Optimality conditions and duality relations for a class of nonlinear multiple objective optimization problems under crisp and fuzzy environment

While dealing with development of mathematical nonlinear optimization problems, optimality conditions and duality results play a crucial role. The optimality conditions (necessary and sufficient) allows one to devise efficient numerical methods for practical solution of a given optimization problem. Duality has been used for many theoretical and computational developments in mathematical programming and in other diverse fields, including engineering, management, bussiness problems and economics. In the crisp programming problems, our purpose is to minimize or maximize a function under some constraints. But in many real situations, the decision maker may not be in a position to say anything about the objective and/or constraint functions precisely but rather can specify them in a fuzzy sense. The present project basically deals with the study of necessary and sufficent conditions, and duality realtions for some multiple objective nonlinear optimization problems under crisp and fuzzy environment. This include multiobjective symmetric dual pair with cone constraints, fractional variational problem with multiple objectives and KKT optimality conditions for fuzzy valued optimization problem. Moreover, to validate the results, some nontrivial examples will also be exemplified (using Maple/Mathematica).