Some Classical Directional Derivatives And Their Use In Optimization
ISSN: 2281-1346DEM Working Paper SeriesSome Classical DirectionalDerivatives and Their Use inOptimizationGiorgio Giorgi(Università di Pavia)# 204 (10-21)Via San Felice, 5I-27100 Paviaeconomiaweb.unipv.it
Some Classical Directional Derivativesand Their Use in OptimizationGiorgio GiorgiAbstract.We give an overview, in nite-dimensioonal Euclidean spaces, of some classical directionalderivatives (usual directional derivatives, Dini directional derivatives, Hadamard directional derivatives) and ofsome classical di erentiability notions. We take into consideration some applications of the said concepts toconvex and generalized convex functions, to nonsmooth unconstrained optimization problems and to nonsmoothconstrained scalar and vector optimization pronlems. Also the axiomatic approach to nonsmooth analysis andnonsmooth optimization problems, proposed by K.-H. Elster and J. Thierfelder, is brie‡y considered.Key Words:Directional derivatives, generalized directional derivatives, nonsmooth analysis, nonsmoothoptimality conditions, nonsmooth vector optimization problems.AMS 2000 Mathematics Subject Classi cation:49K27, 90C20, 90C29, 90C30, 90C46.1. IntroductionIn this paper we give an overview on some classical directional derivatives (usual directional derivatives, Dini directional derivatives, Hadamard directional derivatives) and onsome basic notions about di erentiability (Gâteaux, Fréchet, Hadamard). For simplicity thetreatment will be performed in the n-dimensional Euclidean space Rn : We point out some applications of the said concepts to optimization problems; indeed, in several practical optimizationproblems the involved functions are not everywhere di erentiable. Starting from the seventiesof the last century, the necessity of studying nonsmooth (i. e. non di erentiable) functions,within optimization theory, gave rise to a new mathematical theory, called Nonsmooth Analysis(this term was introduced by the Canadian mathematician F. H. Clarke).However, we shall not be concerned with “modern”directional derivatives, due to Clarke,Rockafellar, Michel and Penot, Demyanov and Rubinov, etc., for which there is an abundantliterature. Only in Section 6 we give some de nitions concerning these more recently proposeddirectional derivatives.The work is organized as follows. Section 2 gives an overview on some classical directionalderivatives and classical di erentiability notions.Department of Economics and Management, Via S. Felice, 5 - 27100 Pavia, (Italy). E-mail: giorgio.giorgi@unipv.it
Section 3 is concerned with the use of directional derivatives in convex and generalizedconvex functions. Section 4 gives some applications of directional derivatives to unconstrainedoptimization problems and constrained optimization problems with a set constraint, whereasSection 5 takes into consideration directional derivatives in constrained optimization problemswith functional constraints. Section 6 gives an overview on the axiomatic approach to nonsmooth analysis, proposed by K.-H. Elster and J. Thierfelder. The nal Section 7 gives someinsights on applications of directional derivatives to vector optimization problems.2. An Overview on Some Classical Directional Derivatives andClassical Di erentiability Introduction NotionsIn this section we want to recall progressively various notions of “classical”directionalderivatives and di erentiability for real-valued functions of several real variables. Many authorsconsider the possibility of “extended-valued functions”, i. e. functions which may assume alsoin nite values; moreover, in Convex Analysis it is customary to consider functions de ned onthe whole space Rn : In the present basic overview we consider real-valued functions de ned onan open set X Rn : More generally, it is possible to consider a set X Rn and a related pointx0 2 int(X):Recall that a derivative is some kind of limit of line segments joining points on the graphof a function. The simplest way to take such a limit is along a line segment containing a pointx0 2 X: This leads to the basic de nition of one-sided (or radial) directional derivative. Letus de ne a direction in Rn as a vector v 2 Rn ; v 6 0 (in some