# Optimality principle

A formal description of various notions of an optimum. Optimality principles normally reflect certain characteristics of an intuitive understanding of stability, profitability and fairness. It is essential that the simultaneous realization of all (or of a sufficient large number of) such characteristics often appears to be impossible, owing to their formal incompatibility. As the theory of optimality principles becomes of an axiomatic nature, new optimality principles arise which do not always possess an intuitive transparency.

Problems of optimality principles arise, for example, when that value of a variable is sought which simultaneously extremizes a number of given functions (the so-called multi-criterion extremal problems). Problems which require non-trivial optimality principles in order to be solved actually arise in game theory (cf. Games, theory of). One of the simplest game-theoretic optimality principles is the minimax principle. Other optimality principles are realized in the form of a core or a von Neumann–Morgenstern solution (cf. Core in the theory of games), a Shapley value, etc.

For the Bellman principle of optimality see Dynamic programming.

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See also Pontryagin maximum principle; Optimal control.

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Optimality principle.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Optimality_principle&oldid=32093