Fundamentals - Preventive Biomechanics - page 113

Table 3.2 Deterministic
optimization algorithms

Direct search methods Gradient methods

Box method Steepest descent

Parallel direct search method Newton's method

Pattern search (Hooke and Jeeve) Quasi-Newton methods

Strategy of Rosenbrock Fletcher-Reeves method

Simplex method Marquardt method

2 3

U 1 ð p Þ
U 2 ð p Þ
:
:
U n ð p Þ

4 5

p 2 Q n U ð p Þ¼ ! extr for and n 2 : ð 3 : 367 Þ

where the U i ,(i ¼ 1 ; ... ; n) denote the single objective functions, p the n-dimen-
sional parameter vector which minimizes/maximizes the objective functions
simultaneously, Q is the parameter domain as given in assignment ( 3.366 ) and U in
this case denotes a comparator function which depends on the set of objective
functions U i .
Again, if the minimum problem is considered ( 3.367 ) can be formulated as
follows. A parameter vector p* [ Q n

is said to be Pareto optimal (or efficient) if

there is no other p [ Q n such that U ð p Þ U ð p Þ .

Generally, a multi-objective problem is approached by combining single
objectives. A possibility is to sum the absolute values of all objectives into one
combined function. Since it is likely that the single objectives have different effects
on the summed scalar value, it is reasonable to weight each objective individually.
Thus, it is possible to avoid a solution that is unequally dominated by one (or more)
objectives. The effect of this weighting is to supply a greater importance, or greater
weight, to those objectives which have smaller expected effects. Such weighting
entails choosing appropriate weight factors, depending on the particular problem.

Weighting included, equation ( 3.367 ) can thus be formulated as follows

f U ð p Þ¼ X

n

p 2 Q n

a i f i ð p Þg! extr i ¼ 1 ; 2 ; ... ; n

for and a i [ 0 and

i ¼ 1

ð 3 : 368 Þ

where the a i denote positive weight factors and Q is the parameter domain as given
in assignment ( 3.366 ).

Optimization methods can be classified aspect wise. A possible distinguishing
characterization is the delineation into deterministic and probabilistic algorithms.
Where probabilistic (randomized) algorithms include, at minimum, one directive
component that acts on the basis of random numbers providing different results with