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.

In this paper, we develop a general analysis for the fixed points of the operators defining the graph splitting methods from [SIAM J. Optim., 34 (2024), pp. 1569-1594] by Bredies, Chenchene and Naldi. We particularize it to the case where the maximally monotone operators are normal cones of closed linear subspaces and provide an explicit formula for the limit points of the graph splitting schemes. We exemplify these results on some particular algorithms, unifying in this way some results previously derived as well as obtaining new ones.

Motivated by Ridgway's proof of the perceptron algorithm, we study a simple subgradient method for convex inequality systems in Hilbert space. Assuming strict feasibility and bounded subgradients, we establish finite termination for several natural step sizes. We also examine what can go wrong without strict feasibility: finite convergence may fail even for one function, and with several functions the method may converge to a point outside the feasible set. The linear setting recovers the classical perceptron algorithm.

In this paper, we unify and improve existing results on characterizing strict and almost stricty convex functions via subdifferential mapping, Moreau envelope, and proximal mappings. In particular, it is shown that if a convex function is subdifferentiable on its domain, then it is strictly convex if and only if its subdifferential is strictly monotone, equivalently, almost strictly monotone. Rockafellar-Wets' characterizations of almost strictly convex functions via almost differentiability of Fenchel conjugates and strict monotonicity of subdifferentials are extended from a finite-dimensional space to a Hilbert space. We also establish similar results for paramonotone operators.

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.

Firmly nonexpansive operators arise naturally as resolvents of monotone operators and as generalizations of projections and proximal mappings in convex optimization and fixed point theory. While their iterates are known to converge weakly to a fixed point, strong convergence is not guaranteed (Genel and Lindenstrauss, 1975). In this paper, we provide a new explicit family of counterexamples in infinite-dimensional Hilbert spaces. In the harmonic case the Cesàro means of the iterates remain bounded away from the unique fixed point. A block-construction variant yields Cesàro means whose norms oscillate in the sense that their liminf is zero while their limsup is positive. These results show that von Neumann's classical mean ergodic theorem for linear operators does not extend to Baillon's nonlinear mean ergodic theorem even in the firmly nonexpansive setting, and they illustrate inherent limitations of averaging techniques in infinite-dimensional optimization.

The Krasnoselskii-Mann iteration is an important algorithm in optimization and variational analysis for finding fixed points of nonexpansive mappings. In the general case, it produces a sequence converging weakly to a fixed point provided the parameter sequence satisfies a divergent-series condition. In this paper, we show that strong convergence holds provided the underlying nonexpansive mapping is linear. This improves on a celebrated result by Baillon, Bruck, and Reich from 1978, where the parameter sequence was assumed to be constant as well as on recent work where the parameters were bounded away from 0 and 1.

In 1996, Meshulam proved that any sequence generated in Euclidean space by randomly projecting onto affine subspaces drawn from a finite collection stays bounded even if the intersection of the subspaces is empty. His proof, which works even for relaxed projections, relies on an ingenious induction on the dimension of the Euclidean space. In this paper, we extend Meshulam's result to the general Hilbert space setting by an induction proof of the number of affine subspaces in the given collection. We require that the corresponding parallel linear subspaces are innately regular – this assumption always holds in Euclidean space. We also discuss the sharpness of our result and make a connection to randomized block Kaczmarz methods.

In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ”relocates” fixed-points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed-point, as is common in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas–Rachford algorithm for finding a zero of the sum of N≥ 2 maximally monotone operators, which does not require the resolvent parameter to be constant across iterations.

Opial’s Lemma is a fundamental result in the convergence analysis of sequences generated by optimization algorithms in real Hilbert spaces. We introduce the concept of Opial sequences–sequences for which the limit of the distance to each point in a given set exists. We systematically derive properties of Opial sequences, contrasting them with the well-studied Fejér monotone sequences, and establish conditions for weak and strong convergence. Key results include characterizations of weak convergence via weak cluster points (reaffirming Opial’s Lemma), strong convergence via strong cluster points, and the behavior of projections onto Opial sets in terms of asymptotic centers. Special cases and examples are provided to highlight the subtle differences in convergence behaviour and projection properties compared to the Fejér monotone case.

In 2023, Güntürk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^γ<+\infty$ for all $γ>0$. We extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in $\mathbb{R}^2$, and for a plane and a non-polyhedral cone in $\mathbb{R}^3$.

In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic polynomial defining its derivative. We also focus on the case where the point we project lies on the vertical line defined by the parabola. Lastly, an explicit formula for the projection onto a higher dimensional parabola is derived.

In 1996, Meshulam proved that every sequence generated by applying projections onto affine subspaces, drawn from a finite collection in Euclidean space, must be bounded. In this paper, we extend his result not only from affine subspaces to convex polyhedral subsets, but also from Euclidean to general Hilbert space. Various examples are provided to illustrate the sharpness of the results.

The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real roots. In this note, we present the roots of the cubic in a form that makes them convenient to use and we also focus on information on the location of the real roots. Armed with this, we provide several applications in optimization where we compute Fenchel conjugates, proximal mappings and projections.

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.

Many algorithms in convex optimization and variational analysis can be analyzed using Fejér monotone sequences. In 2024, Behling, Bello-Cruz, Iusem, Alves Ribeiro, and Santos introduced a new, more general, notion: Fejér* monotonicity. They obtained basic results and discussed applications in optimization. In this work, we complement Behling et al.'s work by presenting a thorough study of Fejér* monotonicity. We reveal striking similarities and differences between these notions, including descriptions of the maximal Fejér* set. Moreover, we also touch upon Opial sequences and quasi-Fejér monotonicity. Throughout this paper, we provide numerous limiting examples and counterexamples.

A cornerstone of convex analysis, established by Rockafellar in 1966, asserts that a set has a potential if and only if it is cyclically monotone. This characterization was generalized to hold for any finite-valued cost function c and lies at the core structure of optimal transport plans. However, this equivalence fails to hold for costs that attain infinite values. In this paper, we explore potentiability for an infinite-valued cost c under the assumption of c-path boundedness, a condition that was first introduced by Artstein-Avidan, Sadovsky and Wyczesany. This condition is necessary for potentiability and it is more restrictive than c-cyclic monotonicity; however, this condition alone does not imply potentiability. We provide general settings and other conditions under which c-path boundedness is sufficient for potentiability, and therefore equivalent. We provide a general theorem for potentiability, requiring no topological assumptions on the spaces or the cost. We then provide sufficiency in separable metric spaces and costs that are continuous in their domain. Finally, we introduce the notion of a c-path bounded extension and use it to prove the existence of potentials for a special class of costs on ℝ^2. We illustrate our discussion and results with several examples.

Stochastic Gradient Descent (SGD) with Polyak's stepsize has recently gained renewed attention in stochastic optimization. Recently, Orvieto, Lacoste-Julien, and Loizou introduced a decreasing variant of Polyak's stepsize, where convergence relies on a boundedness assumption of the iterates. They established that this assumption holds under strong convexity. In this paper, we extend their result by proving that boundedness also holds for a broader class of objective functions, including coercive functions. We also present a case in which boundedness may or may not hold.

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 classic and highly successful splitting method to find minimizers of the sum of two (not necessarily smooth) convex functions. Whereas the underlying theory has matured, one case remains a mystery: the behavior 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. Funding: The research of H. H. Bauschke and W. M. Moursi was partially supported by Discovery Grants of the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2018-03703 and RGPIN-2019-04803], respectively.