NONSMOOTH ANALYSIS AND OPTIMIZATION ON PARTIALLY ORDERED VECTOR SPACES

Interval-Lipschitz mappings between topological vector spaces are defined and compared with other Lipschitz-type operators. A theory of generalized gradients is presented when both spaces are locally convex and the range space is an order complete vector lattice. Sample applications to the theory of nonsmooth optimization are given.


I. INTRODUCTION.
The purpose of this paper is to introduce a broad class of Lipschitz-type operators and to present new results concerning first-order optimality conditions for nonsmooth nonconvex programs in infinite dimensions.
Significant progress in deriving more general optimality conditions for mathematical programming models has been made in recent years as a result of advances in nonsmooth analysis and optimization.The study of nonsmooth problems is motivated in part by the desire to optimize increasingly sophisticated models of complex man- made and naturally occurring systems that arise in areas ranging from economics, operations research, and engineering design to variational principles that correspond to partial differential equations.Results in nonsmooth optimization have expedited understanding of the salient aspects of the classic smooth theory and identified concepts fundamental to optimality that are not intertwined with differentlability assumptions.We mention as examples in this regard the works of Hiriart-Urruty [I], where the convexity of a tangent cone is required for optimality in the nonsmooth case but not when differentiability is assumed, and Clarke [2] where standard assumptions in optimal control are weakened.
First-order optimality conditions have received the most scrutiny and in general are well-understood.In terms of first principles they require, for example, that two problem-specific sets be nonintersecting or that a certain map not be locally surjective.Smoothness is not a fundamental prerequisite for these properties to hold.Analysis serves as the link between the above mentioned conditions and their equivalent expression in useable and verifiable algebraic forms.Research in nonsmooth analysis is motivated in part by the attitude that the essentials of optimality are sufficiently amenable and extensive to allow their application to nondifferentiable (and nonconvex) problems, provided an appropriate analysis is developed.
This paper makes a contribution to nonsmooth analysis and optimization based on these ideas.Our approach and subsequent results, while new in many respects, continue the work of others in extending the applicability of differential calculus.
For example, generalized derivatives are defined in the well-known theory of distributions; however, these derivatives are of little use in optimization since their values are often not well-defined at local extrema.
Since the initial contribution of Clarke, the theory and applications of generalized gradients has grown to such an extent that a survey is beyond the scope of this introduction.For excellent summaries of the theory, motivation and applications of generalized gradients and extensive references we refer the reader to Clarke [2], Hiriart-Urruty [1] and Rockafellar [26]; in addition, Borwein and  Strojwas [27] provide an insightful comparison of several recent directional derivatives and generalized gradients of the same genre as Clarke's gradient.The quasidifferentiable functions.A distinguishing feature of our optimality conditions is that they allow for an infinite-dimensional equality constraint.Ioffe [30] obtains results for problems in Banach spaces with an infinite-dimensional Lipschitz equality constraint operator or finitely many directionally Lipschitzian equality constraint functions.

INTERVAL-LIPSCHITZ MAPPINGS.
Unless specified otherwise, in this section X and V denote, respectively, a linear topological space and an ordered topological vector space.We will denote the zero elements of X and V by 0. We will occasionally make the assumption that the positive cone V+" (v E V" v 0) is normal, that is, there is a neighborhood base of the origin 0 E i such that, for W E W, W (W+V+) n (W-V+).Such neighborhoods are said to be full or satur@ted.Several consequences of normality utilized in the sequel can be found in Peressini [34].We will always make expltcit mention of this assumption when it is being used.
