Stability of Error Bounds for Semi-infinite Convex Constraint Systems
SIAM Journal on Optimization
Stability of Error Bounds for Convex Constraint Systems in Banach Spaces
SIAM Journal on Optimization
Ekeland's Variational Principle for Set-Valued Functions
SIAM Journal on Optimization
The positiveness of lower limits of the Hoffman constant in parametric polyhedral programs
Journal of Global Optimization
SIAM Journal on Optimization
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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).