Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems
Mathematics of Operations Research
A smoothing Newton method for a type of inverse semi-definite quadratic programming problem
Journal of Computational and Applied Mathematics
Correlation stress testing for value-at-risk: an unconstrained convex optimization approach
Computational Optimization and Applications
A Continuation Method for Nonlinear Complementarity Problems over Symmetric Cones
SIAM Journal on Optimization
Nonsingularity Conditions for the Fischer-Burmeister System of Nonlinear SDPs
SIAM Journal on Optimization
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For any function f from $\mathbb R$ to $\mathbb R$, one can define a corresponding function on the space of n × n (block-diagonal) real symmetric matrices by applying f to the eigenvalues of the spectral decomposition. We show that this matrix-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as ($\rho$-order) semismoothness. Our analysis uses results from nonsmooth analysis as well as perturbation theory for the spectral decomposition of symmetric matrices. We also apply our results to the semidefinite complementarity problem, addressing some basic issues in the analysis of smoothing/semismooth Newton methods for solving this problem.