generalized Jacobians, nonsmooth analysis, mean value conditions, optimality

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1 SIAM J. CONTROL OPTIM. c 1998 Society for Industrial and Applied Mathematics Vol. 36, No. 5, pp , September APPROXIMATE JACOBIAN MATRICES FOR NONSMOOTH CONTINUOUS MAPS AND C 1 -OPTIMIZATION V. JEYAKUMAR AND D. T. LUC Abstract. The notion of approximate Jacobian matrices is introduced for a continuous vectorvalued map. It is shown, for instance, that the Clarke generalized Jacobian is an approximate Jacobian for a locally Lipschitz map. The approach is based on the idea of convexificators of realvalued functions. Mean value conditions for continuous vector-valued maps and Taylor s expansions for continuously Gâteaux differentiable functions (i.e., C 1 -functions) are presented in terms of approximate Jacobians and approximate Hessians, respectively. Second-order necessary and sufficient conditions for optimality and convexity of C 1 -functions are also given. Key words. conditions generalized Jacobians, nonsmooth analysis, mean value conditions, optimality AMS subject classifications. 49A52, 90C30, 26A24 PII. S Introduction. Over the past two decades, a great deal of research has focused on the study of first- and second-order analysis of real-valued nonsmooth functions [2, 3, 4, 5, 11, 12, 14, 15, 21, 23, 24, 20, 25, 27, 28, 29, 30, 34, 35]. The results of nonsmooth analysis of real-valued functions now provide basic tools of modern analysis in many branches of mathematics, such as mathematical programming, control, and mechanics. Indeed, the range of applications of nonsmooth calculus demonstrates its basic nature of nonsmooth phenomena in the mathematical and engineering sciences. On the other hand, research in the area of nonsmooth analysis of vector-valued maps has been of substantial interest in recent years [2, 6, 7, 8, 9, 10, 18, 21, 22, 23, 24, 29, 31]. In particular, it is known that the development and analysis of generalized Jacobian matrices for nonsmooth vector-valued maps are crucial from the viewpoint of control problems and numerical methods of optimization. For instance, the Clarke generalized Jacobian matrices [2] of a locally Lipschitz map play an important role in the Newton-based numerical methods for solving nonsmooth equations and optimization problems (see [26] and other references therein, and see also [17, 18, 19] for other applications). Warga [32, 33] examined derivative (unbounded derivative) containers in the context of local and global inverse function theorems as set-valued derivatives for locally Lipschitz (continuous) vector-valued maps. Mordukhovich [21, 22] developed generalized differential calculus for general nonsmooth vector-valued maps using the set-valued derivatives, called coderivatives [9, 21]. Our aim in this paper is to introduce a new concept of approximate Jacobian matrices for continuous vector-valued maps that are not necessarily locally Lipschitz, develop certain calculus rules for approximate Jacobians, and apply the concept to optimization problems involving continuously Gâteaux differentiable functions. This Received by the editors November 8, 1996; accepted for publication (in revised form) October 2, 1997; published electronically July 9, This research was partially supported by a grant from the Australian Research Council. Department of Applied Mathematics, University of New South Wales, Sydney 2052, Australia (jeya@maths.unsw.edu.au). Some of the work of this author was carried out while visiting the Centre for Experimental and Constructive Mathematics at the Simon Fraser University, Canada. Institute for Mathematics, Hanoi, Vietnam (dtluc@thevinh.ac.vn). Some of the work of this author was done while visiting the University of New South Wales. 1815

2 1816 V. JEYAKUMAR AND D. T. LUC concept is a generalization of the idea of convexificators of real-valued functions, studied recently in [4, 5, 13], to vector-valued maps. Convexificators provide twosided convex approximations [30] for real-valued functions. Unlike the set-valued generalized derivatives [9, 21, 22, 32, 33], mentioned above for vector-valued maps, the approximate Jacobian is defined as a closed subset of the space of (n m) matrices for a vector-valued map from R n into R m. Approximate Jacobians not only extend the nonsmooth analysis of locally Lipschitz maps to continuous maps but also unify and strengthen various results of nonsmooth analysis. They also enjoy useful calculus, such as the generalized mean value property and chain rules. Moreover, approximate Jacobians allow us to present second-order optimality conditions in easily verifiable forms in terms of approximate Hessian matrices for C 1 -optimization problems, extending the corresponding results for C 1,1 -problems [7]. The outline of the paper is as follows. In section 2, approximate Jacobian matrices are introduced, and it is shown that for a locally Lipschitz map the Clarke generalized Jacobian is an approximate Jacobian. Various examples of approximate Jacobians are also given. Section 3 establishes mean value conditions for continuous vector-valued maps and provides necessary and sufficient conditions in terms of approximate Jacobians for a continuous map to be locally Lipschitz. Various calculus rules for approximate Jacobians are given in section 4. Approximate Hessian matrices are introduced in section 5, and their connections to C 1,1 -functions are discussed. Section 6 presents generalizations of Taylor s expansions for C 1 -functions. In section 7, second-order necessary and sufficient conditions for optimality and convexity of C 1 -functions are given. 2. Approximate Jacobians for continuous maps. This section contains notation, definitions, and preliminaries that will be used throughout the paper. Let F : R n R m be a continuous function which has components (f 1,...,f m ). For each v R m, the composite function, (vf) :R n R, is defined by (vf)(x) = v, F(x) = m v i f i (x). The lower Dini directional derivative and the upper Dini directional derivative of vf at x in the direction u R n are defined by (vf) (x, u) := lim inf t 0 (vf) + (x, u) := lim sup t 0 i=1 (vf)(x + tu) (vf)(x), t (vf)(x + tu) (vf)(x). t We denote by L(R n, R m ) the space of all (n m) matrices. The convex hull and the closed convex hull of a set A in a topological vector space are denoted by co(a) and co(a), respectively. Definition 2.1. The map F : R n R m admits an approximate Jacobian F (x) at x R n if F (x) L(R n, R m ) is closed, and for each v R m, (2.1) (vf) (x, u) sup Mv,u u R n. M F (x)

