Optimality and Duality with Respect to b-(ℰ,m)-Convex Programming - Bo Yu (Symmetry, 2018)
Noticing that <semantics> E </semantics> -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 <semantics> ( E , m ) </semantics> -convex sets and the b- <semantics> ( E , m ) </semantics> -convex mappings are introduced. The properties concerning operations that preserve the <semantics> ( E , m ) </semantics> -convexity of the proposed mappings are derived. The unconstrained and inequality constrained b- <semantics> ( E , m ) </semantics> -convex programming are considered, where the sufficient conditions of optimality are developed and the uniqueness of the solution to the b- <semantics> ( E , m ) </semantics> -convex programming are investigated. Furthermore, the sufficient optimality conditions and the Fritz⁻John necessary optimality criteria for nonlinear multi-objective b- <semantics> ( E , m ) </semantics> -convex programming are established. The Wolfe-type symmetric duality theorems under the b- <semantics> ( E , m ) </semantics> -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- <semantics> ( E , m ) </semantics> -convex programming.