On the Metric Subregularity for Cyclic Projections via Quotient Spaces and Tangential Approximations

We give a comprehensive proof of metric subregularity for a cyclic composition of projection operators $T=P_{A_1}\cdots P_{A_m}$ in an Euclidean space.  Both the consistent and inconsistent cases are treated. The affine case is handled by a quotient-space reduction that yields a global subregularity inequality.   The nonlinear theory is developed under a $p$-order tangentially projecting approximation ($p$-otpa) assumption, which reduces the geometry to the affine model whenever $p>1$. Smooth embedded manifolds satisfy $2$-otpa, and thus their metric subregularity follows immediately. We also present a non-manifold example of set satisfying $p$-otpa for $p\in(1,2)$. This is joint work with David Russell Luke and Thi Lan Dinh in Institute for Numerical and Applied Mathematics, University of Goettingen, Germany.