3 APPROXIMATE JACOBIANS AND OPTIMIZATION 1817 A matrix M of F (x) is called an approximate Jacobian matrix of F at x. Note that condition (2.1) is equivalent to the condition (2.2) (vf) + (x, u) inf M F (x) Mv,u u Rn. It is worth noting that the inequality (2.1) means that the set F (x)v is an upper convexificator [13, 16] of the function vf at x. Similarly, the inequality (2.2) states that F (x)v is a lower convexificator of vf at x. In the case m = 1, the inequality (2.1) (or (2.2)) is equivalent to the condition (2.3) F (x, u) sup x,u and F + (x, u) inf x F (x) x F (x) x,u ; thus, the set F (x) is a convexificator of F at x. Also note that in the case m =1, condition (2.3) is also equivalent to the condition that for each α R, (2.4) (αf ) (x, u) sup αx,u u R n. x F (x) Similarly, the condition (2.3) is also equivalent to the condition that for each α R, (2.5) (αf ) + (x, u) inf x F (x) αx,u u R n. For applications of convexificators, see [5, 13, 16]. To clarify the definition, let us consider some examples. Example 2.2. If F : R n R m is continuously differentiable at x, then any closed subset Φ(x) ofl(r n, R m ) containing the Jacobian F (x) is an approximate Jacobian of F at x. In this case, for each v R m, (vf) (x, u) = F (x)v, u sup Mv,u u R n. M Φ(x) Observe from the definition of the approximate Jacobian that for any map F : R n R m, the whole space L(R n, R m ) serves as a trivial approximate Jacobian for F at any point in R n. Let us now examine approximate Jacobians for locally Lipschitz maps. Example 2.3. Suppose that F : R n R m is locally Lipschitz at x. Then the Clarke generalized Jacobian C F (x) is an approximate Jacobian of F at x. Indeed, for each v R m, (2.6) (vf)(x) = C F (x)v. Consequently, for each u R n, where (vf) (x, u) = max ξ,u = max Mv,u, ξ (vf)(x) M C F (x) C F (x) =co{ lim F (x n) T : x n Ω, x n x}, n Ω is the set of points in R n where F is differentiable, and the Clarke directional derivative of vf is given by (vf) (x, u) = lim sup x x t 0 v, F(x + tu) F (x ). t

4 1818 V. JEYAKUMAR AND D. T. LUC Since for each u R n, (vf) (x, u) (vf) (x, u) u R n, the set C F (x) is an approximate Jacobian of F at x. For the locally Lipschitz map F : R n R m, the set B F (x) :={ lim n F (x n) T : x n Ω, x n x} is also an approximate Jacobian of F at x. The set B F (x) is known as the B- subdifferential of F at x, which plays a significant role in the development of nonsmooth Newton methods (see [26]). In passing, note that for each v R m, (vf)(x) =co( M (vf)(x)) = co(d F (x)(v)), where the set-valued mapping D F (x) from R m into R n is the coderivative of F at x and M (vf)(x) is the first-order subdifferential of vf at x in the sense of Mordukhovich [22]. However, for locally Lipschitz maps, the coderivative does not appear to have a representation of the form (2.6), which allowed us above to compare approximate Jacobians with the Clarke generalized Jacobian. The reader is referred to [9, 21, 22, 29] for a more general definition and associated properties of coderivatives. A second-order analogue of the coderivative for vector-valued maps is given recently in [10]. Let us look at a numerical example of a locally Lipschitz map where the Clarke generalized Jacobian strictly contains an approximate Jacobian. Example 2.4. Consider the function F : R 2 R 2 F (x, y) =( x, y ). Then F (0) = {( ) ( 1 0, 0 1 ) ( 1 0, 0 1 ) ( 1 0, 0 1 )} is an approximate Jacobian of F at 0. On the other hand, the Clarke generalized Jacobian {( ) } α 0 C F (0) = : α, β [ 1, 1], 0 β which is also an approximate Jacobian of F at 0 and contains F (0). Observe in this example that C F (0) is the convex hull of F (0). However, this is not always the case. The following example illustrates that even for the case where m = 1, the convex hull of an approximate Jacobian of a locally Lipschitz map may be strictly contained in the Clarke generalized Jacobian. Example 2.5. Define F : R 2 R by Then it can easily be verified that F (x, y) = x y. 1F (0) = {(1, 1), ( 1, 1)} and 2F (0) = {(1, 1), ( 1, 1)}

