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Second-order conditions are given which are sufficient to guarantee that a given point be a local minimizer for a real-valued locally Lipschitzian function over a closed set in «-dimensional real Euclidean space. These conditions are expressed in terms of the generalized gradients of Clarke. The conditions provide a very general and unified framework into which many previous first- and second-order theorems fit. 1. Introduction. Let S be a closed subset of «-dimensional space R" and let W be an open set in R". Suppose that the point x* belongs to S D W and let / be a real-valued locally Lipschitzian function on W. We consider here the problem P: Minimize/(x), over x in S D W. We shall present a general theorem which gives second-order conditions that are sufficient to guarantee that x* be a local solution to problem P. This theorem both unifies and extends separate earlier results presented for this problem by the author (2); specific remarks on this point are made below. The theorems in (2) have already been shown to yield both the classical theorem for problems with a finite number of functional constraints and Ioffe's theorem in finite dimensions; it can be shown that they yield other results, such as the sufficiency theorem of Fletcher and Watson (8) for certain nondifferentiable problems containing norms and a theorem proved by the author (1) for a certain class of nonsmooth problems. The second-order sufficiency conditions given here are formulated in terms of the generalized gradients of Clarke. The generalized gradients of / at x are the members of the Clarke subdifferential 3/(x) of /at x. The Clarke subdifferential reduces to the subdifferential of convex analysis when / is convex and it contains only the gradient vector V/(x) when/is C1 near x. We shall give below a brief summary of those basic properties of generalized gradients which we shall use. For more information about generalized gradients, one should examine such earlier papers of Clarke as (4, 5, 6), the work by Rockafellar (23, 20, 21), or the forthcoming book by

A general sufficiency theorem for nonsmooth nonlinear programming