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Upper Hölderian with Explicit Exponent of Solution Mapping with Applications to Ball Constrained Least Squares Problems

This paper extends the Robinson implicit function theorem to the upper Hölderian case with explicit exponent dependence and applies this result to demonstrate that the solution mapping for ball-constrained linear least squares problems under linear perturbation is locally upper Hölder continuous with an exponent of $1/3$ .

Original authors: Yu Wang, Shenglong Hu

Published 2026-08-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu Wang, Shenglong Hu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics and engineering, there is a constant struggle to understand how systems react when the world around them shifts slightly. Imagine a machine designed to find the best possible setting for a complex task, like fitting a curve to a cloud of data points. If you nudge the data just a little bit, you expect the machine's answer to move only a little bit as well. This expectation is the foundation of stability. For decades, mathematicians have relied on a powerful tool called the implicit function theorem to predict this behavior. This theorem acts like a guarantee: if the rules of the system are smooth and well-behaved, a small change in the input will result in a small, predictable change in the output. However, the real world is often messy. Many important problems involve constraints that create sharp corners or sudden breaks in the rules, making the system "rough" rather than smooth. In these rougher landscapes, the old guarantees often fail, leaving scientists without a clear map of how the solution will move when the parameters change.

This uncertainty is particularly acute in a specific type of problem known as the ball-constrained least squares problem. This is a method used to find the best fit for data while forcing the answer to stay within a specific boundary, like a ball of a fixed size. This boundary is crucial in many fields, from statistics to engineering, because it prevents the solution from becoming wildly unstable when the data is noisy or the underlying system is ill-conditioned. For a long time, researchers could only describe the behavior of these solutions when the system was perfectly smooth. When the system hit the edge of the boundary or became irregular, the standard tools broke down, and the behavior of the solution became a mystery. It was known that the solution would still exist, but no one could say exactly how fast or how far it would move in response to a change.

In a new study, a team of mathematicians has bridged this gap by developing a more flexible version of the classic theorem. They extended the theory to handle these "rough" systems, proving that even when the rules are not smooth, the solution still moves in a predictable way, just not in a perfectly straight line. Instead of moving at a constant speed relative to the change, the solution moves at a rate that follows a specific power law. The researchers were able to calculate the exact exponent of this power law, which acts as a precise measure of the system's sensitivity. They found that for the ball-constrained least squares problem, the solution is stable, but its movement is governed by a specific mathematical rhythm. In the most difficult cases, where the system is right at the edge of its stability, the solution moves with an exponent of one-third. This means that if you change the input by a certain amount, the solution will change by the cube root of that amount. This is a slower, more cautious reaction than in smooth systems, but it is a predictable one.

The team did not stop at theory; they applied their new framework to the specific problem of fitting data within a spherical boundary. They demonstrated that their new theorem holds true for these complex scenarios, providing a complete characterization of how the solution behaves under linear perturbations. Their work shows that the solution mapping is what they call "upper Hölder continuous," a technical way of saying the solution stays within a predictable envelope of movement. Crucially, they proved that the exponent governing this movement is exactly one-third in the most critical situations. This finding is significant because it fills a void in the literature where previous methods could only offer vague descriptions or failed entirely. By establishing this explicit exponent, the researchers have provided a concrete tool for analyzing the stability of these problems.

This discovery has immediate implications for how we understand and solve optimization problems. In many practical applications, such as sequential optimization procedures where one problem feeds into the next, knowing the exact rate of stability is essential for designing efficient algorithms. If a computer program knows that a solution will move at a rate of one-third in response to a change, it can adjust its steps accordingly to avoid overshooting or getting stuck. The researchers also showed that their method applies to a broader class of problems, including separable nonlinear least squares, which are common in statistical modeling. By proving that the solution mapping for these problems also follows a predictable Hölder pattern, they have opened the door to more robust convergence analysis for numerical methods.

The study stands as a rigorous extension of a foundational mathematical principle. It does not claim to solve every problem in the field, but it provides a necessary and precise description of behavior in a regime that was previously poorly understood. The authors have shown that even in the presence of sharp constraints and irregularities, the mathematical landscape is not chaotic. There is an order to the movement of solutions, and that order can be quantified with exact numbers. This work transforms a vague intuition about stability into a concrete, calculable fact, offering a new lens through which to view the behavior of constrained systems in science and engineering.

Technical Summary: Upper Hölderian with Explicit Exponent of Solution Mapping with Applications to Ball Constrained Least Squares Problems

Problem Statement
The paper addresses the sensitivity and stability analysis of solution mappings for parameterized generalized equations of the form $S(p) := \{x \in \mathbb{R}^n : 0 \in F(p, x) + T(x)\}$ , where $p$ is a parameter, $F$ is a continuous mapping, and $T$ is a set-valued mapping. While classical results (e.g., the Implicit Function Theorem) and Robinson's theorem provide conditions for the local upper Lipschitz continuity of $S(p)$ , these often fail when the solution mapping is set-valued or lacks single-valued localization, a common scenario in constrained optimization. Specifically, the paper focuses on the Ball-Constrained Least Squares (Ball-LS) problem:
$\min_{x} \frac{1}{2} \|Ax - b\|^2 \quad \text{s.t.} \quad \|x\| \le R,$
where $A \in \mathbb{R}^{m \times n}$ , $b \in \mathbb{R}^m$ , and $R \ge 0$ . The authors note that existing literature often assumes non-degenerate matrices or local constant rank, leaving the quantitative behavior of the solution set under general perturbations (including degenerate cases) largely undetermined. The primary goal is to establish upper Hölderian continuity with explicit exponents for the solution mapping of this problem.

