Fuzzy Optimization with Constraints

摘要

In this paper, we apply neoclassical analysis to problems of constraint optimization. Constraint optimization plays an important role in many areas, especially, in economics and control theory. However, structures of classical calculus used in constraint optimization often imply too many restrictions on optimized functions. Neoclassical analysis extends methods of classical calculus to reflect vagueness, imprecision and uncertainties that arise in computations and measurements. That is why here we apply neoclassical analysis to problems of constraint optimization. In Section 2, fuzzy gradients of functions of three variables are constructed and studied. In Section 3, we obtain various forms of the chain rule for functions of several variables (Theorems 5 - 9), providing an important tool for solving problems of calculus in the context of inexactness, vagueness, uncertainty, and imprecision. In particular, the chain rule for functions of several variables is utilized in the proof of optimization theorems. In Section 4, we develop analytical optimization techniques for real functions of two variables, giving the necessary conditions for finding maximal and minimal values of functions satisfying additional constraints (Theorems 10, 11). In Conclusion, we discuss possibilities for future research.