Stable global well-posedness and global strong metric regularity

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.