DEFINITION 1.The mapping f-X V is interval-Lipschitz at R X if there exists neighborhoods N of R and W of 0 E X, ( > O, two mappings m and M from W into V satisfying m(y) S M(y), and a mapping r from (0,(] x X x X into V satisfying lim r(t,x;y) 0 for all y e W, such that tO XX t-1[f(x+ty) f(x)] [m(y), M(y)] + r(t,x;y) for all x e N, y e W and t e (0,(].If U is an open subset of X, f is locally interval-Lipschitz on U if f is interval-Lipschitz at R for every R E U. If X is a normed space, V=R, and f is Lipschitz at R X in the usual sense, that is, there exist a neighborhood N O of 2 and k R + such that If(x) f(y)[ N k[lx-y[[ for all x, y e N O then f is interval-Lipschitz at 2. Indeed, select a neighborhood N of and a circled neighborhood W of 0 X such that N + W { NO; EXAMPLE I.For X a Banach space and V an order complete Banach lattice, Papageorgiou [28] defines a mapping f" X V to be locally o-Lipschit if for every open bounded subset U of X there is a k V+:-{v V" vO}, the positive cone oZ V, is an open bounded subset of X, then f is locally interval-Lipschitz on U. Indeed, if R U, choose a neighborhood N of R and a circled neighborhood W of 0 E X such that N+W { U. Then for x e N, y e W, and t (0,I], we have [f(x+ty) f(x)[ ktl[y[[;   the same choices for m(.), M(.), and r as in the preceding paragraph show that f is interval-Lipschitz at 2. Since R U was arbitrary, f is locally interval- Lipschitz on U. EXAMPLE 2. If X is a normed vector space, f" X V is strictl.differentiabl at R e X if there exists a continuous linear mapping Vf(R)'X V such that li [f(x) f(z) vf()(x-z)]/llx-zll 0 then lim r(t,x;y) 0 and f is interval-Lipschitz at R.
tO XX EXAMPLE 3. If f:X V is sublinear (i.e., subadditive and positively homogeneous), then f is interval-Lipschitz on X.In fact, if u and z are in X, then by the sublinearity of f, f(u) f(z) f(u z) and -f(z-u) f(u) f(z).Thus, for x and y in X and t > O, -f(-ty) f(x+ty) f(x) f(ty) and the choices rEO, m(y) -f(-y), M(y) f(y) show that f is interval-Lipschitz.EXAMPLE 4A.If V is a vector lattice, Kusraev {31] defines a mapping f: X-V to be Lipschitz at R in X if there exists a neighborhood N O of R and a continuous monotone sublinear operator P" X V such that If(u) f(v) P(u-v) for all u,v in N O Let N be a neighborhood of R and W a circled neighborhood of 0 in X such that N + W { N O Then the sublinearity of P and the choices m(y) -P(y), M(y) P(y) and rO show that f is interval-Lipschitz at i. EXAMPLE 4B.If X is a Banach space, then the inequality in Kusraev's definition of a Lipschitz mapping f at R in Example 4a can be stated as If(u) f (v)   for all u,v E N O and for some k E V+.These Lipschitz mappings are equivalent to the subclass of interval-Lipschitz mappings, called order-Lipschitz mappings, on the Banach space X where m(y) v I, M(y) v 2, and r(.,.;y) 0 for all y W. Indeed, if f is Lipschitz at according to Kusraev, then choosing neighborhoods N of and W of 0 in X such that N + W { N O and selecting m(y) -k, M(y) k, and rO shows that f is order-Lipschitz at R. Conversely, suppose f is order-Lipschitz at with m(y) v I, M(y) v 2, and r(.,.;y) 0 for all y W. Let the real number p > 0 be such that B(R,2p) :-{xEX'JIR-xIJ<2p} C N, B(8,2p) W and choose o > 0 such that p-lo < (.Then for all x,y E B(R,o) we have f(y) f(x) f(x+p'llly-xll.p((y-x)/llYxll) f(x) E p-1 ily_xll[vx,vz] if x y; since pXlly-xll < and p(y-x)/llY-x W, If(y) f(x)l klly-xtl for all x,y E B(,o), where k='l(Ivll + Iv21) v+, and thus f is Lipschitz at R according to Kusraev.REMARK.If X is a Banach space, V is an order complete Banach lattice and f" X V is locally o-Lipschitz according to Papageorgiou [28] (see Example ]), then if int V+ , f is Lipschitz at R according to Kusraev for any R e X.Indeed, let v 0 be in the interior of V+; then [-v O, v O] + R is a (convex) neighborhood of and is (topologically) bounded since the normality of V+ implies that order bounded sets are topologically bounded (Peressini [34, p. 62]. The next example shows that an interval-Lipschitz mapping is no necessarily continuous.EXAMPLE 5. Let (c) be the space of all convergent sequences of real numbers with norm llxJl(R) sup (IXnl) and let W be an open bounded neighborhood of (c) relative to the topology o((c), tl), i.e., the weak topology on (c).Since tl is the dual of (c), tl is norm-determining for (c) (Taylor [35, p. 202]), hence by Taylor [35, p.   208] W is bounded relative to the norm topology.In particular, W is absorbed by B {x: llxll < I}, thus there exists 0 > 0 W { B for all II S 0" Let W 0 oW; then W 0 is order bounded since B {x-llxll I} coincides with [-e,e] in (c), where e (en), e n for all n.Therefore, since f" (c) (c) given by f(x) Ixl is sublinear, for any x E (c) and y E W 0 we have which shows that f(x) Ixl is interval Lipschitz on (c).However, f(x) is not continuous since the dual of (c) is not the sequence space (x-(Xn):X n 0 for all but a finite number of choices of n} (Peressini [34, p. 135]).
