Some Problems About Fuzzy Nonlinear Optimization

With its wide application in many fields of the daily life, the fuzzy optimization becomes more and more important. In real life, fuzzy objective functions and constrains are always nonlinear. Because of the complexity of objective functions and constrains, and the irregularity of the feasible set, it is difficult to find a valid method to solve this kind of problems. Therefore, the research of fuzzy nonlinear optimization problems becomes a hot spot. In this dissertation, our research focuses on the necessary and sufficient conditions of fuzzy nonlinear optimization problems.In traditional mathematics, there is the equivalence among optimal solutions of optimization problems, the nonlinear Kuhn-Tucker condition, the saddle points and the minimax condition of Lagrange operators. This dissertation mainly studies the results in fuzzy nonlinear optimization problems, paralleling to the equivalence. We provide the sufficient conditions for the optimal solutions of fuzzy nonlinear optimization problems under general circumstances, including the saddle points and the minimax theory of Lagrange operators, and the improved nonlinear Kuhn-Tucker condition, which is equivalent to the minimax theory, and prove the sufficiency of these conditions. However, the condition that is parallel to the traditional nonlinear Kuhn-Tucker condition, is necessary but not sufficient. We analyze the reason why this condition is not sufficient. The origin is the difference between the subtraction of L- fuzzy numbers and the traditional one, and the not-always-exist Hukuhara-difference adopted to define theλ- subdifferential of fuzzy functions, therefore theλ- subdifferential only has some properties of the traditional subdifferential. We strengthen this condition, letting L ( z,η? ), the Lagrange operator of z andη? , be bounded above, thus this condition is still sufficient for the optimal solutions of fuzzy nonlinear optimization problems.