Alternating projections in CAT(0) spaces
Copyright policy )Alternating projections in CAT(0) spaces
In this work, the (weak and strong) convergence of the alternating projection method in CAT(0) spaces is proved. More precisely, the following theorem is proven. Theorem 4.1. Let \(X\) be a complete CAT(0) space and \(A,B\subset X\) be closed and convex such that \(A\cap B\neq\emptyset\). Then, for \(x_0\in X\) a starting point, the sequence \((x_n)\) given by: \[ x_{2n-1}=P_A(x_{2n-2}), \quad x_{2n}=P_B(x_{2n-1}) \] converges weakly to a point \(x\in A\cap B\). If, additionally, \(A,B\) are boundedly regular, then the sequence \((x_n)\) converges strongly to \(x\). An application to the problem of minimization of convex functionals in CAT(0) spaces is also included.
- Max Planck Society Germany
- University of Newcastle Australia Australia
convex optimization, Alternating projections, nonpositive curvature, Feasibility problem, Iterative procedures involving nonlinear operators, Non-positive curvature, Special maps on metric spaces, Weak convergence, feasibility problem, algorithm, Fixed-point and coincidence theorems (topological aspects), Applied Mathematics, Nearest point mapping, nearest point mapping, nonexpansive mapping, Convex optimization, Algorithm, strong convergence, non-expansive mapping, non-positive curvature, Non-expansive mapping, CAT(0) space, alternating projections, Applications of operator theory in optimization, convex analysis, mathematical programming, economics, weak convergence, Analysis
convex optimization, Alternating projections, nonpositive curvature, Feasibility problem, Iterative procedures involving nonlinear operators, Non-positive curvature, Special maps on metric spaces, Weak convergence, feasibility problem, algorithm, Fixed-point and coincidence theorems (topological aspects), Applied Mathematics, Nearest point mapping, nearest point mapping, nonexpansive mapping, Convex optimization, Algorithm, strong convergence, non-expansive mapping, non-positive curvature, Non-expansive mapping, CAT(0) space, alternating projections, Applications of operator theory in optimization, convex analysis, mathematical programming, economics, weak convergence, Analysis
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