Three Regimes in the Sharp Quadratic Stability Problem for Dittert's Functional
Year: 2026
Type: preprint
Abstract: We study the sharp global quadratic stability constant for Dittert's functional on the simplex of nonnegative (n\times n) matrices with total entry sum (n). Writing (U_n=J_n/n), we consider [\kappa_n^*=\inf_{A\ne U_n}\frac{\Phi(U_n)-\Phi(A)}{|A-U_n|_F^2}.] Exact witnesses and reproducible numerical searches reveal three distinct low-dimensional regimes. For (n=3,4,5), the best observed finite-distance witnesses are cycle matrices of the form ((P+Q)/2). For (n=6,7), they are replaced by highly symmetric block minimizers. From (n=8) onward, the best observed branch is a nearly uniform matrix with a single zero entry. Two general results are proved for every (n\ge3). First, the local Hessian scale (L_n) is never sharp: (\kappa_n^<L_n). Second, an explicit one-dimensional one-zero family yields the strict upper bound[\kappa_n^\le \widetilde Z_n<Z_n,]where (Z_n) is the doubly stochastic one-zero value. We derive the asymptotic expansion (Z_n/\gamma_n\to1/2) from below and quantify the exponentially small departure of the optimized branch from double stochasticity. The computational archive provides exact rational upper bounds for (n=6,\ldots,9), KKT and Hessian diagnostics, and comparisons of seven prescribed-zero boundary patterns through (n=12). Global optimality beyond the rigorously established (n=3) case remains open.
Author: Kaczorowski, Jarosław
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225792576
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