arXiv subjects

Tran Thanh Tung

Publications and source records attributed to Tran Thanh Tung.

6 recordsLinked to original sources

arXiv2026

The Güntürk-Thao theorem revisited: polyhedral cones and limiting examples

In 2023, Güntürk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite innately regular collection of 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 show that the Güntürk-Thao result is sharp in the sense that it fails for two closed subspaces that is not innately regular. We also 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 nonpolyhedral cone in $\mathbb{R}^3$.

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arXiv2026

Cesàro means of firmly nonexpansive iterates need not converge strongly

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). Strong convergence of Cesàro means of iterates is also known to fail for general nonlinear nonexpansive mappings (Krengel and Lin, 1987). In this paper, we show that this failure persists in the much smaller class of firmly nonexpansive mappings. Using suitable meshes, we construct a new explicit family of counterexamples in infinite-dimensional Hilbert spaces with the origin as the unique fixed point. In the harmonic case, the Cesàro means of the iterates remain bounded away from the origin. Another variant yields Cesàro means that converge strongly to the origin. A third variant presents Cesàro means whose norms oscillate in the sense that their liminf is zero while their limsup is positive. Thus the strong convergence conclusion in von Neumann's linear mean ergodic theorem does not extend to Baillon's nonlinear mean ergodic theorem, even for firmly nonexpansive mappings.

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arXiv2026

Extending Meshulam's result on the boundedness of orbits of relaxed projections onto affine subspaces from finite to infinite-dimensional Hilbert spaces

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.

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arXiv2026

Finite Termination of a Generalized Perceptron Algorithm

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.

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arXiv2026

On a result by Meshulam

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.

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arXiv2025

On the boundedness of the sequence generated by minibatch stochastic gradient descent

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.

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