About the Lipschitz property of the metric projection in the Hilbert space
For a real Banach space \(E\), \(x\in E\), and \(A\subset E\) put \(\rho(x,A)=\mathrm{dist}(x,A)\) and let \(P_Ax=\{a\in A: \|x-a\|=\rho(x,A)\}\) be the metric projection on \(A\). For \(a\in E\) and \(r>0\) denote by \(B_r(a)\) the closed ball of center \(a\) and radius \(r\) and let \(U(A,r)=\{x\in E : \rho(x,A)<r\}\). The set \(A\) is called strongly convex of radius \(R>0\) if it can be written as an intersection of closed balls of radius \(R>0.\) Some basic results on strongly convex analysis are given in the book [\textit{E. S. Polovinkin} and \textit{M. V. Balashov}, Elements of convex and strongly convex analysis. Moskva: Fizmatlit (2007; Zbl 1181.26028)].NEWLINENEWLINE The paper is concerned with the Lipschitz properties of the metric projection on strongly convex subsets of a Hilbert space \(\mathcal H.\) For instance, if \(A\subset \mathcal H\) is bounded, closed and convex and, for some \(0<C<1\) and \(r>0,\, \|P_Ax_0-P_Ax_1\|\leq C\|x_0-x_1\|\), for all \(x_0,x_1\in \mathcal H\setminus U(A,r)\), then \(A\) is strongly convex of radius \(R=Cr/(1-C)\) (Theorem 2.1). In Corollary 2.1 one shows that \(A\) is strongly convex of radius \(R>0\) iff for every \(r>0\), \(P_A\) satisfies the Lipschitz condition \(\|P_Ax_0-P_Ax_1\|\leq R (R+r)^{-1}\|x_0-x_1\|\), for all \(x_0,x_1\in \mathcal H\setminus U(A,r)\). If \(E\) is a strictly convex reflexive Banach space having the Mazur intersection property such that the metric projection is uniformly continuous with some given modulus \(\omega\) on the class of strongly convex sets of radius \(R\), for all \(R>0\), then \(E\) is isomorphic to a Hilbert space (Theorem 2.3). In the last part of the paper, the obtained results are applied to evaluate the rate of convergence of the gradient projection algorithm for the minimization of a differentiable convex function on a strongly convex set.
scientific article; zbMATH DE number 6062842
scientific article