arXiv subjects

Tomáš Roubal

Publications and source records attributed to Tomáš Roubal.

5 recordsLinked to original sources

arXiv2026

Anisotropic Higher-Order Semiregularity of Degenerate Generalized Equations

We give a self-contained triangular formulation of anisotropic higher-order covering and inverse estimates for smooth mappings and generalized equations. For a $C^p$ mapping into a finite-dimensional target, an indexed target decomposition is fixed, the corresponding triangular factor condition is imposed, and a direction satisfying the associated triangular $p$-kernel condition is chosen. Surjectivity of the resulting triangular $p$-factor operator yields a fixed-scale inclusion in which each nonzero block $Y_i$ is covered at order $t^i$, and hence an inverse estimate with exponent $1/i$ on that block. Zero blocks are allowed, and $p$ denotes the highest active grade. The smooth result is a derivative-level graded formulation: it verifies the model directly from the $C^p$ derivatives and records the blockwise fixed-scale inclusion needed for the set-valued analysis. For generalized equations, an auxiliary exact-model theorem isolates the range-transfer mechanism with acyclic correction fibres. The main sufficient criterion uses a closed convex-process inner approximation whose sum with the smooth triangular factor operator is surjective, allowing the set-valued term to supply missing directions.

math.OC↗

arXiv2025

A note on the uniformity of strong subregularity around the reference point

This paper investigates strong metric subregularity around a reference point as introduced by H. Gfrerer and J. V. Outrata. In the setting of Banach spaces, we analyse its stability under Lipschitz continuous perturbations and establish its uniformity over compact sets. Our results ensure that the property is preserved under small Lipschitz perturbations, which is crucial for maintaining robustness in variational analysis. Furthermore, we apply the developed theory to parametric inclusion problems. The analysis demonstrates that the uniformity of strong metric subregularity provides a theoretical foundation for addressing stability issues in parametrized optimization and control applications.

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arXiv2025

On Josephy-Halley method for generalized equations

We extend the classical third-order Halley iteration to the setting of generalized equations of the form \[ 0 \in f(x) + F(x), \] where $f\colon X\longrightarrow Y$ is twice continuously Fréchet-differentiable on Banach spaces and $F\colon X\tto Y$ is a set-valued mapping with closed graph. Building on predictor-corrector framework, our scheme first solves a partially linearized inclusion to produce a predictor $u_{k+1}$, then incorporates second-order information in a Halley-type corrector step to obtain $x_{k+1}$. Under metric regularity of the linearization at a reference solution and Hölder continuity of $f''$, we prove that the iterates converge locally with order $2+p$ (cubically when $p=1$). Moreover, by constructing a suitable scalar majorant function we derive semilocal Kantorovich-type conditions guaranteeing well-definedness and R-cubic convergence from an explicit neighbourhood of the initial guess. Numerical experiments-including one- and two-dimensional test problems confirm the theoretical convergence rates and illustrate the efficiency of the Josephy-Halley method compared to its Josephy-Newton counterpart.

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arXiv2024

On semismooth$^*$ path-following method and uniformity of strong metric subregularity at/around the reference point

This paper investigates a path-following method inspired by the semismooth$^*$ approach for solving algebraic inclusions, with a primary emphasis on the role of uniform subregularity. Uniform subregularity is crucial for ensuring the robustness and stability of path-following methods, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping $ x: \mathbb{R} \longrightarrow \mathbb{R}^n $ that satisfies $ 0 \in F(t, x(t)) $ for each $ t \in [0, T] $, where $ F $ is a set-valued mapping from $ \mathbb{R} \times \mathbb{R}^n $ to $ \mathbb{R}^n $. The paper discusses two approaches: the first considers mappings with uniform semismooth$^*$ properties along continuous paths, leading to a consistent grid error throughout the interval, while the second examines mappings exhibiting pointwise semismooth$^*$ properties at individual points along the path. The uniform strong subregularity framework is integrated into these approaches to strengthen the stability of solution trajectories and improve algorithmic convergence.

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arXiv2022

A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

Ioffe's criterion and various reformulations of it have become a~standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland's variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semi-continuous envelope even in the case of an incomplete target space. The gist is to use the "correct" function to which Ekeland's variational principle is applied; namely, the distance function to the graph of the set-valued mapping under consideration. This approach originates in the notion of graphical regularity introduced by L. Thibault, which is equivalent to the property of metric regularity. Our criteria cover also both metric subregularity and metric semiregularity, which are weaker properties obtained by fixing one of the points in the definition of metric regularity.

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