The following example shows that convex mappings are interval-Lipschitz.EXAMPLE 6.Let X and V be as in Example 1.The mapping f: X V is onve if f(x + (1-)y) f(x) + (1-)f(y) for all E [0,1] and x,y e X.If f is convex and majorized in a neighborhood of x 0 E X, then by Theorem 3.2 in Papageorgiou [17] and Example ], f is interval-Lipschitz on X.We conclude this section with a brief comparison of interval-Lipschitz mappings and two similar Lipschitz-type operators proposed by Thibault [36].Unless specified otherwise, X and V are linear topological vector spaces.Thibault [36] defines a compactly Lipschitzian mapping at a point as follows: f:X-V is compactly [ipschitzian at R X if there is mapping K:X Comp(V):-{nonempty compact subsets of V} and a mapping r:(O,]] x X x X into V such that (i) lim r(t,x;y) 0 for each y X; tO XX (ii) for each y X there is a neighborhood El of R and Q e (0,1] such that t-I[f(x+ty)-f(x)] e K(y) + r(t,x;y) for all x e El and t e (O,Q] This definition does not require the range space to be ordered as in Definition and hence in this respect can be considered more general than our definition.However, the approach taken in this paper and in Thibault [36] (and in many other works as well) to derive a theory of generalized gradients requires that the range space be ordered.In this case, Definition takes explicit account of the order structure.In addition, the order interval [m(y), M(y)] is in general not compact.If V is normal, then the order interval [m(y), M(y)] is bounded and hence by Alaoglu's Theorem is -compact if V is a dual space; however, it is in general not compact for any other stronger topology.From this viewpoint, Definition can be considered somewhat more general than Thibault's definition.
For a mapping f: X V, V an ordered topological vector space, Thibault [36] defines f to be order-Lipschitz at a point R X as follows: there exist mappings and B of X into V and a mapping r:(O,l] x X x X V such that (i) b(x) _< l(x) for all x e X and lim l(x) 0; (ii) lim tO X-X r(t,x;y) 0 for a11 y e X; (iii) for each y X there is a neighborhood 0 of R and r/ e(O,l] such that t-1[f(x+ty)-f(x)] e [h(y),l(y)] + r(t,x;y) for all t e (O,r/], x e El There are no implications between the above definition and Definition without additional technical assumptions.For instance, if f is order-Lipschitz at e X according to Thibault and in addition there is a neighborhood W of 0 ( X with a corresponding neighborhood El of R and r (0,1] such that t-1[f(x+ty)-f(x)] E [h(y),B(y)] + r(t,x;y) for all x E O, t E(0,T), Y ( W, then f is interval-Lipschitz at according to Definition with m h and M Conversely, suppose f is interval-Lipschitz at according to Definition with the additional assumptions that lim M(y) 0 and lim r(t,x;y) 0 for all y6) t$0 X-X y E X (not just for all y E W).There exists an element W 0 of a neighborhood basis of 6) E X such that W 0 _c W with W 0 radial (Peressini [34, p. 162]).Thus, for each y E X there exists >,y > 0 such that y E W 0 for all with lkl _< ),y.Then f is order- Lipschitz at according to Thibault with T min{(,ky,1}.
