5. ANÁLISIS DE DATOS
6.5 ÁREAS DE CONOCIMIENTO LA EDUCACIÓN FÍSICA
ample of new applications, we can look to mathematics. Formal systems in mathematics provide fertile opportunities for novel optimization applications. In most cases, the formal systems10 defined by mathematicians and computer scientists used to study particular problems can easily be encoded in an al gorithmically convenient manner. Defining an objective function over these formal systems will result in entirely new classes of optimization problems. We might, for example, want to build a formal system having certain char acteristics. If we can define an objective quantifying the difference between any formal system and one with the attributes we desire then we can optimize to find such structures. Of course, the optimization will only be successful if our representation of the problem results in a relatively smooth objective land scape. As a concrete example of the power of this approach, we might use some of the algebras that have been developed to model concurrency in com puter systems (Milner, 1999). There is no reason why, in principle, we cannot optimize over representations of mobile concurrent processes11 to construct processes that perform certain tasks in as parallel a manner as possible.
New applications like these will surely bring new questions. How do we even begin to think theoretically of the properties of such spaces when the configurations represent complex formal systems? How do we combat the
9 Interestingly, the field of machine learning is now exploding with new applications by using kernel methods which map many different input spaces (e.g. text documents, bioinformatic sequences, phylogenetic trees, rankings, etc.) to linear vector spaces where the learning algorithms operate. See www.kernel-machines.org for more information.
10These formal systems may be either discrete (discrete groups, algebras, graphs, logics, calculi) or contin uous (continuous groups, algebras, vector spaces).
entropic force resulting in ever larger configurations? Work in this direction has begun (a recent example of some theoretical properties of the search space of trees is found in (Bastert et al., 2001)), and will become more important.
5.
CONCLUSIONS
Optimization has a bright and interesting future if the problems posed here, and the countless others not considered, are any indication. If, in a hundred years, researchers can look back on these problems as we do on David Hilbert’s problems seeing most of them solved, optimization will be a vastly more influ ential field than it is today. The answers to these questions will have spawned myriad new applications and perhaps will have shaped the way we view other disciplines and the natural world itself.
Acknowledgements I would like to thank BiosGroup Inc. and RIACS/NASA
for support, and Mohammed El-Beltagy, Stuart Kauffman, Jose Lobo, and Michele Shouse for input.
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