cases it is useful to consider anormalized direction, i. e. kvk 1).De nition 1. Let X Rn be an open set, let x0 2 X and let f : X ! R. We say that fhas the right-sided directional derivative at x0 in the direction v; if the limitlim t !0f (x0 tv)tf (x0 )exists ( nite or not).The result of the above limit is denoted by D f (x0 ; v): Other notations are used in theliterature, such as f (x0 ; v); Dv f (x0 ); etc. For v 0; D f (x0 ; v) is assumed to be zero.In order that this de nition to make sense we implicitly require that there is some " 0such that 0 5 t 5 " implies that x0 tv 2 X; so that f (x0 tv) is de ned. This will be implicitlyassumed also in the other de nitions of directional derivatives given further. Obviously, if X isopen or, more generally, if x0 2 int(X); this is always possible. The next result shows that theset of the directions v for which D f (x0 ; v) exists is a cone, and that D f (x0 ; v) is positivelyhomogeneous on this cone.Theorem 1.The right-sided directional derivative D f (x0 ; v) is positively homogeneous of2
degree one in v. That is, if D f (x0 ; v) exists, thenD f (x0 ; v) D f (x0 ; v); 8 0:Proof. This follows at once fromf (x0 t v)twheref (x0 ) f (x0 v)f (x0 );! 0 : t ; and letting t;Similarly, we have the following de nition.De nition 2. Let X Rn be an open set, let x0 2 X and let f : X ! R. We say that fhas the left-sided directional derivative at x0 in the direction v; if the limitlimt !0f (x0 tv)tf (x0 ) D f (x0 ; v)exists ( nite or not). For v 0; D f (x0 ; v) is assumed to be zero.Finally, we give the following de nition.De nition 3.Under the same assumptions as before, we say that f has a (bilateral)directional derivative at x0 2 X in the direction v or that f is directionally di erentiable atx0 2 X in the direction v; if the limitf (x0 tv)!0tlimtf (x0 ) Df (x0 ; v)exists.It is quite immediate to note that f : X ! R has a left-sided directional derivative at x0in the dirrection v if and only if f has a right-sided directional derivative at x0 in the direction( v): In this case it holdsD f (x0 ; v) D f (x0 ; v):This explains the fact that in applications (mainly in Convex Analysis and in Optimization)only D f (x0 ; v) is considered. Furthermore, it is immediate that f : X ! R is directionallydi erentiable at x0 2 X in the direction v, if and only ifD f (x0 ; v) D f (x0 ; v);i. e.D f (x0 ; v) D f (x0 ; v);i. e.D f (x0 ; v) D f (x0 ; v) 0:3
It follows also that if Df (x0 ; v) exists, then Df (x0 ; v) Df (x0 ; v); 8 2 R.Example 1. Let be y 2 Rn and let be f : Rn ! R be de ned by f (x) kx yk ; 8x 2 Rn :This function is right-sided directionally di erentiable at every point x0 2 Rn ; in every directionv: The reader is invited to verify that(kvk ; if x0 y; 0D f (x ; v) (x0 t) v; if x0 6 y:kx0 ykRemark 1. De nition 3 is essentially based on a one-dimensional concept: if we put '(t) f (x0 tv); the functioon ' is the restriction of f to the straight line passing through x0 andwith direction v: It is immediate to note that the di erence quotient'(t)'(0)tcoincides with the di erence quotient appearing in De nition 3. Hence, the quantity Df (x0 ; v);if it exists, coincides with the usual derivative '0 (0):Example 2.Compute the directional derivative of the function pf : R2p ! R de ned byf (x; y) x2 y 2 xy x at x0 (1; 1) ; in the direction v (1 2; 1 2) :We have f (x0 tv) '(t) 2 pt2 32 t2 : Hence Df (x0 ; v) '0 (0) p12 3t jt 0 p12 :Remark 2. The existence of the directional derivative at x0 in a certain direction, gives noinformation on the existence of the directional derivatives at x0 in other directions.Example 3. Consider the function f : R2 ! R de ned by(x2; if (x; y) 6 0;x2 y 2f (x; y) 0; if (x; y) 0:The reader is invited to verify that Df (0; e2 ) 0; but Df (0; e1 ) does not exist. Here e1and e2 are the two unit coordinate vectors of R2 : e1 (1; 0) ; e2 (0; 1) :If f : X Rn ! R is directionally di erentiable at x0 2 int(X) in the direction ei ; beinge the i-th unit coordinate vector of Rn ; i. e.iei [0; 0; :::; 1; 0; :::; 0] ;with 1 as the i-th element, we say that f is partially di erentiable at x0 with respect to thei-th variable xi ; and the quantityDf (x0 ; ei )is the partial derivative of f at x0 ; with respect to the i-th variable xi ; and denoted as@f 0(x ) or also fxi (x0 ):@xi4
