Q5: What about all this Optimization stuff?

     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  nothing  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  habe  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 pogramming 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:

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


End of ai-faq/genetic/part3
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