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bibkey: foucartliao2024sprocedure authors: Simon Foucart; Chunyang Liao year: 2024 title: “S-Procedure Relaxation: a Case of Exactness Involving Chebyshev Centers” doi: 10.1007/978-3-031-66497-7_1 url: https://arxiv.org/abs/2310.09677v1 claim: “Section 5, Question 2 asks whether the Chebyshev center is always obtained via quadratic regularization in the real optimal-recovery setting.” strata_touched:

• D5/S3/Analytic/OptimalRecovery/ChebyshevCenterRegularization license: citation-only triage: anchor

Locator

DOI: 10.1007/978-3-031-66497-7_1. Author text: https://arxiv.org/abs/2310.09677v1. Published in Explorations in the Mathematics of Data Science (Springer, 2024). Section 5, “Some Open Questions”, Question 2:

Is the Chebyshev center always obtained via regularization?

The authors explain:

From [10], we know that the answer is affirmative under the specific conditions mentioned above, since the Chebyshev center equals the regularizer $f_{{\tau }_{♯}}$ with parameter ${\tau }_{♯}\in (0,1)$ satisfying the implicit equation (9). Without the specific conditions, we also know from [8] that the answer is affirmative if one considered the complex setting rather than the real setting.

Optimal recovery and regularization

Section 5 takes $Q={\mathrm{Id}}_H$ and the model and error sets (25)

$$\mathcal{K}=\{f\in H:\Vert Pf{\Vert }_H\le \epsilon \},\mathcal{E}=\{e\in {\mathbb{R}}^m:\Vert e{\Vert }_2\le \epsilon \}.$$

The feasible set at data $y$ consists of the $f$ satisfying both $\Vert Pf\Vert \le \epsilon$ and $\Vert y-\Lambda f\Vert \le \epsilon$. The specific conditions are that $P$ is an orthogonal projector and $\Lambda {\Lambda }^{\ast }={\mathrm{Id}}_m$. Equation (8) defines

$$f_{\tau }={\mathrm{argmin}}_f((1-\tau )\Vert Pf{\Vert }^2+\tau \Vert y-\Lambda f{\Vert }^2).$$

Repository-derived counterexample. In $H={\mathbb{R}}^2$, take $P=\mathrm{diag}(1,0)$, $\Lambda =\left(\begin{array}{cc}1/2& -1/2\\ 0& 1\end{array}\right)$, $y=(1,0)$ and both bounds equal to one. The Chebyshev center is $c_{\ast }=(3/5,-1/5)$ with radius $\sqrt{4/5}$. The point $v=(3/5,-7/25)$ has the same squared model norm $9/25$ and a strictly smaller squared data residual, $49/125<50/125$. Consequently $c_{\ast }$ minimizes no such objective with $\tau \in (0,1)$. The theorem D5.S3.Analytic.OptimalRecovery.ChebyshevCenterRegularization.result proves the negation of the universal real assertion. This example retains the projector condition and violates the row-isometry condition.

Related literature and scope

Foucart–Liao, arXiv:2401.11112, concerns the global setting only; Foucart–Hengartner, arXiv:2406.14418, and Lin–Du, arXiv:2506.00704, do not answer this local-center question. These references do not establish an exhaustive priority claim.

ASSUMED-UNVERIFIED: classical work on Chebyshev centers of intersections of two ellipsoids, including Beck–Eldar (2007) and follow-ups, has not been checked for an earlier real instance whose center lies on no regularization path.

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