arXiv subjects

Roger Behling

Publications and source records attributed to Roger Behling.

At least 19 recordsLinked to original sources

arXiv2026

The method of ellipcenters with momentum and relaxation for convex quadratic minimization

The method of ellipcenters (ME) is a recent technique developed for unconstrained minimization. Its iteration relies only on first order information and consists of building a suitable two-dimensional ellipse that tries to capture intrinsic ill-conditioning of the problem. The center of the ellipse is taken as the next iterate, which justifies the name of the method. ME was already shown to converge linearly when the objective function is smooth and strongly convex. The special case when the objective is quadratic was studied in the first paper on ME and is also the subject of our work here. In this paper, we propose two variants of ME: RelaxME which adds in ME an additional relaxation step at each iteration and MomME which embeds momentum in ME. We prove linear convergence of RelaxME and MomME and, in particular, we show convergence of RelaxME and MomME (and also of ME) in one iteration when the matrix of the quadratic form has only two distinct eigenvalues. Finally, we provide the results of numerical experiments which compare on five optimization problems (and different combinations of parameters defining these problems) ME, RelaxME, and MomME, with 3 other optimizers: the conjugate gradient method [10], Barzilai and Borwein gradient method with long step [2], and the gradient method with adaptive spectral step length [7]. On most instances, MomME provides the smallest number of iterations and conjugate gradient and MomME provide the smallest CPU times and similar CPU times. This opens optimistic possibilities for ME with momentum in broader settings.

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arXiv2026

The Method of Ellipcenters for strongly convex minimization

This work is about ME, the Method of Ellipcenters. ME was recently introduced by these very authors as a first order accelerated scheme for unconstrained minimization. Its iterates are all centers of ellipses carefully designed to somehow capture ill-conditioning of the underlying optimization problem. In the first article on ME, we were able to prove that it converges with linear rate when the objective function is quadratic and strongly convex, while here we derive convergence for any differentiable strongly convex objective. This investigation was inspired by the great performance of ME in quadratic minimization against steepest descent with exact line search, FISTA, Barzilai-Borwein and Conjugate Gradient. The experiments we carry out now, make ME even more attractive from the numerical point of view. On top of that, the theory seems promising for quite more general settings.

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arXiv2026

New operator designs for Halpern iterations with explicit rates under H\"older error bounds

We investigate the asymptotic behavior of Halpern-type iterations applied to quasi-nonexpansive operators arising in best approximation problems over the intersection of finitely many closed convex sets in $\mathbb{R}^n$. Assuming a local decrease condition for the underlying operator and standard requirements on the stepsizes $(\alpha_k) \subset (0,1)$, we first prove strong convergence of the Halpern sequence $x_{k+1} = \alpha_k x_0 + (1-\alpha_k) T x_k$ to the best approximation point $x^\star$ in the intersection set, that is, the metric projection of $x_0$ onto that set. Under the additional assumption that the intersection satisfies a H\"older-type error bound with exponent $\gamma \in (0,1]$, we then derive explicit convergence rates for both feasibility and norm error: the distance from $x_k$ to the intersection set decays like $\mathcal O(\alpha_k^{\gamma/(2-\gamma)})$, while the norm error $\|x_k - x^\star\|$ decays like $\mathcal O(\alpha_k^{\gamma/(4-2\gamma)})$. These results apply to most projection-type operators used in convex feasibility problems (including MAP, CRM/SCCRM, Cimmino and 3PM/A3PM) and extend classical convergence analyses of the Halpern-type iterations by providing explicit, geometry-dependent rates governed by H\"older-type error bounds. Our numerical experiments show that Halpern-type iterations combined with most of these projection-type operators are quicker than Dykstra's algorithm to find the projection of a point in an intersection of ellipsoids or in an intersection of polyhedrons.

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arXiv2025

Introducing the method of ellipcenters, a new first order technique for unconstrained optimization

In this paper, we introduce the Method of Ellipcenters (ME) for unconstrained minimization. At the cost of two gradients per iteration and a line search, we compute the next iterate by setting it as the center of an elliptical interpolation. The idea behind the ellipse built in each step is to emulate the original level curve of the objective function constrained to a suitable two-dimensional affine space, which is determined by the current iterate and two appropriate gradient vectors. We present the method for general unconstrained minimization and carry out a convergence analysis for the case where the objective function is quadratic. In this context, ME enjoys linear convergence with the rate being at least as good as the linear rate of the steepest descent (gradient) method with optimal step. In our experiments, however, ME was much faster than the gradient method with optimal step size. Moreover, ME seems highly competitive in comparison to several well established algorithms, including Nesterov's accelerated gradient, Barzilai-Borwein, and conjugate gradient. The efficiency in terms of both time and number of iterations is stressed even more for ill-conditioned problems. A theoretical feature that might be a reason for this is that ME coincides with Newton for quadratic programs in two-dimensional Euclidean spaces, solving them in one single step. In our numerical tests, convergence in one iteration only was also observed for much larger problem sizes.

