**Special Class of Second-Order Non-Differentiable Symmetric Duality Problems with (***<sup>G</sup>***,** *<sup>α</sup>f* **)-Pseudobonvexity Assumptions**

#### **Ramu Dubey 1, Lakshmi Narayan Mishra 2,\* and Rifaqat Ali 3**


Received: 24 June 2019; Accepted: 15 August 2019; Published: 20 August 2019

**Abstract:** In this paper, we introduce the various types of generalized invexities, i.e., *<sup>α</sup>f*-invex/*<sup>α</sup>f*-pseudoinvex and (*<sup>G</sup>*, *<sup>α</sup>f*)-bonvex/(*<sup>G</sup>*, *<sup>α</sup>f*)-pseudobonvex functions. Furthermore, we construct nontrivial numerical examples of (*<sup>G</sup>*, *<sup>α</sup>f*)-bonvexity/(*<sup>G</sup>*, *<sup>α</sup>f*)-pseudobonvexity, which is neither *<sup>α</sup>f*-bonvex/*<sup>α</sup>f*-pseudobonvex nor *<sup>α</sup>f*-invex/*<sup>α</sup>f*-pseudoinvex with the same *η*. Further, we formulate a pair of second-order non-differentiable symmetric dual models and prove the duality relations under *<sup>α</sup>f*-invex/*<sup>α</sup>f*-pseudoinvex and (*<sup>G</sup>*, *<sup>α</sup>f*)-bonvex/(*<sup>G</sup>*, *<sup>α</sup>f*)-pseudobonvex assumptions. Finally, we construct a nontrivial numerical example justifying the weak duality result presented in the paper.

**Keywords:** symmetric duality; second-order; non-differentiable; (*<sup>G</sup>*, *<sup>α</sup>f*)-invexity/(*<sup>G</sup>*, *<sup>α</sup>f*)-pseudoinvexity; (*<sup>G</sup>*, *<sup>α</sup>f*)-bonvexity/(*<sup>G</sup>*, *<sup>α</sup>f*)-pseudobonvexity