If f : XRn ! R admits at x0 all n partial derivatives, then the vector@f 0@f 0(x ); :::;(x )@x1@xn is called the gradient of f at x0 and denoted as rf (x0 ):Note that f may have directional derivatives in all nonzero directions at x0 ; yet not becontinuous at x0 : Note, moreover, that we may not be able to express the directional derivativesof a given function at a point x0 as a linear function of the components of the direction v 2 Rn :Example 4. Let f : R2 ! R be de ned as(f (x; y) xy;x2 yy 6 0; y x2 ;x2 :Observe that f has directional derivatives at (0; 0) in every direction:f (tx; ty)f (0; 0)t t2 xyt2 x2 tyt xy: ytx2If y 6 0; the the limit of this expression is x, as t ! 0; and if y 0; the limit is 0: Thusthe directional derivative exists for every direction (x; y); but the function is not continuos atx0 (0; 0):Some authors call the quantity Df (x0 ; v) the “ rst variation”in the sense of Lagrange off at x0 :De nition 4. Let be f : X ! R, with X open subset of Rn ; and let x0 2 X: If D f (x0 ; v)exists for all v 2 Rn ; then f is said to be weakly Gâteaux di erentiable at x0 or also Gâteauxsemidi erentiable at x0 or also, less frequently, hemi-di erentiable at x0 or also Dini di erentiable at x0 or, simply, directionally di erentiable at x0 :If f is weakly Gâteaux di erentiable at x0 and the function v ! D f (x0 ; v) is linear, thenf is said to be Gâteaux di erentiable at x0 : The quantity D f (x0 ; v) is also called “Gâteauxdi erential”of f at x0 or “Gâteaux derivative”of f at x0 :If f is Gâteaux di erentiable at x0 ; the Gâteaux di erential of f at x0 is given by rf (x0 ) vand we have thereforeD f (x0 ; v) rf (x0 ) v; 8v 2 Rn :In other words, it holdsf (x0 tv) f (x0 ) trf (x0 ) v o(t);for t ! 0 :5
Some authors refer to Df (x0 ; v) the de nitions of weak Gâteaux di erentiability andGâteaux di erentiability. This is, for example, the classical approach of Kolmogorov and Fomin(1980), Kantoroviµc and Akilov (1980), Ortega and Rheinboldt (1970), Vainberg (1956), etc.For what concerns Gâteaux di erentiability, the two de nitions are equivalent: indeed linearityimplies that D f (x0 ; v) D f (x0 ; v) D f (x0 ; v) Df (x0 ; v): We point out that thereis not uniformity of notations and de nitions for what concerns the subjects of the presentsection.Note that Gâteaux di erentiability of f at x0 does not imply continuity of f at x0 :Example 5.De ne f : R2 ! R byf (x; y) Then, for x 6 0;f (tx; ty)f (0; 0)t (y(x2x y 2 ); x 6 0;0; x 0:ty 2 2t (xtx y2)t ty 2(x y 2 ) ! 0:xThus D f (0; v) 0 for every v; so f has a Gâteaux derivative at the origin, namelythe zero linear map. However, f is not continuous at the origin. Indeed, consider, e. g.,v(") ("4 ; "): Then v(") ! 0 as " ! 0; butf (v(")) " 8125(" ") " :"4"Thus f (v(")) ! 1 as " ! 0 and f (v(")) !not exist and in any case f is not continuous at (0; 0):1 as " ! 0 ; so lim f (v(")) does" !0Note, moreover, that the existence of all partial derivatives do not assure the existence ofD f (x0 ; v) for all directions v 2 Rn ; nor the Gâteaux di erentiability of f at x0 :Example 6.Consider the function f : R2 ! R de ned by(x1 x2; if (x1 ; x2 ) 6 (0; 0);(x1 )2 (x2 )2f (x1 ; x2 ) 0; if (x1 ; x2 ) (0; 0):This function possesses at x0 (0; 0) both partial derivatives (i. e. Df (x0 ; e1 ) andDf (x0 ; e2 )), but D f (x0 ; e1 e2 ) does not exist.We now give the classical de nition of Fréchet di erentiability.De nition 5. Let f : X ! R, with X Rn open set (or, more generally, X Rn arbitraryand x0 2 int(X)). We say that f is Fréchet di erentiable at x0 (or, simply, di erentiable atx0 ); if there exists a vector a 2 Rn ; depending only from the point x0 ; such thatlimv !0f (x0 v)f (x0 )kvk6a v 0;