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arXiv2025

Basis pursuit by inconsistent alternating projections

Basis pursuit is the problem of finding a vector with smallest $\ell_1$-norm among the solutions of a given linear system of equations. It is a well-known convex relaxation of the sparse affine feasibility problem, where sparse solutions to underdetermined systems are sought. Since basis pursuit admits a linear programming reformulation, standard LP solvers are directly applicable. We instead address the basis pursuit directly in its $\ell_1$-minimization form, without LP reformulation, via a scheme that uses alternating projections in its subproblems. These subproblems are designed to be inconsistent in the sense that they relate to two non-intersecting sets. Recently in [R. Behling, Y. Bello-Cruz and L.-R. Santos, SIAM J. Optim., 31 (2021), pp. 2863-2892], inconsistency coming from infeasibility has been shown to accelerate convergence of alternating projections. We deliberately enforce this inconsistency by constructing subproblems whose feasible sets are disjoint by design. We prove that the resulting $\ell_1$-radii converge linearly to the optimal value, and that when the solution is unique, all generated sequences converge linearly to it at a rate governed by a natural error bound between the feasible set and the optimal $\ell_1$-ball. The proposed method is numerically competitive against state-of-the-art open-source solvers on synthetic and real-world instances.

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arXiv2025

On circumcentered direct methods for monotone variational inequality problems

Circumcentered techniques have been shown to significantly accelerate projection-based methods for convex feasibility problems. Motivated by this success, we propose two direct methods with circumcenter acceleration for solving variational inequality problems involving two classes of operators: paramonotone and monotone. Both schemes rely on approximate projections onto separating halfspaces, thereby avoiding computationally expensive exact projections. We establish convergence results for both methods and conduct numerical experiments, demonstrating that the proposed algorithms outperform classical methods, such as the extragradient algorithm, by orders of magnitude in terms of computational time, particularly when the feasible set is a complex intersection of convex sets.

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arXiv2025

Parallel Polyhedral Projection Method for the Convex Feasibility Problem

In this paper, we introduce and study the Parallel Polyhedral Projection Method (3PM) and the Approximate Parallel Polyhedral Projection Method (A3PM) for finding a point in the intersection of finitely many closed convex sets. Each iteration has two phases: parallel projections onto the target sets (exact in 3PM, approximate in A3PM), followed by an exact or approximate projection onto a polyhedron defined by supporting half-spaces. These strategies appear novel, as existing methods largely focus on parallel schemes like Cimmino's method. Numerical experiments demonstrate that A3PM often outperforms both classical and recent projection-based methods when the number of sets is greater than two. Theoretically, we establish global convergence for both 3PM and A3PM without regularity assumptions. Under a Slater condition or error bound, we prove linear convergence, even with inexact projections. Additionally, we show that 3PM achieves superlinear convergence under suitable geometric assumptions.

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arXiv2025

Parallelizing the Circumcentered-Reflection Method

This paper introduces the Parallelized Circumcentered Reflection Method (P-CRM), a circumcentric approach that parallelizes the Circumcentered Reflection Method (CRM) for solving Convex Feasibility Problems in affine settings. Beyond feasibility, P-CRM solves the best approximation problem for any finite collection of affine subspaces; that is, it not only finds a feasible point but directly computes the projection of an initial point onto the intersection. Within a fully self-contained scheme, we also introduce the Framework for the Simultaneous Projection Method (F-SPM) which includes Cimmino's method as a special case. Theoretical results show that both P-CRM and F-SPM achieve linear convergence. Moreover, P-CRM converges at a rate that is at least as fast as, and potentially superior to, the best convergence rate of F-SPM. As a byproduct, this also yields a new and simplified convergence proof for Cimmino's method. Numerical experiments show that P-CRM is competitive compared to CRM and indicate that it offers a scalable and flexible alternative, particularly suited for large-scale problems and modern computing environments.

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arXiv2024

Fej\'er* monotonicity in optimization algorithms

Fej\'er monotonicity is a well-established property often observed in sequences generated by optimization algorithms. In this paper, we study an extension of this property, called Fej\'er* monotonicity, which was initially proposed in [SIAM J. Optim., 34(3), 2535-2556 (2024)]. We discuss and explore its behavior within Hilbert spaces as a tool for optimization algorithms. Additionally, we investigate weak and strong convergence properties of this novel concept. Through illustrative examples and insightful results, we contrast Fej\'er* with weaker notions of quasi-Fej\'er-type monotonicity.

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arXiv2023

A finitely convergent circumcenter method for the Convex Feasibility Problem

In this paper, we present a variant of the circumcenter method for the Convex Feasibility Problem (CFP), ensuring finite convergence under a Slater assumption. The method replaces exact projections onto the convex sets with projections onto separating halfspaces, perturbed by positive exogenous parameters that decrease to zero along the iterations. If the perturbation parameters decrease slowly enough, such as the terms of a diverging series, finite convergence is achieved. To the best of our knowledge, this is the first circumcenter method for CFP that guarantees finite convergence.

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arXiv2022

A successive centralized circumcenter reflection method for the convex feasibility problem

In this paper we present the successive centralization of the circumcenter reflection scheme with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Under additional smoothness assumptions, we prove the superlinear convergence. Numerical experiments confirm the efficiency of our method.