Unless specified otherwise, in this section X denotes a locally convex Hausdorff topological vector space and V denotes a locally convex ordered topological vector space, that is, V is a Hausdorff locally convex topological vector space and an ordered vector space with a convex positive cone V+ -{v V" v > 0} that is closed.
We also assume V is an order complete vector lattice for its order structure, that is, sup(u,v) exists for all u,v in V and sup B exists for each nonempty subset B of V that is order bounded above.
The subdifferential of an interval-Lipschitz mapping will be defined in terms of a directional derivative which we now introduce.DEFINITION 2. If f: X V is interval-Lipschitz at , the generalized directional derivative of f at R in the direction y E X, denoted f(R;y), is given by fo(;y) where T(R) is a neighborhood base of in X.If X is a Banach space, V--R, and f is Lipschitz at R (which implies f is interval-Lipschitz at ), then f(x;.) coincides with Clarke's qeneralized directional derivative at ; see Clarke [2,[22][23][24][25].If V is an order complete Banach lattice and f is locally o-Lipschitz (see Example I) then f(R;.) also coincides with the generalized o-directional derivative of f at R in the direction y defined by Papageorgiou [28].The Clarke derivative of f at R defined by Ku@raev [31] coincides with f(R;.) if the range space and the filter in Kusraev [31] are, respectively, order complete and limited to the neighborhood filter of R.
The next two results exhibit properties of fo(;y) as a mapping of y e X. PROPOSITION I.The mapping y fo(;y) is a sublinear mapping from X to V that satisfies f(R;y) < M(y) for all y e W and f(R;-y)-(-f)(R;y) for every y X.
PROOF.The proof of the sublinearity of fo(;.)follows that for real-valued Lipschitz functions, while f(R;y) < M(y) for all y W follows directly from Definitions and 2. For any given y E X, there exists ey > 0 such that ey W for lel < ey; hence f(R;yy) eyf(;y) < M(eyy), so f(R;y) _< elM(eyy) and thus fo(;y) E V.  f(k;-y) REMARK.Note that since f(k;-) is sublinear, by Example 3 it is interval- Lipschitz on X.
The next result exhibits several sufficient conditions for f(R;.) to be a continuous mapping.For f" X V we define the epigraph of f, denoted epi f, by epi f:-{(x,v) X x Vlv f(x)}.Recall that the positive cone V+ in V is normal if there exists a neighborhood basis of 0 E V such that W (W+V+)(W-V+) for all W (Peressini [34, p. 61]).
PROPOSITION 2. If the positive cone V+ of V is normal, then each of the following conditions implies that f(k;.) is continuous" (i) int epi fo(;.) is nonempty; (ii) lim M(y) 0 where the convergence is an order convergence; yO (iii) M(.) is continuous at ( E X. PROOF.(i) Since the order intervals in V are bounded in the topology of V and fo(R;.) is convex, fo(R;.) is continuous on X if it is bounded above in a neighborhood of one point (Valadier [IO, p. 71]).But int epi fo(R;.) is included in the set of (y,v) E X x V such that f(k;.) is bounded above by v in a neighborhood of y.
(iii) Since fo(;y) M(y) for each y E W and fo(;.) is convex, the continuity of f(R;.) at B E X follows directly from Borwein [16, Prop. 2.3] since M(.) is assumed continuous at 0 E X.The continuity of fo(R;.) on X follows as in part (ii).
The continuity of fo(R;.)leads to several results concerning the subdifferential.Hence we make the following definition.DEFINITION 3. The mapping f: X V is reqular at R E X if f is interval- Lipschitz at R and if fo(R;.) is a continuous mapping from X to V.
Denote by L(X,V) the vector space of linear mappings from X to V. (X,V) denotes the space of continuous linear mappings from X to V; s(X,V) denotes the latter space endowed with the topology of pointwise convergence.DEFINITION 4. Let f: XV be interval-Lipschitz at R X.The @ubdifferential of f at R, denoted af(k), is defined as follows: af(k):={T (X,V)IT(y S f(k;Y) V y E X).
If f is Lipschitz at and V=R, the above definition coincides with Clarke's If we ignore the topological structure on X and V and deal only with the algebraic structures, then we can define the alqebraic subdifferential of f at R, denoted aaf(R) thus aaf(R)-=(T L(X,V)IT(y < f(R,y) V y e X}.
