Fuzzy optimization of real functions
The main goal of this paper is to develop such means of analysis that allows us to reflect and model vagueness and uncertainty of our knowledge, which result from imprecision of measurement and inaccuracy of computation. To achieve this goal, we use here neoclassical analysis to problems of optimization. Neoclassical analysis extends the scope and results of the classical mathematical analysis by applying fuzzy concepts to conventional mathematical objects, such as functions, sequences, and derivatives. Basing on the theory of fuzzy limits, we construct a fuzzy extension for the classical theory of differentiation in the context of computational mathematics. It is done in the second part of this paper, going after introduction. Two kinds of fuzzy derivatives of real functions are considered: weak and strong ones. In addition, we introduce and study extended fuzzy derivatives, which may take infinite values. In the third part of this paper, fuzzy derivatives are applied to a study of maxima and minima of real functions. Different conditions for maxima and minima of real functions are obtained. Some of them are the same or at least similar to the conditions for the differentiable functions, while others differ in many aspects from those for the standard differentiable functions. Many classical results are obtained as direct corollaries of propositions for fuzzy derivatives, which are proved in this paper. Such results as the Fuzzy Intermediate Value theorem, Fuzzy Fermat's theorem, Fuzzy Rolle's theorem, and Fuzzy Mean Value theorem are proved. These results provide better theoretical base for computational methods of optimization.
fuzzy derivative extended fuzzy derivative fuzzy maximum fuzzy critical point fuzzy minimum