$ E $-Optimality and $ E $-duality results for multiobjective variational problems and application to the cake-eating problem

In this paper, we introduce a new class of generalized convexity for an $ E $-differentiable multiobjective variational problem, and to solve this problem, an associated multiobjective $ E $-variational problem is formulated with the help of an $ E $-operator. The necessary and sufficient optimality conditions are derived under the assumption of $ E $-convexity. The so-called Wolfe and Mond-Weir $ E $-dual problems are defined for the considered $ E $-differentiable multiobjective variational programming problem, and several Wolfe and Mond-Weir $ E $-duality theorems are derived under the $ E $-convexity assumption. Furthermore, to emphasize the significance of the results of this study, we considered a cake-eating problem, and the solution obtained by constructing its associated $ E $-differentiable problem and the sufficiency theorem. Throughout the paper, non-trivial illustrations are presented to support the findings.