Guang-Ri Piao, Zhe Hong, Kwan Deok Bae, Do Sang Kim, A characterization of the $\varepsilon$-normal set and its application in robust convex optimization problems

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DOI: 10.23952/jnva.7.2023.6.01

Volume 7, Issue 6, 1 December 2023, Pages 897-907

Abstract. Let $C:=\{x\in {\mathbb R}^n \colon g(x, v)\leqq 0, \ \forall v\in \mathcal V\},$ where $g \colon {\mathcal R}^n\times {\mathcal R}^p\rightarrow {\mathcal R}$ is a continuous function such that, for all $v\in {\mathcal R}^p,$ $g(\cdot, v)$ is a convex function, and $\mathcal{V}\subset {\mathcal R}^p$ is some uncertain set. In this paper, under the satisfaction of the robust characteristic cone constraint qualification, we first propose a represented form of the $\epsilon$ -normal set to the convex set $C$ at a considered point $\bar x\in C.$ Then, the proposed result is applied to formulate a (necessary and sufficient) approximate optimality theorem for a quasi $(\alpha, \epsilon)$ -solution to the robust counterpart of a convex optimization problem in the face of data uncertainty.

How to Cite this Article:
G.R. Piao, Z. Hong, K.D. Bae, D.S. Kim, A characterization of the $\varepsilon$ -normal set and its application in robust convex optimization problems, J. Nonlinear Var. Anal. 7 (2023), 897-907.