5 APPROXIMATE JACOBIANS AND OPTIMIZATION 1819 are approximate Jacobians of F at 0, whereas B F (0) = {(1, 1), ( 1, 1), (1, 1), ( 1, 1)} and C F (0) = co({(1, 1), ( 1, 1), (1, 1), ( 1, 1)}). It is also worth noting that co( 1F (0)) co( M F (0)) = C F (0). Clearly, this example shows that certain results, such as mean value conditions and necessary optimality conditions that are expressed in terms of F (x), may provide sharp conditions even for locally Lipschitz maps (see section 3). Let us now present an example of a continuous map where the Clarke generalized Jacobian does not exist, whereas approximate Jacobians are quite easy to calculate. Example 2.6. Define F : R 2 R 2 by F (x, y) =( x sgn(x)+ y, y sgn(y)+ y ), where sgn(x) =1forx>0, 0 for x = 0, and 1 for x<0. Then F is not locally Lipschitz at (0, 0), and so the Clarke generalized Jacobian does not exist. However, for each c R, the set F (0, 0) = {( α 1 0 β ) ( α 1, 0 β ) } : α, β c is an approximate Jacobian of F at (0, 0). 3. Generalized mean value theorems. In this section we derive mean value theorems for continuous maps in terms of approximate Jacobians and show how locally Lipschitz vector-valued maps can be characterized using approximate Jacobians. Theorem 3.1. Let a, b R n and F : R n R m be continuous. Assume that for each x [a, b], F (x) is an approximate Jacobian of F at x. Then F (b) F (a) co( F ([a, b])(b a)). Proof. Let us first note that the right-hand side above is the closed convex hull of all points of the form M(b a), where M F (ζ) for some ζ [a, b]. Let v R m be arbitrary and fixed. Consider the real-valued function g :[0, 1] IR g(t) = v, F(a + t(b a)) F (a)+t(f (a) F (b)). Then g is continuous on [0, 1] with g(0) = g(1). So g attains a minimum or a maximum at some t 0 (0, 1). Suppose that t 0 is a minimum point. Then, for each α R, g (t 0,α) 0. It now follows from direct calculations that g (t 0,α)=(vF) (a + t 0 (b a),α(b a)) + α v, F(a) F (b). Hence, for each α R, (vf) (a + t 0 (b a),α(b a)) α v, F(b) F (a).

6 1820 V. JEYAKUMAR AND D. T. LUC Now, by taking α = 1 and α = 1, we obtain that (vf) (a + t 0 (b a),a b) v, F(b) F (a) (vf) (a + t 0 (b a),b a). By (2.1), we get inf Mv,b a v, F(b) F (a) sup Mv,b a. M F (a+t 0(b a)) M F (a+t 0(b a)) Consequently, and so v, F(b) F (a) co( F (a + t 0 (b a))v)(b a), (3.1) v, F(b) F (a) co( F ([a, b])v)(b a). Since this inclusion holds for each v R m, we claim that F (b) F (a) co( F ([a, b])(b a)). If this is not so, then it follows from the separation theorem p, F (b) F (a) ɛ> sup p, u u co( F ([a,b])(b a)) for some p R m since co( F ([a, b])(b a)) is a closed convex subset of R m. This implies p, F (b) F (a) > sup{α : α co( F ([a, b])p)(b a)}, which contradicts (3.1). Similarly, if t 0 is a maximum point, then g + (t 0,α) 0 for each α R. Using the same line of arguments as above, we arrive at the same conclusion, and so the proof is complete. Corollary 3.2. Let a, b R n and F : R n R m be continuous. Assume that F (x) is a bounded approximate Jacobian of F at x for each x [a, b]. Then (3.2) F (b) F (a) co( F ([a, b])(b a)). Proof. Since for each x [a, b], F (x) is compact, the set co( F ([a, b])(b a) =co{ F ([a, b])(b a)} is closed, and so the conclusion follows from Theorem 3.1. In the following corollary we deduce the mean value theorem for locally Lipschitz maps (see [1, 6]) as a special case of Theorem 3.1. Corollary 3.3. Let a, b R n and F : R n R m be locally Lipschitz on R n. Then (3.3) F (b) F (a) co( C F ([a, b])(b a)).

