On an optimal problem of bilinear forms
Naihuan Jing, Yibo Liu, Jiacheng Sun, Chengrui Zhao, Haoran Zhu
TL;DR
The paper tackles a simplified, Grothendieck-constant–related bilinear optimization by formulating it as a generalized eigenvalue problem and solving it via generalized normal equations derived from Lagrange multipliers. It establishes a duality between solutions of $(A^T \Lambda^{-1} A - M)\boldsymbol{y}=0$ and $(A M^{-1} A^T - \Lambda)\boldsymbol{x}=0$, with the optimal value given by $\mathrm{tr}(\Lambda)=\mathrm{tr}(M)$ and $\Lambda, M$ representing squared generalized singular values; the framework recovers standard SVD in the diagonal case $\Lambda=M=\lambda I$. In the two-dimensional setting, the authors obtain explicit maxima and bounds, including $\max B(\boldsymbol{u},\boldsymbol{v}) = n^2 d \sigma_1(A)$ in the higher-dimensional construction and closed-form results such as $|a_{11}\pm a_{22}|$ for real $2\times 2$ symmetric $A$, along with conditions relating the generalized eigenvalues when off-diagonal entries are nonzero. The work provides an elementary, structurally revealing approach to a component of the Grothendieck-constant problem and clarifies the role of generalized singular values in optimal bilinear forms.
Abstract
We study an optimization problem originated from the Grothendieck constant. A generalized normal equation is proposed and analyzed. We establish a correspondence between solutions of the general normal equation and its dual equation. Explicit solutions are described for the two-dimensional case.