Continuity of Generalized Gradients and Multipliers Under Perturbations

We prove that the multiplier rule, given by Clarke for mathematical programming problems with locally Lipschitz data, is stable under epi-convergent data perturbations in the finite-dimensional case for global solutions. This stability result is derived from a continuity theorem for generalized gradients extending the classical formula lim ∇f_n = ∇(lim f_n).

Continuous dependence theorems for multipliers in the Fritz-John form proved in this paper extend stability results of Gauvin, Robinson and others for nonlinear programming problems with smooth data.