Much is known about when a locally optimal solution depends in a
single-valued Lipschitz continuous way on the problem's parameters, including
tilt perturbations. Much less is known, however, about when that solution and a
uniquely determined multiplier vector associated with it exhibit that
dependence as a primal-dual pair. In classical nonlinear programming, such
advantageous behavior is tied to the combination of the standard strong
second-order sufficient condition (SSOC) for local optimality and the linear
independent gradient condition (LIGC) on the active constraint gradients. But
although second-order sufficient conditions have successfully been extended far
beyond nonlinear programming, insights into what should replace constraint
gradient independence as the extended dual counterpart have been lacking.
The exact answer is provided here for a wide range of optimization problems
in finite dimensions. Behind it are advances in how coderivatives and strict
graphical derivatives can be deployed. New results about strong metric
regularity in solving variational inequalities and generalized equations are
obtained from that as well