Directional differentiability of metric projections onto moving sets at boundary points (Q1102525)

Let \(\Omega: X\rightrightarrows Y\) be a set-valued mapping between finite-dimensional normed spaces X and Y. Associated with \(\Omega\) are the distance function \(d_{\Omega}(x,y):=\inf \{\| y-\omega \|:\omega\in \Omega (x)\}\) and the metric projection \(\Pi_{\Omega}(x,y):=\{\omega \in \Omega (x):\| y-\omega \| =d_{\Omega}(x,y)\}\). This paper concerns the conical approximation of \(\Omega\) near a boundary point \((x_ 0,y_ 0)\) of gph \(\Omega\), under the assumption that \(\Omega\) is pseudo-Lipschitz continuous at \((x_ 0,y_ 0)\) [see \textit{R. T. Rockafellar}, Nonlinear Anal., Theory Methods Appl. 9, 867-885 (1985; Zbl 0573.54011)]. Its main result establishes the equivalence of ``tangential differentiability'' of the multifunction \(\Omega\) at \((x_ 0,y_ 0)\); ``conical approximability'' of the graph of \(\Omega\) ; and directional differentiability (in every direction) of the distance function \(d_{\Omega}\). When these conditions hold, it shows that one can approximate the multifunction \(\Pi_{\Omega}\) of metric projection on \(\Omega\) near \((x_ 0,y_ 0)\) by taking the metric projection \(\Pi_{D\Omega (x_ 0,y_ 0)}\) on the approximating cone to gph\(\Omega\). These results are applied to the multifunction \[ \Omega (x):=\{y: g_ i(x,y)\leq 0,\quad i=1,...,q;\quad g_ i(x,y)=0,\quad i=q+1,...,p\}. \]