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Symmetry 2018, 10(12), 774; https://doi.org/10.3390/sym10120774

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

1,2
,
1
and
1,2,*
1
Department of Mathematics, College of Science, China Three Gorges University, Yichang 443002, China
2
Three Gorges Mathematical Research Center, China Three Gorges University, Yichang 443002, China
*
Author to whom correspondence should be addressed.
Received: 12 November 2018 / Revised: 8 December 2018 / Accepted: 17 December 2018 / Published: 19 December 2018
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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. View Full-Text
Keywords: (,m)-convex sets; b-(,m)-convex mappings; optimality conditions; duality theorems (,m)-convex sets; b-(,m)-convex mappings; optimality conditions; duality theorems
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Yu, B.; Liao, J.; Du, T. Optimality and Duality with Respect to b-(,m)-Convex Programming. Symmetry 2018, 10, 774.

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