7 APPROXIMATE JACOBIANS AND OPTIMIZATION 1821 Proof. In this case the Clarke generalized Jacobian C F (x) is a convex and compact approximate Jacobian of F at x. Hence, the conclusion follows from Corollary 3.2. Note that even for the case where F is locally Lipschitz, Corollary 3.2 provides a stronger mean value condition than condition (3.3) of Corollary 3.3. To see this, let n =2,m=1,F (x, y) = x y, a =( 1, 1), and b =(1, 1). Then condition (3.2) of Corollary 3.2 is verified by F (0) = {(1, 1), ( 1, 1)}. However, condition (3.3) holds for C F (0), where C F (0) = co({(1, 1), ( 1, 1), (1, 1), ( 1, 1)}) F (0). As a special case of the above theorem, we see that if F is real-valued, then an asymptotic mean value equality is obtained. This was shown in [13]. Corollary 3.4. Let a, b X and F : R n R be continuous. Assume that, for each x [a, b], F (x) is a convexificator of F. Then there exist c (a, b) and a sequence {x k } co( F (c)) such that F (b) F (a) = lim k x k,b a. Proof. The conclusion follows from the proof of Theorem 3.1 by noting that a convexificator F (x) is an approximate Jacobian of F at x. We now see how locally Lipschitz functions can be characterized using the above mean value theorem. We say that a set-valued mapping G : R n L(R n, R m )islocally bounded at x if there exist a neighborhood U of x and a positive α such that A α for each A G(U). Recall that the map G is said to be upper semicontinuous at x if for each open set V containing G(x) there is a neighborhood U of x such that G(U) V. Clearly, if G is upper semicontinuous at x and if G(x) isbounded, then G is locally bounded at x. Theorem 3.5. Let F : R n R m be continuous. Then F has a locally bounded approximate Jacobian map F at x if and only if F is locally Lipschitz at x. Proof. Assume that F (y) is the approximate Jacobian of F for each y in a neighborhood U of x and that F is locally bounded on U. Without loss of generality, we may assume that U is convex. Then there exists α>0 such that A α for each A F (U). Let x, y U. Then [x, y] U, and by the mean value theorem, Hence, This gives us that F (x) F (y) co( F ([x, y])(x y)) co( F (U)(x y)). F (x) F (y) x y max{ A : A F (U)}. F (x) F (y) α x y, and so F is locally Lipschitz at x. Conversely, if F is locally Lipschitz at x, then the Clarke generalized Jacobian can be chosen as an approximate Jacobian for F, which is locally bounded at x.

8 1822 V. JEYAKUMAR AND D. T. LUC 4. Calculus rules for approximate Jacobians. In this section, we present some basic calculus rules for approximate Jacobians. We begin by introducing the notion of regular approximate Jacobians which are useful in some applications. Definition 4.1. The map F : R n R m admits a regular approximate Jacobian, F (x) at x R n if F (x) L(R n, R m ) is closed, and for each v R m, (4.1) (vf) + (x, u) = sup Mv,u u R n, M F (x) or equivalently, (4.2) (vf) (x, u) = inf M F (x) Mv,u u Rn. Note that in the case m = 1, this definition collapses to the notion of the regular convexificator studied in [13]. Thus, a closed set h(x) R n is a regular convexificator of the real-valued function h at x if for each u R n, h (x, u) = inf ξ,u and ξ h(x) h+ (x, u) = sup ξ,u. ξ h(x) It is evident that these equalities follow from (4.1) by taking F = h and v = 1 and v = 1, respectively. It is immediate from the definition that if F is differentiable at x, then { f(x)} is a regular approximate Jacobian of F at x. However, if F is locally Lipschitz at x, then the Clarke generalized Jacobian C F (x) is not necessarily a regular approximate Jacobian of F at x. It is also worth noting that if 1F (x) and 2F (x) are two regular approximate Jacobians of F at x, then co( 1F (x)) = co( 2F (x)). In passing, we note that if F is locally Lipschitz on a neighborhood U of x, then there exists a dense set K U such that F admits a regular approximate Jacobian at each point of K. By Rademacher s theorem, the dense subset can be chosen as the set where F is differentiable. Theorem 4.2 (Rule 1). Let F and H be continuous maps from R n to R m. Assume that F (x) is an approximate Jacobian of F at x and H(x) is a regular approximate Jacobian of H at x. Then the set F (x)+ H(x) is an approximate Jacobian of F + H at x. Proof. Let v R m,u R n be arbitrary. By definition, v, F + H (x, u) = lim inf t 0 v, F(x + tu) F (x)+h(x + tu) H(x). t Let {t n } be a sequence of positive numbers converging to 0 such that v, F + H v, F(x + t n u) F (x)+h(x + t n u) H(x) (x, u) = lim. n t n Further, let {s n } be another sequence of positive numbers converging to 0 such that v, F (x, u) = lim inf t 0 Then we have v, F(x + tu) F (x) t v, F(x + s n u) F (x) = lim. n s n v, F(x + s n u) F (x) lim sup Mv, u n s n M F (x)

