arXiv subjects

Jordan Collard

Publications and source records attributed to Jordan Collard.

3 recordsLinked to original sources

arXiv2026

Finite Recovery by Extrapolation for the Douglas--Rachford Method on Affine Subspaces

We completely describe the dynamics of the Douglas--Rachford (DR) method for solving affine subspace feasibility. We characterise the affine hull of the resulting iterates, and we show how the corresponding search space's dimension is an explicit function of the number of distinct nontrivial principal angles. This dimension can be much smaller than the ambient dimension, helping to explain why extrapolation can be so effective. Using this characterisation, we extend the Lyapunov surrogate method (LSM) to higher extrapolation orders and prove that it recovers the DR limit exactly from finitely many iterates. We show that LSM and a least-squares extrapolation (LeSE) share the affine search spaces used by the generalized minimal residual method (GMRES) applied to the DR fixed-point equation (DR-GMRES). The dimension characterisation gives bounds on the number of steps required for exact recovery by all three methods. We develop adaptive implementations that estimate the search-space dimension from differences between successive DR iterates and use this estimate to select the extrapolation order. Numerical experiments show how ill-conditioning affects recovery accuracy, and we identify cases where adaptive extrapolation runs faster than DR-GMRES.

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arXiv2026

Resolvent Moreau identities without monotonicity: theory and applications to Gabay duality, Douglas--Rachford and ADMM

Duality is most often defined as a relationship between convex functions. If those functions are nonconvex, classical duality breaks down. Notwithstanding, we show that another kind of duality still exists, not between the functions themselves, but between the so-called resolvent operators used to solve associated problems. In fact, this duality-like relationship holds for any set-valued mapping, and is a generalization of the Moreau's identity. We use this duality to study existing operator schemes and to design new ones. In particular, we show that the duality-like relationship Daniel Gabay illuminated between the Douglas--Rachford splitting (DR) and the Alternating Direction Method of Multipliers (ADMM) extends to nonmonotone inclusion problems. We use this relationship to provide explicit counterexamples to the convergence of ADMM in several open cases, by studying the (easier to analyse) DR scheme. Motivated by our observations, we design a class of convergent resolvent homotopy schemes and use them to solve nonconvex-regularised least absolute deviations problems. This important problem class has received little attention in the literature, since the convex component of the objective does not enjoy strong convexity.

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arXiv2025

Quadratic Convergence of a Projection Method for a Plane Curve Feasibility Problem

Under conditions that prevent tangential intersection, we prove quadratic convergence of a projection algorithm for the feasibility problem of finding a point in the intersection of a smooth curve and line in $\mathbb{R}^2$. This nonconvex problem has been studied in the literature for both Douglas-Rachford algorithm (DR) and circumcentered reflection method (CRM), because it is prototypical of inverse problems in signal processing and image recovery. This result highlights the potential of extrapolated methods to meaningfully accelerate convergence in structured feasibility problems. Numerical experiments confirm the theoretical findings. Our work lays the foundations for extending such results to higher dimensional problems.

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