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arXiv2022

A circumcentered-reflection method for finding common fixed points of firmly nonexpansive operators

The circumcentered-reflection method (CRM) has been recently proposed as a methodology for accelerating several algorithms for solving the Convex Feasibility Problem (CFP), equivalent to finding a common fixed-point of the orthogonal projections onto a finite number of closed and convex sets. In this paper, we apply CRM to the more general Fixed Point Problem (denoted as FPP), consisting of finding a common fixed-point of operators belonging to a larger family of operators, namely firmly nonexpansive operators. We prove than in this setting, CRM is globally convergent to a common fixed-point (supposing at least one exists). We also establish linear convergence of the sequence generated by CRM applied to FPP, under a not too demanding error bound assumption, and provide an estimate of the asymptotic constant. We provide solid numerical evidence of the superiority of CRM when compared to the classical Parallel Projections Method (PPM). Additionally, we present certain results of convex combination of orthogonal projections, of some interest on its own.

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arXiv2021

Circumcentric directions of cones

Generalized circumcenters have been recently introduced and employed to speed up classical projection-type methods for solving feasibility problems. In this note, circumcenters are enforced in a new setting; they are proven to provide inward directions to sets given by convex inequalities. In particular, we show that circumcentric directions of finitely generated cones belong to the interior of their polars. We also derive a measure of interiorness of the circumcentric direction, which then provides a special cone of search directions, all being feasible to the convex region under consideration.

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arXiv2021

On the centralization of the circumcentered-reflection method

This paper is devoted to deriving the first circumcenter iteration scheme that does not employ a product space reformulation for finding a point in the intersection of two closed convex sets. We introduce a so-called centralized version of the circumcentered-reflection method (CRM). Developed with the aim of accelerating classical projection algorithms, CRM is successful for tracking a common point of a finite number of affine sets. In the case of general convex sets, CRM was shown to possibly diverge if Pierra's product space reformulation is not used. In this work, we prove that there exists an easily reachable region consisting of what we refer to as centralized points, where pure circumcenter steps possess properties yielding convergence. The resulting algorithm is called centralized CRM (cCRM). In addition to having global convergence, cCRM converges linearly under an error bound condition, and superlinearly if the two target sets are so that their intersection have nonempty interior and their boundaries are locally differentiable manifolds. We also run numerical experiments with successful results.

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arXiv2021

Circumcentering approximate reflections for solving the convex feasibility problem

The circumcentered-reflection method (CRM) has been applied for solving convex feasibility problems. CRM iterates by computing a circumcenter upon a composition of reflections with respect to convex sets. Since reflections are based on exact projections, their computation might be costly. In this regard, we introduce the circumcentered approximate-reflection method (CARM), whose reflections rely on outer-approximate projections. The appeal of CARM is that, in rather general situations, the approximate projections we employ are available under low computational cost. We derive convergence of CARM and linear convergence under an error bound condition. We also present successful theoretical and numerical comparisons of CARM to the original CRM, to the classical method of alternating projections (MAP) and to a correspondent outer-approximate version of MAP, referred to as MAAP. Along with our results and numerical experiments, we present a couple of illustrative examples.

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arXiv2020

Infeasibility and error bound imply finite convergence of alternating projections

This paper combines two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names such as Kaczmarz and von Neumann, MAP has the ability to track a pair of points realizing minimum distance between two given closed convex sets. Unfortunately, MAP may suffer from arbitrarily slow convergence, and sublinear rates are essentially only surpassed in the presence of some Lipschitzian error bound, which is our first ingredient. The second one is a seemingly unfavorable and unexpected condition, namely, infeasibility. For two non-intersecting closed convex sets satisfying an error bound, we establish finite convergence of MAP. In particular, MAP converges in finitely many steps when applied to a polyhedron and a hyperplane in the case in which they have empty intersection. Moreover, the farther the target sets lie from each other, the fewer are the iterations needed by MAP for finding a best approximation pair. Insightful examples and further theoretical and algorithmic discussions accompany our results, including the investigation of finite termination of other projection methods.

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arXiv2020

The circumcentered-reflection method achieves better rates than alternating projections

We study the convergence rate of the Circumcentered-Reflection Method (CRM) for solving the convex feasibility problem and compare it with the Method of Alternating Projections (MAP). Under an error bound assumption, we prove that both methods converge linearly, with asymptotic constants depending on a parameter of the error bound, and that the one derived for CRM is strictly better than the one for MAP. Next, we analyze two classes of fairly generic examples. In the first one, the angle between the convex sets approaches zero near the intersection, so that the MAP sequence converges sublinearly, but CRM still enjoys linear convergence. In the second class of examples, the angle between the sets does not vanish and MAP exhibits its standard behavior, i.e., it converges linearly, yet, perhaps surprisingly, CRM attains superlinear convergence.

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arXiv2020

On the Circumcentered-Reflection Method for the Convex Feasibility Problem

The ancient concept of circumcenter has recently given birth to the Circumcentered-Reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.

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