Given a real function on a Euclidean space, we consider its "robust regularization": the value of this new function at any given point is the maximum value of the original function in a fixed neighbourhood of the point in question. This construction allows us to impose constraints in an optimization problem *robustly*, safeguarding a constraint against unpredictable perturbations in variables or data. After outlining some examples, we consider in particular a function that is locally Lipschitz on the complement of a suitably well-behaved (for example, semi- algebraic or prox-regular) small set, and satisfies a growth condition near the set. We show that, around any given point, the robust regularization is eventually locally Lipschitz once the size of the neighbourhood is sufficiently small. Our result applies in particular to the pseudospectral abscissa of a square matrix, a useful function in robust stability theory.
Technical report, Simon Fraser University, September 2002