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 ψ -tilt-stable global minimum and stable global φ -well-posedness with ψ and φ being the so-called admissible functions. We adopt global strong metric regularity of the subdifferential mapping ∂ ^ f of the objective function f with respect to an admissible function ψ and prove that the global strong metric regularity of ∂ ^ f at 0 with respect to ψ implies the stable global φ -well-posedness of f with 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 ψ -tilt-stable global minimum and stable global φ -well-posedness. Our results are new even in the convexity case.