The directional subdifferential of the difference of two convex functions (Q628751)

Let \(X\) be a normed vector space with dual \(X^{\ast }\), let \(g,h:X\longrightarrow \mathbb{R\cup \{+\infty \}}\) be convex, let \(f:=g-h,\) with the convention \((+\infty )-(+\infty )=+\infty \), and let \(\overline{x}\in X\) be such that \(g\) and \(h\) are finite at \(\overline{x}\). The main results are the equalities \(\partial^F(\overline{x})=\partial g(\overline{x})\boxminus \partial h(\overline{x})\) and \(\partial^{D}(\overline{x})=\partial g(\overline{x})\boxminus \partial h(\overline{x})\), with \(\partial^{F}\), \(\partial \) and \(\partial^D\) denoting Fréchet, Fenchel-Moreau and Dini-Hadamard subdifferential, respectively, and \(\boxminus\) denotes the star difference: \(\partial g(\overline{x})\boxminus \partial h(\overline{x}):=\{x^{\ast }\in X^{\ast }:x^{\ast}+\partial h(\overline{x})\subseteq \partial g(\overline{x})\}\). Those equalities are proved to hold under suitable generalized dissipativity assumptions on \(\partial h\). It turns out that for the formula for the Dini-Hadamard subdifferential to hold it suffices that \(h\) be continuous at \(\overline{x}\). Extensions to differences of approximately and directionally approximately starshaped functions are also provided.