Variational analysis of an extended eigenvalue problem
This paper considers for a symmetric matrix \(B\) the problem: maximize \(\mu_ a : = x^ tBx - 2a^ tx\) subject to \(x^ tx = 1\). For the solution it is necessary to computers the Lagrange multiplier \(\lambda (a)\) as extended eigenvalue \(\lambda (a) = \max \{\lambda \mid (B - \lambda I)x = a\); \(x^ tx = 1\}\) of the pair \(\{B,a\}\). Nonsmooth analysis is required to discuss the function \(\lambda(a)\) in order to derive a multiplier result on the sensitivity near 0. The authors prove continuous differentiability on a large set of points and that \(\lambda (\cdot)\) is Lipschitz near 0 and regular at 0. They characterize the generalized gradient and directional derivatives at 0.
scientific article; zbMATH DE number 769036