REMARK.The subdifferential cf(R) can be empty; indeed, if f is linear and discontinuous, then af() since fo(;y) f(y) for all y e X. PROPOSITION 3. The subdifferential af() of f at is convex and satisfies -af() a(-f)().
PROOF.The convexity of af() follows directly from the definition; -af(R) a(-f)(R) is a consequence of the relation fo(;_y) (_f)o(;y), for all y e X, proved in Proposition I.
PROPOSITION 4. If f is regular at R and V+ is normal, then af(R) aaf(R), that is, af(R) is the set of all linear mappings T:X V such that T(y) f(R;y) for all y e X.
PROOF.Suppose T: X V is a linear mapping satisfying T(y) _< f(R;y) for all y e X.By the inearity of T, -T(y) T(-y) < f(R;-y), thus -f(R;-y) < T(y) _< f(R;y).Since V+ is normal and fo(;.) is continuous, lim T(y) 0 and hence T is continuous on X.
THEOREM I.Under the assumption of Proposition 4, the subdifferential af(R) is a nonempty, closed, convex, equicontinuous subset of Ls(X,V)with fo(R;y) max{T(y)IT e af(R)) If, in addition, the order intervals in V are compact, then af(R) is compact in Ls(X,V) PROOF.The subdifferential af(R) is the convex subdifferential of fo(;.) at zero.Then since f is assumed regular at R, the results follow from Theoreme 6 and Corollaire 7 in Valadier [I0].
REMARK.Theorem provides a connection between the subdifferential of Definition 4 and the quasidifferential of Pschenichnyi [18].A real-valued function defined on a topological vector space E is quasidifferentiable at e E in the Thus, by Theorem I, if the real-valued function f defined on X (a locally convex Hausdorff spaced with normal cone) is interval-Lipschitz and regular at R with f'(R;d) f(;d), then f is quasidifferentiable at .
REMARK.It is natural to consider a comparison of af() and acf(), the qonvex subdifferential of f at R, and to compare af() with the Frechet or Gateaux derivative of f at .By Theorem 3.2 in Papageorgiou [28] the subclass of interva1-Lipschitz mappings known as locally o-Lipschitz mappings (see Example I) has a subdifferential af(R) such that af() acf(R when f is convex.Similarly, a locally o-Lipschitz mapping f: XY that is continuously Gateau iffrentiable for [I'll1 on Y, where llylll'= inf{k IYl ke) (e is the strong unit on tile Banach lattice Y), satisfies af(R) {f'(R)) by Papageorgiou [28, Th. 3.3]. 4. OPTIMALITY CONDITIONS In this section we show that our approach to the local analysis of nonsmooth operators introduced in Sections 2 and 3 has relevance to mathematical programming.In particular, we give necessary and sufficient optimality conditions for nondifferentiable programming problems with real-valued objective functions and constraints consisting of either an arbitrary set or an arbitrary set and a vectorvalued operator.While the results are related to those obtained in Kusraev [31] and Thibault [36], where the objective functions are vector-valued, our assumptions and proof techniques are somewhat different.Specifically, Kusraev's vector-valued mappings are Lipschitz with the absolute value operator while Thibault's mappings are "compactly Lipschitzian" [36, Def.1.1].In addition, our proof of the Kuhn-Tucker necessary conditions (Theorem 2), which recalls a paper of Guignard [37], does not xplicitly use the assumptions that the range space of the constraint operator is an ordered space.This raises the possibility of substituting for the generalized gradient of the constraint operator g at R any closed convex subset Fg(R), say, of Ls(X,V that satisfies the conditions we require of the generalized gradient.This approach could generate various closed convex-valued multifunctions as in Ioffe [29] (where such multifunctions are called fans) and lead to necessary conditions which have as special cases the necessary conditions of Clarke [24], Hiriart-Urruty [I, 38, 39] and Ioffe [40].Ioffe [30] has in fact used the concept of fan to develop more general necessary conditions.
Let X be a Banach space, V as described at the beginning of Section 3, S a nonempty subset of X, and f an extended real-valued function on X which, unless stated otherwise, is assumed to be finite and interval-Lipschitz at R S.
