Lipschitz Continuity for Constrained Processes

We study Lipschitz continuity properties for “constrained processes”. As applications of our general theory, we consider mathematical programs and optimal control problems. We show that if thegradients of the binding constraints satisfy an independence condition, then the solution and the dual multipliers of a convex mathematical program are a Lipschitz continuous function of the data. Similarly, it is proved that the optimal control and the dual multipliers for strictly convex control problems with convex constraints on the state and the control are Lipschitz continuous in time. In both applications, estimates of the Lipschitz constant are given.