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The inherent human desire to optimize is cerebrated in the famous Dante
quotation:
All that is superfluous displeases God and Nature All that displeases God and Nature is evil. In engineering, optimal projects are considered beautiful and rational, and the far-from-optimal ones are called ugly and meaningless. Obviously, every engineer tries to create the best project and he/she relies on optimization methods to achieve the goal. |
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The general principle by Maupertuis
proclaims:
If there occur some changes in nature, the amount of action necessary for this change must be as small as possible. ![]() This principle proclaims that the nature always finds the "best" way to reach a goal. It leads to an interesting inverse optimization problem: Find the essence of optimality of a natural "project." |
Optimization theory is developed by ingenious and creative people, who
regularly appeal to vivid common sense associations, formulating them in
a general mathematical form. For instance, the theory steers numerical
searches through
canyons
and passes (saddles), towards the
peaks;
it fights the curse of dimensionality,
models
evolution,
gambling,
and other human passions. The optimizing algorithms themselves are mathematical
models of intellectual and intuitive decision making.
The essence of an optimization problem is: Catching
a black cat in a dark room in minimal time.
(A constrained optimization
problem corresponds to a room full of furniture.)
A light, even dim, is needed: Hence optimization methods explore assumptions about the character of response of the goal function to varying parameters and suggest the best way to change them. The variety of a priori assumptions corresponds to the variety of optimization methods. This variety explains why there is no silver bullet in optimization theory. |
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