On the Sensitivity Analysis of Hoffman Constants for Systems of Linear Inequalities

Relying on a general variational method developed by the authors and Lucchetti [Nonlinear Anal., to appear] (the origin of which goes back to Ioffe [Trans. Amer. Math. Soc., 251 (1979), pp. 61--69]), we give a formula for the best Hoffman constant $\sigma=\inf_{x\notin P_{A,b}}\frac{\|(Ax-b)^+\|_\infty}{d(x,P_{A,b})}$, where $P_{A,b}=\{x:Ax\le b\}$ is a nonempty polyhedron in ${\mathbb R}^n$. We also sharpen some results of Luo and Tseng [SIAM J. Matrix Anal. Appl., 15 (1994), pp. 636--659] by characterizing the continuity set of some Hoffman constants and by pointing out their locally Lipschitzian character. We apply these results to the study of the behavior of the solution set of a linear program $\inf_{Ax\le b}u^{\scriptscriptstyle T}x$ with respect to (A,b,u).