9 and lim sup n Consequently, APPROXIMATE JACOBIANS AND OPTIMIZATION 1823 v, H(x + s n u) H(x) v, H + (x, u) = sup Mv,u. s n M H(x) v, F + H v, F(x + s n u) F (x) (x, u) lim n sup M F (x) s n Mv,u + sup N H(x) + v, H(x + s nu) H(x) Nv,u = sup Pv,u. P F (x)+ H(x) Since u and v are arbitrary, we conclude that F (x) + H(x) is an approximate Jacobian of F + H at x. Note that as in the case of convexificators of real-valued functions [18], the set F (x)+ H(x) is not necessarily regular at x. Theorem 4.3 (Rule 2). Let F : R n R m and H :IR m R l be continuous maps. Assume that F (x) is a bounded approximate Jacobian of F at x and H(x) is a bounded approximate Jacobian of H at F (x). If the maps F and H are upper semicontinuous at x and F (x), respectively, then H(F (x)) F (x) is an approximate Jacobian of H F at x. Proof. Let w R l and u R m be arbitrary. Consider the lower Dini directional derivative of w, H F at x: w, H F (x, u) = lim inf t 0 s n w, H(F (x + tu)) H(F (x)). t By applying the mean value theorem (see Theorem 3.1) to H and F, we obtain F (x + tu) F (x) tco( F ([x, x + tu])u), H(F (x + tu)) H(F (x)) co( H([F (x, ), F(x + tu)])(f (x + tu) F (x))) It now follows from the upper semicontinuity of F and H that for an arbitrary small positive ɛ we can find t 0 > 0 such that for t (0,t 0 )wehave F ([x, x + tu]) F (x)+ɛb 1, H([F (x), F(x + tu)]) H(F (x)) + ɛb 2, where B 1 and B 2 are the unit balls in L(R n, R m ) and L(R m, R l ), respectively. Using these inclusions, we obtain w, H(F (x + tu)) H(F (x)) w, A, t where A := co(( H(F (x)) F (x)+ɛ( H(F (x))b 1 + B 2 F (x)) + ɛ 2 B 2 B 1 )u). Since H(F (x)) and F (x) are bounded, we can find α>0 such that M α for all M H(F (x)) or M F (x). Consequently, w, H F (x, u) sup Mw,u +2ɛ u + ɛ 2 u. M H(F (x)) F (x) As ɛ is arbitrary, we conclude that H(F (x)) F (x) is an approximate Jacobian of H F at x.

10 1824 V. JEYAKUMAR AND D. T. LUC 5. Approximate Hessian matrices. In this section, unless stated otherwise, we assume that f : R n R is a C 1 - function, that is, a continuously Gâteaux differentiable function, and introduce the notion of approximate Hessian for such functions. Note that the derivative of f, which is denoted by f, is a map from R n to R n. Definition 5.1. The function f admits an approximate Hessian f(x) 2 at x if this set is an approximate Jacobian to f at x. Note that f(x) 2 = f(x) and the matrix M f(x) 2 is an approximate Hessian matrix of F at x. Clearly, if f is twice differentiable at x, then 2 f(x) isa symmetric approximate Hessian matrix of f at x. Let us now examine the relationships between the approximate Hessians and the generalized Hessians, studied for C 1,1 -functions, that is, Gâteaux differentiable functions with locally Lipschitz derivatives. Recall that if f : R n R is C 1,1, then the generalized Hessian in the sense of Hiriart-Urruty, Strodiot, and Hien Nguyen [7] is given by 2 Hf(x) =co{m : M = lim n 2 f(x n ),x n, x n x}, where is the set of points in R n where f is twice differentiable. Clearly, H 2 f(x) is a nonempty convex compact set of symmetric matrices. The second-order directional derivative of f at x in the directions (u, v) R n R n is defined by f (x; u, v) = lim sup y x s 0 f(y + su),v f(y),v. s Since (v f) (x, u) f (x; u, v), for each (u, v) R n and f (x; u, v) = max Mu,v = max Mv,u, M 2 H f(x) M 2 H f(x) H 2 f(x) is an approximate Hessian of f at x. The generalized Hessian of f at x as a set-valued map, f(x) :R n R n, which was given in Cominetti and Correa [3], is defined by f(x)(u) ={x R n : f (x; u, v) x,v v R n }. It is known that the mapping (u, v) f (x; u, v) is finite and sublinear and that f(x)(u) is a nonempty, convex, and compact subset of R n, and for each x, u, v R n, Moreover, for each u R n, f (x; u, v) = max{ x,v : x f(x)(u)}. f(x)(u) = 2 Hf(x)u. If f is twice continuously differentiable at x, then the generalized Hessian f(x)(u) is a singleton for every u IR n. In [34, 35], another generalized second-order directional derivative and a generalized Hessian set-valued map for a C 1,1 function f at x were given as follows: f (x; u, v) = sup z R n lim sup s 0 f(x + sz + su),v f(x + sz),v, s

