Nonsmooth calculus in finite dimensions

The notion of subgradient, originally defined for convex functions, has in recent years been extended, via the “upper subderivative,” to cover functions that are not necessarily convex or even continuous. A number of calculus rules have been proven for these generalized subgradients. This paper develops the finite-dimensional generalized subdifferential calculus for (strictly) lower semicontinuous functions under considerably weaker hypotheses than those previously used. The most general finite-dimensional convex subdifferential calculus results are recovered as corollaries. Other corollaries given include new necessary conditions for optimality in a nonsmooth mathematical program. Various chain rule formulations are considered. Equality in the subdifferential calculus formulae is proven under hypotheses weaker than the usual “subdifferential regularity” assumptions.