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# Q5: What about all this Optimization stuff?

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Just  think of an OPTIMIZATION problem as a black box.  A large black
box. As large as, for example, a Coca-Cola vending machine.  Now,  we
don't  know  anything  about the inner workings of this box, but see,
that there are some regulators to play with, and of course  we  know,
that we want to have a bottle of the real thing...

Putting  this  everyday problem into a mathematical model, we proceed
as follows:

(1) we label all the regulators with x and a number starting from  1;
the  result  is  a  vector  x, i.e. (x_1,...,x_n), where n is the
number of visible regulators.

(2) we must find an objective function, in this case it's obvious, we
want  to  get k bottles of the real thing, where k is equal to 1.
[some might want a "greater or equal"  here,  but  we  restricted
ourselves to the visible regulators (we all know that sometimes a
"kick in the right place" gets use more than 1,  but  I  have  no
idea how to put this mathematically...)]

(3) thus,  in  the  language  some mathematicians prefer to speak in:
f(x) = k = 1. So, what we have here  is  a  maximization  problem
presented  in  a  form we know from some boring calculus lessons,
and  we  also  know  that  there  at  least   a   dozen   utterly
uninteresting  techniques to solve problems presented this way...

What can we do in order to solve this problem?
We can either try to gain more knowledge or exploit what  we  already
know  about  the interior of the black box. If the objective function
turns out to be smooth and differentiable,  analytical  methods  will
produce the exact solution.

If  this  turns  out  to  be impossible, we might resort to the brute
force method of enumerating the entire SEARCH SPACE.   But  with  the
number  of  possibilities  growing  exponentially in n, the number of
dimensions (inputs), this method becomes  infeasible  even  for  low-
dimensional spaces.

Consequently,  mathematicians  have  developed  theories  for certain
kinds of problems leading  to  specialized  OPTIMIZATION  procedures.
These  algorithms  perform  well  if  the  black  box  fulfils  their
respective prerequisites.  For example, Dantzig's  simplex  algorithm
(Dantzig  66)  probably  represents  the  best known multidimensional
method capable of efficiently finding the global optimum of a linear,
hence  convex, objective function in a search space limited by linear
constraints.  (A USENET FAQ on linear programming  is  maintained  by
John  W.  Gregory  of  Cray  Research,  Inc. Try to get your hands on
"linear-programming-faq" (and  "nonlinear-programming-faq")  that  is
posted monthly to sci.op-research and is mostly interesting to read.)

Gradient strategies are no longer tied to these  linear  worlds,  but
they  smooth their world by exploiting the objective function's first
partial derivatives one has to supply in  advance.  Therefore,  these
algorithms  rely on a locally linear internal model of the black box.

Newton   strategies   additionally   require   the   second   partial
derivatives, thus building a quadratic internal model.  Quasi-Newton,
conjugate gradient and variable metric  strategies  approximate  this
information during the search.

The  deterministic  strategies  mentioned  so  far  cannot  cope with
deteriorations, so the search will stop if  anticipated  improvements
no  longer  occur.  In a multimodal ENVIRONMENT these algorithms move
"uphill" from their respective starting points. Hence, they can  only
converge to the next local optimum.

Newton-Raphson-methods  might  even  diverge if a discrepancy between
their internal assumptions and reality occurs.  But of course,  these
methods  turn  out  to  be  superior  if  a  given task matches their
requirements. Not relying on derivatives, polyeder strategy,  pattern
search  and  rotating coordinate search should also be mentioned here
because they  represent  robust  non-linear  optimization  algorithms
(Schwefel 81).

Dealing with technical optimization problems, one will rarely be able
to write down the objective function in a closed form.  We often need
a SIMULATION model in order to grasp reality.  In general, one cannot
even  expect  these  models   to   behave   smoothly.   Consequently,
derivatives  do  not  exist. That is why optimization algorithms that
can successfully  deal  with  black  box-type  situations  have  been
developed.  The  increasing  applicability is of course paid for by a
loss of "convergence  velocity,"  compared  to  algorithms  specially
designed  for  the given problem.  Furthermore, the guarantee to find
the global optimum no longer exists!

But why turn to nature when looking for more powerful algorithms?
In the attempt to create tools  for  various  purposes,  mankind  has
copied,  more  often instinctively than geniously, solutions invented
by nature.  Nowadays, one can prove in some cases that certain  forms
or structures are not only well adapted to their ENVIRONMENT but have
even reached the optimum (Rosen 67). This is due to the fact that the
laws  of  nature  have  remained  stable  during the last 3.5 billion
years. For instance, at branching points the measured  ratio  of  the
diameters in a system of blood-vessels comes close to the theoretical
optimum provided by the laws of fluid dynamics  (2^-1/3).   This,  of
course,  only  represents  a  limited,  engineering  point of view on
nature. In general, nature performs adaptation, not optimization.

The idea to imitate basic principles of natural processes for optimum
seeking  procedures  emerged  more than three decades ago (cf Q10.3).
Although these  algorithms  have  proven  to  be  robust  and  direct
OPTIMIZATION  tools, it is only in the last five years that they have
caught the researchers' attention. This is due to the fact that  many
people  still look at organic EVOLUTION as a giantsized game of dice,
thus ignoring the fact that  this  model  of  evolution  cannot  have
worked:  a human germ-cell comprises approximately 50,000 GENEs, each
of which consists of about 300 triplets of  nucleic  bases.  Although
the  four  existing  bases  only  encode  20  different  amino acids,
20^15,000,000, ie circa 10^19,500,000 different GENOTYPEs had  to  be
tested in only circa 10^17 seconds, the age of our planet. So, simply
rolling the dice could not have produced  the  diversity  of  today's
complex living systems.

Accordingly,   taking   random   samples  from  the  high-dimensional
parameter space of an objective function in order to hit  the  global
optimum  must  fail  (Monte-Carlo  search). But by looking at organic
evolution as a  cumulative,  highly  parallel  sieving  process,  the
results  of  which pass on slightly modified into the next sieve, the
amazing  diversity  and  efficiency  on  earth  no   longer   appears
miraculous.  When  building a model, the point is to isolate the main
mechanisms which have led  to  today's  world  and  which  have  been
subjected  to  evolution  themselves.  Inevitably, nature has come up
with a mechanism allowing INDIVIDUALs  of  one  SPECIES  to  exchange
parts of their genetic information (RECOMBINATION or CROSSOVER), thus
being able to meet changing environmental conditions in a better way.

Dantzig,  G.B.  (1966)  "Lineare  Programmierung  und Erweiterungen",
Berlin: Springer. (Linear programming and extensions)

Kursawe, F. (1994) " Evolution strategies: Simple models  of  natural
processes?",  Revue Internationale de Systemique, France (to appear).

Rosen,  R.  (1967)  "Optimality  Principles  in  Biologie",   London:
Butterworth.

Schwefel,  H.-P.  (1981) "Numerical Optimization of Computer Models",
Chichester: Wiley.

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Copyright (c) 1993-1997 by J. Heitkoetter and D. Beasley, all  rights
reserved.

This  FAQ  may be posted to any USENET newsgroup, on-line service, or
BBS as long as it  is  posted  in  its  entirety  and  includes  this
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End of ai-faq/genetic/part3
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Parent document is top of "FAQ: comp.ai.genetic part 3/6 (A Guide to Frequently Asked Questions)"
Previous document is "Q4.1: What about Alife systems, like Tierra and VENUS?"