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.
Highlights
Convexity, as well as generalized convexity, has a vital position in optimality and has many consequences in different aspects of mathematical programming
Iqbal et al [3] discussed a new family of sets as well as a new group of mappings named geodesic E -convex sets together with geodesic E -convex mappings, which are defined on a Riemannian manifold
Inspired by the above work and based on the work in [15,16,17], we introduce a new family of generalized convex sets as well as generalized convex mappings, named (E, m)-convex sets and b-(E, m)-convex mappings, and develop some interesting properties of this family of sets and mappings, respectively
Summary
As well as generalized convexity, has a vital position in optimality and has many consequences in different aspects of mathematical programming. Proposition 2 states a sufficient condition for g being a b-(E , 1)-convex mapping, but the converse may fail to hold To illustrate this fact, let us construct an example as follows. Suppose that g : M → R is a nonnegative strictly b-(E , m)-convex mapping on an (E , m)-convex set M, the global optimal solution to the b-(E , m)-convex programming (P ) is unique. Holds for every μ ∈ M and certain fixed m ∈ [0, 1], mE (μ) is the optimal solution to the b-(E , m)-convex programming (P ) corresponding to g on M. Based on Theorem 8, we present the following sufficient optimality conditions for multi-objective b-(E , m)-convex programming ( MP). (Sufficient optimality condition) Let E : M → M be a surjective mapping and M be an (E , m)-convex set. Noticing that μ ∈ E (M), (h ◦ E )(μ) ≤ 0, the proof is finished
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