11 APPROXIMATE JACOBIANS AND OPTIMIZATION 1825 f(x)(u) ={x X : f (x; u, v) x,v v X}. It was shown that the mapping (u, v) f (x; u, v) is finite and sublinear; f(x)(u) is a nonempty, convex, and compact subset of R n ; and f(x)(u) is singled-valued for each u IR n if and only if f is twice Gâteaux differentiable at x. Further, for each u R n, f(x)(u) f(x)(u) = H 2 f(x)u. If for each (u, v) Rn the function y f (y; u, v) is upper semicontinuous at x, then f(x)(u) = 2 Hf(x)u. The following proposition gives us necessary and sufficient conditions in terms of approximate Hessians for a C 1 -function to be C 1,1. Proposition 5.2. Let f : R n R be a C 1 -function. Then f hasalocally bounded approximate Hessian map f 2 at x if and only if f is C 1,1 at x. Proof. This follows from Theorem 3.5 by taking F as f. We complete this section with an example showing that for a C 1,1 function the approximate Hessian may be a singleton which is contained in the generalized Hessian of Hiriart-Urruty, Strodiot, and Hien Nguyen [7]. Example 5.3. Let g be an odd, linear piecewise continuous function on R as follows. g(x) =x for x 1 and g(0)=0;g(x) =2x 1 for x [ 1 2, 1]; g(x) = 1 2 x for x [ 1 6, 1 2 ]; g(x) =2x 1 6 for x [ 1 12, 1 6 ]; g(x) = 1 4 x for x [ 1 60, 1 12 ], etc. Let Define G(x) = x 0 g(t)dt, x R. f(x, y) =G(x)+ y2 2. Then the function f is a C 1,1 function, and the generalized Hessian of f at (0, 0) is {( ) } Hf(0) 2 α 0 = : α [0, 2]. 0 1 However, the approximate Hessian of f at (0, 0) is the singleton {( )} f(0) = Generalized Taylor s expansions for C 1 -functions. In this section, we see how Taylor s expansions can be obtained for C 1 - functions using approximate Hessians. Theorem 6.1. Let f : R n R be continuously Gâteaux differentiable on R n ; let x, y R n. Suppose that for each z [x, y], 2 f(z) is an approximate Hessian of f at z. Then there exists ζ (x, y) such that f(y) f(x)+ f(x),y x co 2 f(ζ)(y x), (y x). Proof. Let h(t) =f(y + t(x y)) + t f(y + t(x y)),y x at2 f(y), where a = 2(f(x) f(y) + f(x),y x ). Then h(0)=0,h(1) = f(x) f(y) +

12 1826 V. JEYAKUMAR AND D. T. LUC f(x),y x + 1 2a = 0, and h is continuous. So h attains its extremum at some γ (0, 1). Suppose that γ is a minimum point of h. Now, by necessary conditions, we have for all v R Then 0 h (γ; v) = lim inf λ 0 + h(γ + λv) h(γ) λ h (γ; v) 0. f(y +(γ + λv)(x y)) f(y + γ(x y)) = lim λ 0 + λ lim a(γ + λv) 2 aγ 2 λ 0 + λ (γ + λv) f(y +(γ + λv)(x y)),y x γ f(y + γ(x y)),y x + lim inf λ 0 + λ = v f(y + γ(x y)),x y + aγv + v f(y + γ(x y)),y x f(y +(γ + λv)(x y)),y x f(y + γ(x y)),y x +γ lim inf λ 0 + λ f(y +(γ + λv)(x y)),y x f(y + γ(x y)),y x = aγv + γ lim inf. λ 0 + λ Let ζ = y + γ(x y). Then ζ (x, y), and for v = 1 we get f(y + γ(x y)+λ(x y)),y x f(y + γ(x y)),y x 0 aγ + γ lim inf λ 0 + λ a + sup M(y x),x y. M 2f(ζ) This gives us that a inf x),y x. M 2 f(ζ) M(y Similarly, for v = 1, we obtain f(y + γ(x y)+λ(y x)),y x f(y + γ(x y)),y x 0 aγ + γ lim inf λ 0 + λ a + sup M(y x),y x ; M 2f(ζ) thus, Hence, it follows that and so a sup M(y x),y x. M 2f(ζ) inf x),y x a M 2 f(ζ) M(y sup M(y x),y x, M 2f(ζ) a co 2 f(ζ)(y x), (y x) ;

13 APPROXIMATE JACOBIANS AND OPTIMIZATION 1827 thus, (6.1) f(y) f(x) f(x),y x = a co 2 f(ζ)(y x), (y x). The case where γ is a maximum point of h also yields the same condition (6.1). The details are left to the reader. Corollary 6.2. Let f : R n R be continuously Gâteaux differentiable on R n and x, y R n. Suppose that for each z [x, y], f(z) 2 is a convex and compact approximate Hessian of f at z. Then there exist ζ (x, y) and M ζ f(ζ) 2 such that f(y) =f(x)+ f(x),y x M ζ(y x),y x. Proof. It follows from the hypothesis that for each z [x, y], 2 f(z) is convex and compact, and so the co in the conclusion of the previous theorem is superfluous. Thus, the inequalities give us that inf x),y x a M 2 f(ζ) M(y sup M(y x),y x M 2f(ζ) a 2 f(ζ)(y x), (y x). Corollary 6.3 (see [7]). Let f : R n R be C 1,1 and x, y R n. Then there exist ζ (x, y) and M ζ H 2 f(ζ) such that f(y) =f(x)+ f(x),y x M ζ(y x),y x. Proof. In this case, the conclusion follows from the above corollary by choosing the generalized Hessian H 2 f(x) as an approximate Hessian of f for each x. 7. Second-order conditions for optimality and convexity of C 1 -functions. In this section, we present second-order necessary and sufficient conditions for optimality and convexity of C 1 -functions using approximate Hessian matrices. Consider the optimization problem (P) minimize f(x) subject to x R n, where f : R n R is a continuously Gâteaux differentiable function on R n. We say that a map F : R n R m admits a semiregular approximate Jacobian F (x) at x R n if F (x) L(R n, R m ) is closed, and for each v R m, (vf) + (x, u) sup Mv,u u R n. M F (x) Similarly, the C 1 -function f : R n R admits a semiregular approximate Hessian 2 f(x) atx if this set is a semiregular approximate Jacobian to f at x.