Consider the problem: minimize f(x), subject to x E S; R is a local minimum of f on S if f is finite at and if there exists a neighborhood N of R such that f(x) f(R) for every x e S n N; R is a minimum of f on S if f is finite at R and f(x) f() for every x e S. The contingent cone of S at x o clS (closure of S), denoted K(S;xo), is defined as follows: K(S;Xo)'={d Xl3t n > O,{Xn) c S, x n x o with d lim tn(Xn-Xo) ) {d Xl3tn O, d n d with x o + tndn e S for all n} The (Clarke) tangent cone of S at x o E cIS, denoted (S;Xo), is the following set" #(S;Xo)'= (d e XI for every {Xn} c clS B X n x o and for every (t n) B t n O, 3{dn} B d n d with x n + tndn E S for all n} K(S:Xo) is a closed cone and /(S;Xo) is a closed convex cone with (S;xo) c: K(S;Xo).
The closure of the convex hull of K(S;x o) is denoted P(S;Xo).The polar cone of a nonempty set A C X is given by A:={x * E X*Ix*(x) 0 x E A}, where X* is the topological dual of X; if A , A:=X * If A* C X* is nonempty, the prepolar of A* is (A*)" (x E Xlx*(x) 0 y x* A* E A*} If (A*) =X.A((A*)) is a weak*-closed (weakly closed) convex cone in X*(X).
We begin our study of optimality with three results that give necessary conditions for a vector R to be a local minimum.PROPOSITION 5.If is a local minimum of f on S=X, then 0 af().
PROOF.Consider a sequence {tn} C (0,1] converging to 0 and select neighborhoods N of R and W of 0 E X, a constant ( > o, and m, M and r satisfying Definition ]. We may assume f(x) f() for all x E N. For each y E W there exists n o such that t1[f(+tn y) f()] r(t n, ;y) E [m(y), M(y)] and + tnY E N for all n n o In addition, there exists a convergent subsequence (tIn Since W is radial, we conclude fo(;y) _> 0 for y E X and hence that 0 E af().
REMARK.Proposition 5 is related to a necessary condition for an unconstrained optimum of a quasidifferentiable function on En.A real-valued function f on E n is quasidifferentiable at x if f is directionally differentiable at x and if there exists convex compact sets _f(x) and af(x) in E n such that f'(x;d) max <v,d> + min <w,d> vEf(x) w(f(x) (Demyanov and Rubinov [41]).Polyakova [42] has shown that -af() C f(R) is a necessary condition for to be a minimum of a quasidifferentiable function f on En.By Theorem 1, if the real-valued function f on E n is order-Lipschitz and regular at R, then f is quasidifferentiable at with af() {0} and f() af(), thus the optimality condition immediately above reduces to the condition in Proposition 5" 0 E af().However, Proposition 5 is applicable in the broader context of infinite dimensional spaces.In addition Proposition 5 generalizes several results in the literature obtained for Lipschitz functions on a Banach space, e.g., Clarke [2], Ioffe [30,40] and Thibault [36].for all y E K(S;), then fo(;y) >_ 0 for all y E K(S;).
PROOF.Suppose y E K(S;) and let {tn} and {Yn} be the sequences corresponding to (4.1).In addition, choose N, W, (, m, M and r satisfying Definition with m and M continuous.There exists n such that + tnYn E N for all n > nl, hence R + tnYn e S n N for all n _> n (4.2) by (4.1).We may assume f(R) _< f(x) for all x e S n N. Since W is radial, corresponding to y and each Yn there exists ey > 0 and e n > 0, respectively, such that ey E W for le < ey and ey n E W for lel -< e n.Hence there exists n 2 such that t1[f(R+tnenYn) f(R)] r(tn,R;enYn) e [m(enYn), M(nYn)] for all n _> n 2 and thus (4.2) and (4.3) hold for all n > no:=max{nl,n2}.Since {Yn} converges to y, there exists a sequence of en'S that converges to ey.In addition, since each [m(enYn), M(nYn)} is compact and m and M are continuous, there exists a convergent subsequence tIn) which implies f(R;y) 0.
