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In this paper, we present the concepts of generalized derivative, directional generalized derivative, subdifferential and conjugate for  n -dimensional fuzzy-number-valued functions and discuss the characterizations of generalized derivative and directional generalized derivative by, respectively, using the derivative and directional derivative of crisp functions that are determined by the fuzzy mapping. Furthermore, the relations among generalized derivative, directional generalized derivative, subdifferential and convexity for  n -dimensional fuzzy-number-valued functions are investigated. Finally, under two kinds of partial orderings defined on the set of all  n -dimensional fuzzy numbers, the duality theorems and saddle point optimality criteria in fuzzy optimization problems with constraints are discussed.

Generalized derivatives and optimization problems for n -dimensional fuzzy-number-valued functions