Stable global well-posedness and global strong metric regularity

Xi Yin Zheng () and Jiangxing Zhu ()
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Xi Yin Zheng: Yunnan University
Jiangxing Zhu: Yunnan University

Journal of Global Optimization, 2022, vol. 83, issue 2, No 9, 359-376

Abstract: Abstract In this paper, in contrast to the literature on the tilt-stability only dealing with local minima, we introduce and study the $$\psi $$ ψ -tilt-stable global minimum and stable global $$\varphi $$ φ -well-posedness with $$\psi $$ ψ and $$\varphi $$ φ being the so-called admissible functions. We adopt global strong metric regularity of the subdifferential mapping $${{\hat{\partial }}} f$$ ∂ ^ f of the objective function f with respect to an admissible function $$\psi $$ ψ and prove that the global strong metric regularity of $$\hat{\partial }f$$ ∂ ^ f at 0 with respect to $$\psi $$ ψ implies the stable global $$\varphi $$ φ -well-posedness of f with $$\varphi (t)=\int _0^t\psi (s)ds$$ φ ( t ) = ∫ 0 t ψ ( s ) d s and that if f is convex then the converse implication also holds. Moreover, we establish the relationships between $$\psi $$ ψ -tilt-stable global minimum and stable global $$\varphi $$ φ -well-posedness. Our results are new even in the convexity case.

Keywords: Stable well-posedness; Tilt stability; Metric regularity; Subdifferential; 90C31; 49K40; 49J52 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01100-4

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