Gauss-Newton method for convex composite optimizations on Riemannian manifolds
The notions of quasi-regularity for \(p_0 \in M\) (a Riemannian manifold), quasi-regular radius \(r_{p_0}\) and quasi-regular bound function \(\beta{p_0}\) attached to a quasi-regular point \(p_0\) are extended to Riemannian manifolds. The authors show that if the initial point \(p_0\) is a quasi-regular point with \((r_{p_0}, \beta_{p_0})\) and if \(DF\) (\(F\) is a nonlinear Fréchet differentiable mapping from \(\mathbb R^m\) to \(\mathbb R^n\), \(DF(p)\) denotes the derivative of \(F\) at \(p\)) satisfies a Lipschitz type condition, then the Gauss-Newton sequence \(\{p_n\}\) generated by the algorithm \(R (\eta, \Delta, p_0)\) converges at a quadratic rate to some \(p^*\) with \(F(p^*) \in C\) (in particular, \(p^*\) solves: \(\min_{p\in M} f(p):=h(F(p))\), where \(h\) is a real-valued convex function on \(\mathbb R\)). Furthermore, two applications to special cases, one is for the case of regularities of Riemannian manifolds and the other is for the case when \(C\) is a cone and \((DF (p_0)(.)-C)\) is surjective, are proposed. Main result: The authors extend the Gauss-Newton method to a Riemannian manifold to solve the convex composite optimization on the Riemannian manifold \(M\)which is formulated as follows: \[ \min_{p \in M} f(p):=h(F(p)).\tag{1} \] The majorizing function technique to establish the semi-local convergence of sequences generated by the Gauss-Newton method for the convex composite function \(h_0F\) on the Riemannian manifold is used. The extended Gauss-Newton for the convex composite optimization problem on Riemannian manifold (1) is defined. The authors extend the notion of the quasi-regular point to Riemannian manifolds and specialize own results in two important cases: the classical Lipschitz condition and the \(\gamma\)-condition (implies the \(L\)-average Lipschitz condition) which have been used extensively in the study of convergence of Newton's method both in Banach spaces and Riemannian manifolds. If \(F\) satisfies the \(\gamma\)-condition at \(p_0\) in \(B (p_0, r)\), \(DF\) satisfies the \(L\)-average Lipschitz condition in \(B (p_0, r)\) with \(L\) given by \(L(u) = \frac{2 \gamma}{(1- \gamma u)^3}\) for each \(u\) with \(0 \leq u < \frac{1}{\gamma}\). For a regular point \(p_0\) of \(F(p) \in C\) there exist constants \(r>0\) and \(\beta_0 >0\) such that \(\Lambda (p) \neq \emptyset\) and \(d (0, \Lambda (p)) \leq \beta (d(p_0,p))d(F(p),C)\) for all \(p \in B (p_0,r)\) holds for \(r\) and \(\beta (.)= \beta_0\), consequently, \(p_0\) is a quasi-regular point with the quasi-regular radius \(r_{p_0} \geq r\) and the quasi-regular bound function \(\beta_{p_0} (.) \leq p_0\) on \([0,r]\).
scientific article; zbMATH DE number 6084586
scientific article