14 1828 V. JEYAKUMAR AND D. T. LUC Of course, every semiregular approximate Hessian to f at x is an approximate Hessian at x. For a C 1,1 function f : R n R, the generalized Hessian, H 2 f(x), of f at x is a bounded semiregular approximate Hessian of f at x since (v f) + (x, u) f (x; u, v) = max Mu,v = max Mv,u. M 2 H f(x) M 2 H f(x) Theorem 7.1. For the problem (P), let x R n. Assume that 2 f( x) is a semiregular approximate Hessian of f at x. (i) If x is a local minimum of (P), then f( x) =0, and for each u IR n, sup Mu,u 0. M 2f( x) (ii) If x is a local maximum of (P), then f( x) =0, and for each u R n, inf M 2 f( x) Mu,u 0. Proof. Let u R n. Since x is a local minimum of (P), there exists δ>0 such that for each s [0,δ], f( x + su) f( x). Then, by the mean value theorem, for each s (0,δ], there exists 0 <t<ssuch that f( x + tu),u 0. So, there exists a positive sequence {t n } 0 such that f( x + t n u),u 0. Now, as f( x) = 0, it follows that (u f) + ( x; u) = lim sup s 0 0. f( x + su),u f( x),u s Since 2 f(x) is a semiregular approximate Hessian of f at x, wehave and hence, (u f) + ( x; u) sup Mu,u, M 2f( x) sup Mu,u 0. M 2f( x) On the other hand, if f attains a local maximum at x, then it follows by the similar arguments as above that for each u R n, inf M 2 f( x) Mu,u 0. Note in this case that it is convenient to use the inequality (u f) ( x, u) inf M 2 f( x) Mu,u.

15 APPROXIMATE JACOBIANS AND OPTIMIZATION 1829 Let us look at a numerical example to illustrate the significance of the optimality conditions obtained in the previous theorem. Example 7.2. Define f : R 2 R by f(x, y) = 2 3 x y2. Then f is C 1 but is not C 1,1 since the gradient ( ) f(x, y) = x sgn(x),y is not locally Lipschitz at (0, 0). Evidently, (0, 0) is a minimum point of f, f(0, 0) = (0, 0), and 2 f(0) = {( α ) } : α 0 is a semiregular approximate Hessian of f at (0, 0). And for each u =(u 1,u 2 ) R 2, sup Mu,u = sup{αu 2 M 2f(0) 1 + u 2 2 : α 0} 0. Hence, the statement (i) of Theorem 7.1 is verified. However, the generalized Hessians [7] do not apply to this function. Corollary 7.3. For the problem (P), let x R n. Suppose that f( x) 2 is a bounded semiregular approximate Hessian of f at x. (i) If x is a local minimum of (P), then f( x) =0, and for each u R n there exists a matrix M f( x) 2 such that Mu,u 0. (ii) If x is a local maximum of (P), then f( x) =0, and for each u R n there exists a matrix M f( x) 2 such that Mu,u 0. Proof. Since f( x) 2 is closed and bounded, it follows from Theorem 7.1 that f( x) = 0, and for each u IR n, max M 2 f( x) Mu,u 0, and so the first conclusion holds. The second conclusion similarly follows from Theorem 7.1. We now see how optimality conditions for the problem (P ) where f is C 1,1 follows from Corollary 7.3 (cf. [7]). Corollary 7.4. For the problem (P), assume that the function f is C 1,1 and x R n. (i) If x is a local minimum of (P), then f( x) =0, and for each u R n there exists a matrix M H 2 f( x) such that Mu,u 0. (ii) If x is a local maximum of (P), then f( x) =0, and for each u R n there exists a matrix M H 2 f( x) such that Mu,u 0. Proof. The conclusion follows from Corollary 7.3 by choosing H 2 f( x) asthe semiregular bounded approximate Hessian f( x) 2 off at x. Clearly, the conditions of Theorem 7.1 are not sufficient for a local minimum, even for a C 2 -function f. The generalized Taylor s expansion is now applied to obtain a version of second-order sufficient condition for a local minimum. For related results, see [34, 16].

