Lagrange multipliers for nonconvex generalized gradients with quality, inequality, and set constraints

摘要

A Lagrange multiplier rule for finite dimensional Lipschitz problems that uses a nonconvex generalized gradient is proven. This result uses either both the linear generalized gradient and the generalized gradient of Mordukhovich or the linear generalized gradient and a qualification condition involving the pseudo-Lipschitz behavior of the feasible set under perturbations. The optimization problem includes equality constraints, inequality constraints, and a set constraint. This result extends known nonsmooth results for the Lipschitz case.