OPTIMIZATION OF FAULT DIAGNOSIS IN HELICOPTER HEALTH AND USA(72)

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Problems, for which a point derivative can be calculated, can be solved using steepest descent methods. More general methods are Direct Search and Evolutionary Optimization. These methods do not require the object function to be point derivable, nor even continuous.
A.4.1 Trust Region
Trust
Region
[34]
is
a
frequently
used
steepest
descent
algorithm.
The
al-
gorithm approximates the object function f(x) by a trivial function d(x), normally the .rst few terms of a Taylor series, which provides a reasonably accurate approximation within a region, the trust region, surrounding the initial current position. Once the position minimizing d(x) has been found, this point is accepted as the new current position, provided that it provides lower output from f(x) than the previous position. Each time this criteria is meet, the trust region is expanded. Upon failure, i.e. the previous position outperforms the new one, the current position is kept and the trust region contracted.
1.
Approximate f(x) by trivial function d(x) in region R around x0

2.
Find the value for x, x1, which minimizes d(x)

3.
If f(x1) <f(x0) then set x0 = x1, expand R and goto 1

A.4. NONLINEAR OPTIMIZATION
4. Else contract R and goto 1
The process iterates until some stopping criterion is meet. This will normally be the step length converging to zero, max number of iterations, or max execution time. All examples in this study uses a step length of less than 10.8 as stopping criterion.

A.4.2 Evolutionary Optimization
Evolutionary
algorithms
[15]
exist
in
several
variants.
This
study
makes
use
of a method which does not require discretization of the parameters, and uses the same number of individuals in each generation for a limited number of generations. The algorithm is initiated by creating a population of parameter combinations which are chosen on random from an initial range of possible values. Once performance is tested for all individuals, the best individuals,
i.e. the best combinations of parameters, are labeled elite individuals and transferred to the next generation. The other individuals for the next gener-ation are generated from the remaining population of the current generation by either crossing or mutating. Crossing means that two children are created from two parents by choosing each parameter for each child from one of the parents. Which parameter is chosen from which parent is determined by a binary random process. The probability that an individual is chosen as a parent is proportional with its rank among the individuals. Consequently, the best individuals will normally be chosen as parents several times within the same generation. Finally, a set of individuals are mutated. This means that they are randomly displaced in parameter space, driven by a Gaussian random process. Once the parameters for the next generation are ready, the performance of the next generation is calculated. This process iterates until a performance goal or the maximum number of generations is reached.
　
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