A note of Lipschitz constants for solutions of linear inequalities and equations
This paper is concerned with Lipschitz constants for systems of the form \[ Ax\leq b,\quad Cx = d\tag{1} \] in which \(A\) and \(C\) are matrices of order \(m\times n\) and \(k \times n\), respectively, and \((b,d) \in R^m \times R^k\) is treated as a parameter vector. For a given parameter vector \(M(b,d)\) denotes the set of all \(x\) that satisfy the system (1). In 1952, \textit{A. J. Hoffman} [J. Res. Nat. Bur. Standards 49, 263-265 (1952; MR 14-455)] showed that the solution set mapping \((b,d) \mapsto M(b,d)\) of (1) is Lipschitzian on its effective domain, \(\text{dom} M: = \{(b,d) \mid M(b,d)\neq \emptyset\}\). This means that there exists a constant \(\rho= \rho_{\alpha \beta} (A,C)>0\) such that \[ h_\beta \bigl(M(b,d), M(b',d')\bigr) \leq\rho\left |\begin{matrix} b & - & b' \\ d & - & d' \end{matrix} \right|_\alpha \quad \forall (b,d),\;(b',d')\in \text{dom} M \tag{2} \] where \(|\cdot|_\alpha\) and \(|\cdot |_\beta\) are any two norms on \(R^{m+k}\) and \(R^n\), respectively, and \(h_\beta\) is the Hausdorff metric induced by \(|\cdot |_\beta\). This paper belongs to the literature dealing with estimates of the Lipschitz constant \(\rho\). Its principal contribution is a theorem to the effect that a sharp Lipschitz constant due to \textit{W. Li} [SIAM J. Control Optimization 32, No. 1, 140-153 (1994; Zbl 0792.90080)] is the same as the one used by \textit{S. M. Robinson} [Linear Algebra Appl. 6, 69-81 (1973; Zbl 0283.90028)] and that it agrees with one considered by the authors [Wiss. Z. PH Halle-Köthen XXVI(8), 13-17 (1988; MR 91f:65123)].
scientific article; zbMATH DE number 931954
scientific article