Optimality and Duality with Respect to b-(ε, m)-Convex Programming

Noticing that epsilon-convexity, m-convexity and b-invexity have similar structures in their definitions, there are some possibilities to treat these three class of mappings uniformly. For this purpose, the definitions of the (epsilon, m)-convex sets and the b-(epsilon, m)-convex mappings are introduced. The properties concerning operations that preserve the (epsilon, m)-convexity of the proposed mappings are derived. The unconstrained and inequality constrained b-(epsilon,m)-convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b-(epsilon, m)-convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz-John necessary optimality criteria for nonlinear multi-objective b-(epsilon, m)-convex programming are established. The Wolfe-type symmetric duality theorems under the b-(epsilon, m)-convexity, including weak and strong symmetric duality theorems, are also presented. Finally, we construct two examples in detail to show how the obtained results can be used in b-(epsilon, m)-convex programming.