Optimality and Duality with Respect to <i>b</i>-(<i>ℰ</i>,<i>m</i>)-Convex Programming

Abstract

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

Keywords

• (<i>ℰ</i>,<i>m</i>)-convex sets
• <i>b</i>-(<i>ℰ</i>,<i>m</i>)-convex mappings
• optimality conditions
• duality theorems