arXiv subjects

Walaa M. Moursi

Publications and source records attributed to Walaa M. Moursi.

At least 19 recordsLinked to original sources

arXiv2026

Best Approximation Optimal Control for Infeasible Double Integrator and Douglas--Rachford Algorithm

We consider the problem of finding (in some sense) the best approximation control for an infeasible double integrator. The control function is constrained by upper and lower bounds that are too tight and thus cause infeasibility. The infeasibility is characterized by a gap function (representing the separation between two constraint sets) whose squared ${\cal L}^2$-norm is to be minimized to find the best approximation control solution. First, we review the existing results for problems involving a general linear control system. Then, for the infeasible double integrator problem, we present an analytical solution for the bang--bang control with at most one switching. The infinite-dimensional optimization problem is reduced to the problem of solving two algebraic equations in two variables, to compute the switching time and gap function. We discuss numerical approaches to solving the system of equations. Finally, we describe the (relaxed) Douglas--Rachford algorithm for the double integrator problem and carry out numerical experiments to illustrate the implementation of the algorithm and test performance.

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arXiv2026

Range of the displacement operator of PDHG with applications to quadratic and conic programming

Primal-dual hybrid gradient (PDHG) is a first-order method for saddle-point problems and convex programming introduced by Chambolle and Pock. Recently, Applegate et al.\ analyzed the behavior of PDHG when applied to an infeasible or unbounded instance of linear programming, and in particular, showed that PDHG is able to diagnose these conditions. Their analysis hinges on the notion of the infimal displacement vector in the closure of the range of the displacement mapping of the splitting operator that encodes the PDHG algorithm. In this paper, we develop a novel formula for this range using monotone operator theory. The analysis is then specialized to conic programming and further to quadratic programming (QP) and second-order cone programming (SOCP). A consequence of our analysis is that PDHG is able to diagnose infeasible or unbounded instances of QP and of the ellipsoid-separation problem, a subclass of SOCP.

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arXiv2026

Strong Convergence of FISTA for Affinely Constrained Convex Quadratic Minimization

In October 2025, research by Boţ, Fadili, and Nguyen, and by Jang and Ryu, led to the seminal result that Beck and Teboulle's FISTA converges weakly to a minimizer of the sum of two convex functions resolving a long-standing open problem. The first strong convergence result was obtained in November 2025 by Moursi, Naguib, Pavlovic, and Vavasis for affinely constrained convex minimization provided certain closedness conditions hold. In this paper, we prove strong convergence in the affine-quadratic case without any closedness assumption. Specializing this to the unconstrained case, we obtain the strong convergence of Nesterov's accelerated gradient method when applied to a convex quadratic objective function.

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arXiv2026

Eckstein-Ferris-Pennanen-Robinson duality revisited: paramonotonicity, total Fenchel-Rockafellar duality, and the Chambolle-Pock operator

Finding zeros of the sum of two maximally monotone operators involving a continuous linear operator is a central problem in optimization and monotone operator theory. We revisit the duality framework proposed by Eckstein, Ferris, Pennanen, and Robinson from a quarter of a century ago. Paramonotonicity is identified as a broad condition ensuring that saddle points coincide with the closed convex rectangle formed by the primal and dual solutions. Additionally, we characterize total duality in the subdifferential setting and derive projection formulas for sets that arise in the analysis of the Chambolle-Pock algorithm within the recent framework developed by Bredies, Chenchene, Lorenz, and Naldi.

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arXiv2026

Understanding FISTA's weak convergence: A step-by-step introduction to the 2025 milestone

Beck and Teboulle's FISTA for finding the minimizer of the sum of two convex functions is one of the most important algorithms of the past decades. While function value convergence of the iterates was known, the actual convergence of the iterates remained elusive until October 2025 when Jang and Ryu, as well as Boţ, Fadili, and Nguyen proved weak convergence. In this paper, we provide a gentle self-contained introduction to the proof of their remarkable result.