Methodology
The methodology proceeds in two main stages:

Extension of Robinson's Theorem: The authors extend Robinson's implicit function theorem for generalized equations from the upper Lipschitzian case to the upper Hölderian case.
- They define a set-valued mapping $F$ to be upper Hölder continuous with exponent $\alpha$ and modulus $\lambda$ if $F(p) \subseteq F(p_0) + \lambda \|p - p_0\|^\alpha B_Y$ .
- Theorem 3.1 establishes conditions under which the solution mapping of a generalized equation is locally upper Hölder continuous. Key assumptions include the outer semi-continuity of $T$ , differentiability of $F$ , and specific properties of the "partial linearization" of the mapping.
- A critical technical component is the handling of the case where $\alpha < 1$ . The authors introduce a specific assumption (A4) regarding the Hölder continuity of the derivative of $F$ to offset the local growth rate when composing mappings, a necessity because standard contraction arguments used in Lipschitz cases do not directly apply to Hölder cases with $\alpha < 1$ .
- The proof utilizes fixed-point arguments (Kakutani's theorem), projection operators, and estimates involving the directed Hausdorff metric.
Application to Ball-LS: The generalized theorem is applied to the Ball-LS problem by formulating it as a parametric generalized equation involving the normal cone of the ball constraint.
- The authors analyze the solution mapping $S(A, b, R)$ by treating $A, b, R$ as parameters.
- They derive explicit formulas for the solution set and the optimal Lagrange multiplier under linear perturbations.
- The analysis distinguishes three regimes based on the value of a specific quantity $\tilde{\Delta}_{A,b}(0)$ $\tilde{Δ}_{A, b} (0)$ (related to the unconstrained solution's distance from the ball boundary):
  - Strictly interior solution ( $\tilde{\Delta} > 0$ ): The solution is unique and Lipschitz continuous ( $\alpha = 1$ ).
  - Strictly exterior solution ( $\tilde{\Delta} < 0$ ): The solution is Hölder continuous with exponent $\alpha = 1/2$ .
  - Boundary solution ( $\tilde{\Delta} = 0$ ): The solution is Hölder continuous with exponent $\alpha = 1/3$ .

Key Contributions

Theoretical Extension: The paper establishes Theorem 3.1, an extension of Robinson's implicit function theorem to the upper Hölderian setting with explicit exponent characterization. This fills a gap in the literature regarding the explicit determination of Hölder exponents for generalized equations.
Explicit Exponents for Ball-LS: The authors provide a complete characterization of the upper Hölderian properties of the Ball-LS solution mapping. They prove that the solution mapping is locally upper Hölder continuous with an explicit exponent of $1/3$ in the critical case where the unconstrained solution lies exactly on the boundary of the ball and the constraint is active.
Tightness of Results: Through Example 4.3, the authors demonstrate that the exponent $1/3$ is tight. They construct a specific case where the solution mapping behaves asymptotically as $\|z\|^{1/3}$ under perturbation, confirming that a better exponent cannot be guaranteed in the general case.
Extension to Separable NLS: The framework is applied to separable nonlinear least squares (NLS) problems with ball constraints, showing that if the matrix-valued mapping $G(y)$ is upper Hölder continuous with exponent $\beta$ , the solution mapping inherits an exponent of $\alpha\beta$ .

Results

Theorem 3.1: Proves the existence of a neighborhood where the solution mapping of a generalized equation is upper Hölder continuous, provided the linearized mapping satisfies specific inverse Hölder properties and the derivative of the function satisfies a complementary Hölder condition.
Proposition 4.1: Characterizes the solution map of Ball-LS under linear perturbation, identifying the exponent $\alpha$ as $1$, $1/2$ , or $1/3$ depending on the geometric relationship between the unconstrained solution and the constraint boundary.
Theorem 4.5: Establishes the upper Hölderian continuity of the solution mapping for the general parametric Ball-LS problem $P(A, b, R)$ with the explicit exponent $1/3$ in the critical degenerate case.
Proposition 4.6: Extends these results to separable NLS problems, linking the Hölder exponent of the solution to the product of the exponent of the parameterized matrix function and the exponent derived from the Ball-LS analysis.

Significance and Claims
The paper claims that its results are of independent interest because they provide a complete quantitative characterization of the solution set for Ball-LS, a fundamental problem in numerical linear algebra and optimization, particularly in scenarios where standard regularity conditions (like non-degeneracy or constant rank) fail.

The authors highlight that while previous works (e.g., by Jiang and Li) established Hölder stability for ball-constrained quadratic programming with an exponent of $1/4$ , this paper achieves a tighter exponent of $1/3$ for the specific case of Ball-LS through refined analysis. This improvement is significant for the convergence rate analysis of numerical algorithms solving these problems. The extension of Robinson's theorem to the Hölderian case is presented as a tool applicable to other complex systems, such as polynomial systems, where only Hölder continuity can be expected. The paper concludes by noting that these findings lay the groundwork for future studies on general quadratic constrained quadratic programs (QCQP) and optimization on manifolds.

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