Primal-dual Solution Perturbations in Convex Optimization

Solutions to optimization problems of convex type are typically characterized by saddle point conditions in which the primal vector is paired with a dual ‘multiplier’ vector. This paper investigates the behavior of such a primal-dual pair with respect to perturbations in parameters on which the problem depends. A necessary and sufficient condition in terms of certain matrices is developed for the mapping from parameter vectors to saddle points to be single-valued and Lipschitz continuous locally. It is shown that the saddle point mapping is then semi-differentiable, and that its semi-derivative at any point and in any direction can be calculated by determining the unique solutions to an auxiliary problem of extended linear-quadratic programming and its dual. A matrix characterization of calmness of the solution mapping is provided as well.