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arXiv2026

Automated Disentangling Analysis of Skin Colour for Lesion Images

Machine-learning models applied to skin images often have degraded performance when the skin colour captured in images (SCCI) differs between training and deployment. These discrepancies arise from a combination of 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 -- a simplification commonly adopted by prior work. To mitigate such colour mismatches, 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. Ultimately, our work promotes equitable diagnosis through creating diverse training datasets that include different skin tones and image-capturing conditions.

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arXiv2025

Accelerated Proximal Gradient Methods in the affine-quadratic case: Strong convergence and limit identification

Recent works by Bot-Fadili-Nguyen (arXiv:2510.22715) and by Jang-Ryu (arXiv:2510.23513) resolve long-standing iterate convergence questions for accelerated (proximal) gradient methods. In particular, Bot-Fadili-Nguyen prove weak convergence of discrete accelerated gradient descent (AGD) iterates and, crucially, convergence of the accelerated proximal gradient (APG) method in the composite setting, with extensions to infinite-dimensional Hilbert spaces. In parallel, Jang-Ryu establish point convergence for the continuous-time accelerated flow and for discrete AGD in finite dimensions. These results leave open which minimizer is selected by the iterates. We answer this in the affine-quadratic setting: when initialized at the same point, the difference between the proximal gradient (PGM) and APG iterates converges weakly to zero. Consequently, APG converges weakly to the best approximation of the initial point in the solution set. Moreover, under mild assumptions on the parameter sequence, we obtain strong convergence of APG. The result is tight: a two-dimensional example shows that coincidence of the APG and PGM limits is specific to the affine-quadratic regime and does not hold in general.

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arXiv2025

On the Bredies-Chenchene-Lorenz-Naldi algorithm

Monotone inclusion problems occur in many areas of optimization and variational analysis. Splitting methods, which utilize resolvents or proximal mappings of the underlying operators, are often applied to solve these problems. In 2022, Bredies, Chenchene, Lorenz, and Naldi introduced a new elegant algorithmic framework that encompasses various well known algorithms including Douglas-Rachford and Chambolle-Pock. They obtained powerful weak and strong convergence results, where the latter type relies on additional strong monotonicity assumptions. In this paper, we complement the analysis by Bredies et al. by relating the projections of the fixed point sets of the underlying operators that generate the (reduced and original) preconditioned proximal point sequences. We also obtain strong convergence results in the case of linear relations. Various examples are provided to illustrate the applicability of our results.

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arXiv2024

On the Douglas-Rachford and Peaceman-Rachford algorithms in the presence of uniform monotonicity and the absence of minimizers

The Douglas-Rachford and Peaceman-Rachford algorithms have been successfully employed to solve convex optimization problems, or more generally find zeros of monotone inclusions. Recently, the behaviour of these methods in the inconsistent case, i.e., in the absence of solutions has triggered significant consideration. It has been shown that under mild assumptions the shadow sequence of the Douglas-Rachford algorithm converges weakly to a generalized solution when the underlying operators are subdifferentials of proper lower semicontinuous convex functions. However, no convergence behaviour has been proved in the case of Peaceman-Rachford algorithm. In this paper, we prove the convergence of the shadow sequences associated with the Douglas-Rachford algorithm and Peaceman-Rachford algorithm when one of the operators is uniformly monotone and $3^*$ monotone but not necessarily a subdifferential. Several examples illustrate and strengthen our conclusion. We carry out numerical experiments using example instances of optimization problems.

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arXiv2024

Infeasible and Critically Feasible Optimal Control

We consider optimal control problems involving two constraint sets: one comprised of linear ordinary differential equations with the initial and terminal states specified and the other defined by the control variables constrained by simple bounds. When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the ``best approximation'' optimal control minimizing the distance (and thus finding the ``gap'') between the two sets is of bang--bang type, with the ``gap function'' playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang--bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.