16 1830 V. JEYAKUMAR AND D. T. LUC Theorem 7.5. For the problem (P), let x R n. Assume that for each x in a neighborhood of x, 2 f(x) is a bounded approximate Hessian of f at x. If f( x) =0 and for 0 <α<1, each u R n satisfies u 0;then the following holds: (7.1) ( M co( 2 f( x + αu))), Mu,u 0. Then x is a local minimum of (P). Proof. Suppose that x is not a local minimum of (P ). Then there exists a sequence {x n } such that x n x, x n x as n +, and f(x n ) <f( x) for each n. Let x n = x + u n, where u n 0. From the generalized Taylor expansion, Theorem 6.1, there exists 0 <α n < 1 such that f(x n ) f( x)+ f( x),x n x co 2 f( x + α n u n )(u n ),u n. Thus, there exists M n co( f( x 2 + α n u n )) such that f(x n )=f( x) + M n u n,u n, and so M n u n,u n < 0. This contradicts (7.1). Hence, x is a local minimum of (P). The following theorem gives us second-order sufficient optimality conditions for a strict local minimum. Theorem 7.6. For the problem (P), let x R n. Assume that, for each x in a neighborhood of x, f(x) 2 is a bounded approximate Hessian of f at x. If f( x) =0 and for 0 <α<1, each u R n satisfies u 0, then the following holds: (7.2) ( M co( 2 f( x + αu))), Mu,u > 0. Then x is a strict local minimum of (P). Proof. The method of proof is similar to the one given above for Theorem 7.5 and so it is omitted. We now see how the mean value theorem of section 3 and approximate Hessians can be used to characterize convexity of C 1 - functions. Theorem 7.7. Let f : R n R be a continuously Gâteaux differentiable function. Assume that 2 f(x) is an approximate Hessian of f for each point x R n. If the matrices M 2 f(x) are positive semidefinite for each x R n, then f is convex. Proof. Let x, u R n. Then, by the mean value theorem, and so, f(x + u) f(x) co( 2 f([x, x + u])u), f(x + u) f(x), u co( 2 f([x, x + u])u),u. Thus, there exist z [x, x + u] and M co( 2 f(z)) such that It follows by the assumption that f(x + u) f(x), u = Mu,u. f(x + u) f(x), u 0. Since x, u R n are arbitrary, we get that f is monotone in the sense that for each x, u R n, f(x + u) f(x), u 0.

17 APPROXIMATE JACOBIANS AND OPTIMIZATION 1831 The conclusion now follows from the standard result of convex analysis that f is convex if and only if f is monotone. Corollary 7.8. Let f : R n R be C 1,1. Then f is convex if and only if for each x R n, the matrices M H 2 f(x) are positive semidefinite. Proof. Since f is C 1,1 for each x R n, H 2 f(x) is an approximate Hessian of f at x. Hence, it follows from Theorem 7.7 that f is convex. Conversely, assume that f is convex. Let be a set of points in R n on which f is twice differentiable. Then, each matrix M of { lim n 2 f(x n ) : {x n }, x n x} is positive semidefinite as it is a limit of a sequence of positive semidefinite matrices. Hence, each matrix M of Hf(x) 2 =co{ lim n 2 f(x n ) : {x n }, x n x} is also positive semidefinite. Acknowledgments. The authors are grateful to the referees for their detailed comments and valuable suggestions which have contributed to the final preparation of the paper. The first author is grateful to Professor Jonathan Borwein for his helpful comments on the earlier version of the paper and for certain useful references. The second author wishes to thank the first author for his kind invitation and hospitality. REFERENCES [1] F. H. Clarke, Necessary Conditions for Pproblems in Optimal Control and Calculus of Variations, Ph.D. Thesis, University of Washington, Seattle, [2] F. H. Clarke, Optimization and nonsmooth analysis, Wiley-Interscience, New York, [3] R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control and Optim., 28 (1990), pp [4] V. F. Demyanov and V. Jeyakumar, Hunting for a smaller convex subdifferential, J. Global Optim., 10 (1997), pp [5] V. F. Demyanov and A. M. Rubinov, Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt am Main, [6] J. -B. Hiriart-Urruty, Mean value theorems for vector valued mappings in nonsmooth optimization, Numer. Funct. Anal. Optim., 2 (1980), pp [7] J. B. Hiriart-Urruty, J. J. Strodiot, and V. Hien Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with C 1,1 data, Appl. Math. Optim., 11 (1984), pp [8] A. D. Ioffe, Nonsmooth Analysis: differential calculus of nondifferential mappings, Trans. Amer. Math. Soc., 266 (1981), pp [9] A. D. Ioffe, Approximate subdifferentials and applications I: The finite dimensional theory, Trans. Amer. Math. Soc., 281 (1984), pp [10] A. D. Ioffe and J. -P. Penot, Limiting subhessians and limiting subjects and their calculus, Trans. Amer. Math. Soc., 349 (1997), pp [11] V. Jeyakumar, On optimality conditions in nonsmooth inequality constrained minimization, Numer. Funct. Anal. Optim., 9 (1987), pp [12] V. Jeyakumar, Composite nonsmooth programming with Gâteaux differentiability, SIAM J. Optim., 1 (1991), pp [13] V. Jeyakumar and D. T. Luc, Nonsmooth Calculus, Minimality and Monotonicity of Convexificators, Applied Mathematics Research Report AMR96/29, University of New South Wales, Australia, 1996, submitted. [14] V. Jeyakumar and X. Q. Yang, Convex composite multi-objective nonsmooth programming, Math. Progr., 59 (1993), pp [15] V. Jeyakumar and X. Q. Yang, Convex composite minimization with C 1,1 functions, J. Optim. Theory Appl., 86 (1995), pp

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