On the Metric Subregularity for Cyclic Relaxed Douglas-Rachford Operators via Spectral Theory and Differentiability

Date: 21 May 2026 (Thursday)

Venue: Room 507, A6, Institute of Mathematics

Time: 9:30 - 11:00

Abstract: We study metric subregularity of the residual mapping $I-T$ associated with a cyclic relaxed Douglas--Rachford ($\CDR_\lambda$) operator $T$ in a finite dimension.

First, for linear operators we provide a spectral characterization of the linear subregularity modulus in terms of the smallest singular value of $I-T$ restricted to $(\Fix T)^\perp$.

Second, for nonlinear operators generated by smooth manifolds we combine differentiability of the projector (Lewis--Malick) with linearization to reduce local behavior of $\CDR_\lambda$ to a cyclic composition of relaxed Douglas--Rachford mappings on tangent subspaces. In the consistent case, tangent-space transversality forces $\Fix DT(\bar x)=\{0\}$ and yields linear metric subregularity. In the inconsistent case, local linearization is performed along a fixed cycle of shadow points.  Each block derivative of the cyclic relaxed Douglas--Rachford operator acts on the tangent spaces of the active sets, and the fixed space of the full derivative is characterized via an intersection of the kernels of the corresponding shape operators. This yields an exact formula for $\Fix DT(\bar x)$ and allows us to deduce linear metric subregularity under active-tangent transversality.

Finally, we formulate metric subregularity with nonlinear gauge and show that analytic fixed-point mappings satisfy a H\"older-type error bound via a \L ojasiewicz inequality, yielding nonlinear-gauge subregularity even when linear subregularity fails.

This is joint work with David Russell Luke and Thi Lan Dinh, Institute for Numerical and Applied Mathematics, University of Göttingen.