tables and irregular grids are microwindow-specific and are precalculated. The algorithms for the computation of both the optimised look-up tables and the irregular grids relative to the microwindows used in MIPAS retrieval have been developed at Oxford University (see Ref.s [14] and [15]). 

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