REMARK.The assumption in Proposition 6 concerning f(R;y) plays a role similar to "condition ()" imposed by Hiriart-Urruty [38, p. 89] to obtain the same necessary optimality condition.
It is customary to express optimality conditions in terms of the polar cones of the cones of displacement.A result of this type is presented below.Recall first that if C is a nonempty subset of X, the distance function dc: X R, defined by dc(x) inf[ilx-clll c E C}, is a globally Lipschitz function on X with Lipschitz constant I. PROPOSITION 7. Let be a local minimum of f on S. If f is regular at , then 0 e af(x) + ((s;R)) .
PROOF.Since f is interval-Lipschitz at R, choose neighborhoods N and W, mappings m, M and r, and ( > 0 that satisfy Definition 1.We will first show that there exists a neighborhood N O of over which R minimizes f(x) + p-](IM() + r(i,x;))l + Im()) + r(t,x;))l)ds(x) for some ) e W and some e(0,(], where p > 0 is such that B(0,2p) S W. By way of contradiction, suppose this result is false.Then there exists a sequence {xn} converging to R such that f(Xn) + P-I(IM(Y) + r(t,xn;Y)l + lm(y) + r(t,xn;Y)l) ds(xn) < f(R) for y e g and t e (0,(].
There exists n o such that dS(Xn) > 0 for n n o since otherwise x n belongs to S and the above inequality contradicts the local optimality of .Since ds(xn) converges to 0 as n (R), we can choose n sufficiently large so that f is order-Lipschitz at x n in a neighborhood of radius 2dS(Xn) and with the same neighborhood W, mappings M, m and r, and ( > 0 mentioned at the beginning of the proof.There exists s n e S such that Jls n Xnl min{p(, (l+)dS(Xn)), where e (0,1) satisfies f(Xn) + p'l(IM(Y + r(t,xn;Y)l+Im(y) + r(t,Xn:Y)l)(1+)dS(Xn) < f(R) for y W and t (0,(].Since s n x n + toYo where t o p-]IISn-Xnl ( and y0-= pllSn-Xnll-1(Sn-Xn W with IJSn-Xnl < 2dS(Xn), we have f{s n) f{x n) + to(IM(yo) + r(to,Xn;Yo) l+Im(y o) + r(to,Xn;YO)l) < f{x n) + p-I(IM(Yo) + r(to,Xn;YO) l+Im(y O) + r(to,Xn;Yo) l)(l+I)ds(xn) < f(R) which contradicts the local optimality of R. Thus R is a local minimum of f(x) + p-I(IM()) + r(t,x;))l+Im()) + r(t,x;))l)ds(xn) for some ) W and t (0,(].Since im r(t,x;)) O, we have tO X-X k'-p'l(IM())l+Ir(l,R;)I+Im())l+Ir(l,R;))l) > p'l(IM( + r(t,x;))l+Im()) + r(t,x;))l); therefore R is also a local minimum of f(x) + kds(x).Finally, since a(f1+f2)(R) C aft(R) + af2(R) where fl and f2 are interval-Lipschitz at and fl is regular at R, by Proposition 5 and Clarke [2,   Prop.2.4, p. 51] we conclude that 0 a(f(R) + kds(R)) C af(R) + kads(R) c af(R) + (y(s,R)) If f is Lipschitz, then a stronger necessary condition than the one in Proposition 7 can be obtained.PROPOSITION 8. Let R be a local minimum of f on S, where f is Lipschitz at R, and M a convex cone contained in K(S;R); then oaf(R) +M PROOF.Since f is assumed Lipschitz at R, the result follows directly from Theorems 7 and 8 in Hiriart-Urruty [38].
then for x e N, y E W and t e (0,1], [f(x+ty)-f(x)[ S tk[[y[[ and the choices m(y) -kl[y[[, M(y) k[lY[[, rO show that f is interval-Lipschitz at .Below we provide additional sample classes of operators that are interval-Lipschitz.
aO exists for all d E and if ] a nonempty weak*-closed subset Mf(R) of E f'(R;d) Max{x*(d) Ix e Mf(R)}.

PROPOSITION 6 .
If is a local minimum of f on S, m and M in Definition are