Walaa M. Moursi

Heinz H. Bauschke, Walaa M. Moursi

The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results.

Heinz H. Bauschke, Warren L. Hare, Walaa M. Moursi

The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counter-examples are presented, including some concerning the celebrated Brezis-Haraux theorem.

Walaa M. Moursi, Lieven Vandenberghe

The Douglas-Rachford method is a popular splitting technique for finding a zero of the sum of two subdifferential operators of proper closed convex functions; more generally two maximally monotone operators. Recent results concerned with linear rates of convergence of the method require additional properties of the underlying monotone operators, such as strong monotonicity and cocoercivity. In this paper, we study the case when one operator is Lipschitz continuous but not necessarily a subdifferential operator and the other operator is strongly monotone. This situation arises in optimization methods which involve primal-dual approaches. We provide new linear convergence results in this setting.

Wenbo Yang, Eman Rezk, Walaa M. Moursi et al.

Machine-learning models working on skin images often have degraded performance when the skin colour captured in images (SCCI) differs between training and deployment. Such differences arise from entangled environmental factors (e.g., illumination, camera settings), and intrinsic factors (e.g., skin tone) that cannot be accurately described by a single "skin tone" scalar. To mitigate such colour mismatch, we propose a skin-colour disentangling framework that adapts disentanglement-by-compression to learn a structured, manipulable latent space for SCCI from unlabelled dermatology images. To prevent information leakage that hinders proper learning of dark colour features, we introduce a randomized, mostly monotonic decolourization mapping. To suppress unintended colour shifts of localized patterns (e.g., ink marks, scars) during colour manipulation, we further propose a geometry-aligned post-processing step. Together, these components enable faithful counterfactual editing and answering an essential question: "What would this skin condition look like under a different SCCI?", as well as direct colour transfer between images and controlled traversal along physically meaningful directions (e.g., blood perfusion, camera white balance), enabling educational visualization of skin conditions under varying SCCI. We demonstrate that dataset-level augmentation and colour normalization based on our framework achieve competitive lesion classification performance.

Walaa M. Moursi

The forward-backward splitting technique is a popular method for solving monotone inclusions that has applications in optimization. In this paper we explore the behaviour of the algorithm when the inclusion problem has no solution. We present a new formula to define the normal solutions using the forward-backward operator. We also provide a formula for the range of the displacement map of the forward-backward operator. Several examples illustrate our theory.

Heinz H. Bauschke, Minh N. Dao, Walaa M. Moursi

The Douglas-Rachford algorithm is a simple yet effective method for solving convex feasibility problems. However, if the underlying constraints are inconsistent, then the convergence theory is incomplete. We provide convergence results when one constraint is an affine subspace. As a consequence, we extend a result by Spingarn from halfspaces to general closed convex sets admitting least-squares solutions.

Heinz H. Bauschke, Graeme R. Douglas, Walaa M. Moursi

In 1971, Pazy presented a beautiful trichotomy result concerning the asymptotic behaviour of the iterates of a nonexpansive mapping. In this note, we analyze the fixed-point free case in more detail. Our results and examples give credence to the conjecture that the iterates always converge cosmically.

Heinz H. Bauschke, Walaa M. Moursi

The Douglas-Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas-Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas-Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas-Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.