On global properties of lower semicontinuous quadratically minorized functions

Abstract

We use the framework of a type of abstract convexity (${\Phi }_{lsc}$-convexity) to investigate properties of lower semicontinuous quadratically minorized functions in Hilbert spaces. A new result, which states that, for every local ${\Phi }_{lsc}$-subgradient there exists a global one is proved and plays a crucial role in our considerations. We deliver conditions for abstract subdifferentiability (${\Phi }_{lsc}$-subdifferentiability) of locally $C^{1,1}$ functions, twice continuously differentiable functions, prox-regular functions and paraconvex functions. As an application we establish a new sufficient and necessary condition for minimax equality for ${\Phi }_{lsc}$-convex functions. This new condition is expressed in therms of ${\Phi }_{lsc}$-subdifferential.