Objective Functions

As each elder typically produces several offspring in every generation and the offspring themselves again produce further offspring in later generations this approach leads to an exponential increase in population size at the beginning. Due to the fairly large individual step sizes chosen at the beginning of the optimization this also leads to a dispersal of the population across the definition range of the variables initially.

To limit computational effort, some restrictions to population size or growth have to be imposed. These restrictions are a limit on the size of the population set by the user, the requirement of all individuals being unique and the Metropolis Criterion which becomes increasingly restrictive as the quality of the best individual increases and the “tempera-ture” parameter is decreased. The effect this last criterion has is again based on an analogy to nature. As computational resources are plentiful at the beginning, all new individuals are accepted into the population to gain a maximum in diversity. As the bounds on com-putation time and memory come into sight, only the better individuals located at the better local optima are sufficiently well adapted to have a chance to create offspring which will be located relatively close to their successful elders thus concentrating the search on the more promising areas of the definition space.

For the algorithm to work well, a good choice of the “temperature” parameter and of an appropriate cooling schedule are very important. As a cooling schedule, a simple repet-itive multiplication with a factor smaller than 1 was chosen. The choice of the initial

“temperature” was delayed until population size has reached 100 and was then based on a multiple of the standard deviation associated with the function values achieved within the population. In the case that the Metropolis Criterion was not limiting enough leading to a population size at the limit permitted by the user, the “temperature” parameter was adjusted accordingly.

3.2 Objective Functions such as an SQP/Newton method or (in rather few cases so far) on global search methods like the global evolution strategy presented in Chapter 3.1.

Whenever using such techniques however the question of how the algorithm is to decide which parameter set leads to a better fit has to be addressed. Usually the approach here is to define an objective function to be minimized which is formulated as the sum of least squares:

δ =

NP

X

i=1

q^exp_i −q^calc_i 2

(3.6) In this equation, q denotes some (arbitrary) quantity, superscripts exp and calc denote experimentally measured and calculated values, i is an index that goes from 1 to the number of measurement points,NP.

If the measurement data were ideal — meaning without measurement error — the sum should be zero if the ideal parameter set is found and if the equation being fitted is capable of describing the phenomena measured completely (which is often assumed). Unluckily, ideal measurement data are rarely available. Instead, the measurements show a certain amount of scattering around the “true” values, which are not known. What is typically known however (or what can be estimated) is the standard deviation associated with the measurement. This becomes important, when different types of measurements are to be fitted simultaneously. The above terms on the right hand side of Equation 3.6 should then be normalized by dividing them through their variance before being summed up to weight them correctly:

δ =

NP

X

i=1

q^exp_i −q^calc_i 2

σ²_qi (3.7)

The disadvantage of the above approach lies in the fact that typically one has to insert some measured quantities into the equation whose parameters are being optimized to be able to calculate theq^calc_i values. To demonstrate what is meant the Antoine Equation is chosen:

log10

P^sat_i Pa

!

= Ai− Bi T K +Ci

(3.8) In Equation 3.8, P^sat_i is the saturation pressure in Pa, T is the temperature in K and Ai, Bi and Ci are the parameters of substance i to be fitted to the data. The measurement

data will consist of several sets of pressure/temperature values. As both of these values are measured experimentally, they are not the true values (which are unknown) but are assumed to be normally distributed around the true values with estimated variances for the pressures ofσ²_Pand for the temperatures ofσ²_T. In many cases, one of the two values (typically the temperature value) is inserted into Equation 3.8, the according pressure for the current parameter set is calculated and the sum of squares according to Equation 3.7 is computed using this calculated value considering only the variance of the pressure to weight the squared differences. What is being ignored is the fact that the calculated pressure has a variance of its own due to the temperature variance of the temperature measurement value inserted into the equation.

In many cases in which the accuracy of the temperature measurement is sufficiently high and the sensitivity of the calculated value with respect to the variance of the inserted measured temperature value is not too high, this will produce good results. Cases in which this is not the case can easily be constructed, however. To address this problem, the

“true” temperature and pressure values are added into the fitting procedure as additional variables. The new objective function to be minimized for this case is now:

δ =

NP

X

i=1

P^exp_i −P^t_i2

σ²_P

i

+

T_i^exp−T_i^t2

σ²_T

i

(3.9)

Superscriptst denote the “true” values. The equation to be fitted can now be seen as a set of equality constraints of the optimization, the parameters sought are the parameters of these equality constraints. As the “true” values have to exactly fulfill Equation 3.8, and the equation is easily solved for one value if the other is known, either the “true”

temperature or pressure values can be calculated directly from the respective other value allowing to reduce the number of variables of the optimization. Whether such a reduction is possible depends on the nature of the equation to be fitted, though.

This second approach to defining the objective function to be minimized when searching for the best set of parameters has the drawback of usually requiring more computation time because of the additional variables to be optimized (the “true” values). In some cases, however, a significant improvement in the achieved fit was observed without having to resort to global optimization techniques.