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arXiv2023

Optimal Control Duality and the Douglas-Rachford Algorithm

We explore the relationship between the dual of a weighted minimum-energy control problem, a special case of linear-quadratic optimal control problems, and the Douglas-Rachford (DR) algorithm. We obtain an expression for the fixed point of the DR operator as applied to solving the optimal control problem, which in turn devises a certificate of optimality that can be employed for numerical verification. The fixed point and the optimality check are illustrated in an example optimal control problem.

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arXiv2023

Accelerated gradient descent: A guaranteed bound for a heuristic restart strategy

The $O(1/k^2)$ convergence rate in function value of accelerated gradient descent is optimal, but there are many modifications that have been used to speed up convergence in practice. Among these modifications are restarts, that is, starting the algorithm with the current iteration being considered as the initial point. We focus on the adaptive restart techniques introduced by O'Donoghue and Candès, specifically their gradient restart strategy. While the gradient restart strategy is a heuristic in general, we prove that applying gradient restarts preserves and in fact improves the $O(1/k^2)$ bound, hence establishing function value convergence, for one-dimensional functions. Applications of our results to separable and nearly separable functions are presented.

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arXiv2023

Regularized Nonsmooth Newton Algorithms for Best Approximation

We consider the problem of finding the best approximation point from a polyhedral set, and its applications, in particular to solving large-scale linear programs. The classical projection problem has many various and many applications. We study a regularized nonsmooth Newton type solution method where the Jacobian is singular; and we compare the computational performance to that of the classical projection method of Halperin-Lions-Wittmann-Bauschke (HLWB). We observe empirically that the regularized nonsmooth method significantly outperforms the HLWB method. However, the HLWB has a convergence guarantee while the nonsmooth method is not monotonic and does not guarantee convergence due in part to singularity of the generalized Jacobian. Our application to solving large-scale linear programs uses a parametrized projection problem. This leads to a \emph{stepping stone external path following} algorithm. Other applications are finding triangles from branch and bound methods, and generalized constrained linear least squares. We include scaling methods that improve the efficiency and robustness.

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arXiv2023

How averaged is the composition of two linear projections?

Projection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this note, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura-Yamada bound for the modulus of the composition is sharp.

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arXiv2022

How to project onto the intersection of a closed affine subspace and a hyperplane

Let $A$ be a closed affine subspace and let $B$ be a hyperplane in a Hilbert space. Suppose we are given their associated nearest point mappings $P_A$ and $P_B$, respectively. We present a formula for the projection onto their intersection $A\cap B$. As a special case, we derive a formula for the projection onto the intersection of two hyperplanes. These formulas provides useful information even if $A\cap B$ is empty. Examples and numerical experiments are also provided.

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arXiv2022

The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces

The Douglas-Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case, i.e., when a solution does not exist. The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas-Rachford splitting operator and the corresponding minimal displacement vector. In this paper, we provide a formula for the range of the Douglas-Rachford splitting operator in (possibly) infinite-dimensional Hilbert space under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.

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arXiv2022

Strongly nonexpansive mappings revisited: uniform monotonicity and operator splitting

The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of the class of uniformly monotone operators and their corresponding reflected resolvents. We show that the latter is closely related to the class of the strongly nonexpansive operators introduced by Bruck and Reich. Connections to duality via inverse operators are systematically studied. We provide applications to Douglas-Rachford and Peaceman-Rachford methods. Examples that illustrate and tighten our results are presented.

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arXiv2021

On the Douglas-Rachford algorithm for solving possibly inconsistent optimization problems

More than 40 years ago, Lions and Mercier introduced in a seminal paper the Douglas-Rachford algorithm. Today, this method is well recognized as a classical and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. While the underlying theory has matured, one case remains a mystery: the behaviour of the shadow sequence when the given functions have disjoint domains. Building on previous work, we establish for the first time weak and value convergence of the shadow sequence generated by the Douglas-Rachford algorithm in a setting of unprecedented generality. The weak limit point is shown to solve the associated normal problem which is a minimal perturbation of the original optimization problem. We also present new results on the geometry of the minimal displacement vector.

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