Evolution and Optimum Seeking

Particular Problems and Methods of Solution 9
normally the type Gj(x) 0: Points on the edge of the (closed) allowed space are thereby
permitted. A di erent situation arises if the constraint isgiven as a strict inequality of
the form Gj(x) > 0: Then the allowed space can be open if Gj(x) iscontinuous. If for
Gj(x) 0, with other conditions the same, the minimum lies on the border Gj(x) =0,
then for Gj(x) > 0, there is no true minimum. One refers here to an in mum, the
greatest lower bound, at which actually Gj(x) = 0. In analogous fashion one distinguishes
between maxima andsuprema (smallest upper bounds). Optimization in the following
means always to nd a maximum or a minimum, perhaps under inequality constraints.
2.2.2 Static Versus Dynamic Optimization
The term static optimization means that the optimum is time invariant or stationary.
It is su cient to determine its position and size once and for all. Once the location
of the extremum has been found, the search isover. In many cases one cannot control
all the variables that in uence the objective function. Then it can happen that these
uncontrollable variables change with time and displace the optimum (non-stationary case).
The goal of dynamic optimization 2 is therefore to maintain an optimal condition in
the face of varying conditions of the environment. The search for the extremum becomes
a more or less continuous process. According to the speed of movement of the optimum,
it may be necessary, instead of making the slow adjustment of the independent variables
by hand{as for example in the EVOP method (see Chap. 2, Sect. 2.2.1), to give the task
to a robot or automaton.
The automaton and the process together form a control loop. However, unlike conventional
control loops this one is not required to maintain a desired value of a quantity
but to discover the most favorable value of an unknown and time-dependent quantity.
Feldbaum (1962), Frankovic et al. (1970), and Zach (1974) investigate in detail such automatic
optimization systems, known as extreme value controllers or optimizers. In each
case they are built around a search process. For only one variable (adjustable setting) a
variety ofschemes can be designed. It is signi cantly more complicated for an optimal
value loop when several parameters have tobeadjusted.
Many of the search methods are so very costly because there is no a priori information
about the process to be controlled. Hence nowadays one tries to build adaptive control
systems that use information gathered over a period of time to set up an internal model
of the system, or that, in a sense, learn. Oldenburger (1966) and, in more detail, Zypkin
(1970) tackle the problems of learning and self-learning robots. Adaptation is said to take
place if the change in the control characteristics is made on the basis of measurements
of those input quantities to the process that cannot be altered{also known as disturbing
variables. If the output quantities themselves are used (here the objective function) to
adjust the control system, the process is called self-learning or self-adaptation. The latter
possibility is more reliable but, because of the time lag, slower. Cybernetic engineering is
concerned with learning processes in a more general form and always sees or even seeks
links with natural analogues.
An example of a robot that adapts itself to the environment is the homeostat of Ashby
2 Some authors use the term dynamic optimization in a di erent way than is done here.