SOLUTION SENSITIVITY FOR KARUSH–KUHN–TUCKER SYSTEMS WITH NONUNIQUE LAGRANGE MULTIPLIERS
This paper is devoted to quantitative stability of a given primal-dual solution of the Karush– Kuhn–Tucker system subject to parametric perturbations. We are mainly concerned with those cases when the dual solution associated to the base primal solution is nonunique. Starting with a review of known results regarding the Lipschitz-stable case, supplied by simple direct justiﬁcations based on piecewise analysis, we then proceed with new results for the cases of Holder (square-root) stability. Our results include characterizations of asymptotic behavior and upper estimates of perturbed solutions, as well as some sufficient conditions for (the speciﬁc kinds of) stability of a given solution subject to directional perturbations. We argue that Lipschitz stability of strictly complementary multipliers is highly unlikely to occur, and we employ the recently introduced notion of a critical multiplier for dealing with Holder stability.