i. e.lim 0f (x)x !xf (x0 ) a (xkx x0 kx0 ) 0:It is well-known that the vector a is unique and that a rf (x0 ): In this case the quantityrf (x0 ) v is also called the Fréchet di erential of f at x0 : The previous de nition is alsoequivalent to the following condition:f (x0 v) f (x0 ) rf (x0 ) v o(kvk);for v ! 0 2 Rn : Moreover, it can be shown that Fréchet di erentiability at x0 is equivalentto:For every " 0; there exists 0 such that for every v satisfying kvk ; it holdsf (x0 v)rf (x0 ) v 5 " kvk :f (x0 )This relation will be useful to introduce (see further) the stronger notion of strict di erentiability.Another equivalent condition of Fréchet di erentiability at x0 is the following one (see, e.g., Nashed (1971)).Rn open set and x0 2 X: Then f is FréchetTheorem 2.Let f : X ! R, with Xdi erentiable at x0 if and only iflim t !0f (x0 tv)tf (x0 ) rf (x0 ) vand the convergence is uniform, with respect to v; for v varying in a bounded set of Rn (forexample on B fv : kvk 1g).The following results are well-known.Theorem 3.at x0 ; then:Let be f : XRn ! R, X open, and let x0 2 X: If f is Fréchet di erentiablei) f is continuous at x0 (this is an immediate consequence of the de nition of Fréchet di erentiability).ii)f is directionally di erentiable at x0 in every direction v 2 Rn and it holdsDf (x0 ; v) rf (x0 ) v; 8v 2 Rn :Consequently, f is Gâteaux di erentiable at x0 (the vice-versa does not hold).iii) If ftj g ; j 2 N, is a sequence on the interval (0; 1) converging to zero, and fv j g ; j 2 N,a sequence on Rn f0g ; converging to v 2 Rn ; thenj1f (x0 tj v j )! 1 tjlim7f (x0 ) rf (x0 ) v:
iv)There exist two numbers r 0 and c 0 such that N (x0 ; r)X andf (x0 ) 5 c kvk ; 8v 2 N (0; r):f (x0 v)We give now the de nition of strictly di erentiable functions, a property stronger thanFréchet di erentiability.De nition 6.Let be f : XRn ! R, X open, and let x0 2 X: Then f is strictly0di erentiable at x if there exists a vector a 2 Rn ; which will be the gradient rf (x0 ); such thatf (x tv)tlimx ! x0v !vt ! 0 f (x) rf (x0 ) v; 8v 2 Rn ;or. equivalently,limx ! x0x ! x0f (x)f (x) rf (x0 ) (xkx xkx) 0; x 6 x;i. e.f (x) f (x) rf (x0 ) (xx) o(kxxk):Some authors (e. g. Pourciau (1980), Nijenhuis (1974), Ortega and Rheinboldt (1970)) usethe term “strongly di erentiable”. The above conditions are in turn equivalent to the followingone (see, e. g., Alexéev, Tikhomirov and Fomine (1982)):For every " 0 there exists 0 such that for all x1 and x2 verifying the inequalitiesx1x2x0 ;x0 we have the following inequality:f (x1 )f (x2 )rf (x0 ) (x1x2 ) 5 " x1x2 :The following implications are well known.Strict di erentiability at x0Fréchet di erentiability at x0 )Gâteaux di erentiability at x0 : ) )Moreover,Strict di erentiability at x0 ) Continuity of f on a neighborhood of x0 :8
Fréchet di erentiability at x0 ) Continuity of f at the point x0 :The above implications cannot be reversed. See Example 5 and the following two examples,taken from Alexéev, Tikhomirov and Fomine (1982).Example 7.Consider the function f : R2 ! R given by(1; if x1 (x2 )2 ; x2 0;f (x1 ; x2 ) 0; at all other points.This function is Gâteaux di erentiable at the origin (0; 0); but it is not continuous at thesame point.Example 8.Consider the function f : R ! R given by(x2 ; if x is rational;f (x) 0; if x is not rational.At x0 0 this function is Fréchet di erentiable, but it is not strictly di erentiable at thesame point, as it is not continuous for x 6 0:De nition 7.Let be f : XRn ! R, X open, and let x0 2 X:i) The function f is said to be of C 1 -class at x0 ; and denoted by f 2 C 1 (x0 ); if its gradientrf (x) exists in a neighborhood of x0 and is continuous at x0 :ii) The function f is said to be of C 1 -class on X; and denoted by f 2 C 1 (X); if rf (x) iscontinuous for all x 2 X: In this case f is also said to be continuously di erentiable on X:The following su cient condition for Fréchet di erentiability is well known.Theorem 4.Let be f : XRn ! R, X open, and let x0 2 X: If f 2 C 1 (x0 ); then f is0Fréchet di erentiable at x and also strictly di erentiable at x0 : If f 2 C 1 (X); then f is Fréchetdi erentiable on X:The second part of the previous theorem can be made more precise, on the ground of thefollowing result (see, e. g., Rockafellar and Wets (2009)).Theorem 5. Let be f : Xdi erentiable on X:Rn ! R, X open. Then f 2 C 1 (X) if and only if f is strictlyAnother classical notion, useful for further considerations, is the de nition of Lipschitzcontinuous functions and locally Lipschitz continuous functions.De nition 8. Let X Rn be a nonempty set and f : X ! R. The function f is said tobe Lipschitz over X (or Lipschitz continuous over X) if there exists a real number k 0 suchthat, for every x1 ; x2 2 X we havef (x1 )f (x2 ) 5 k x19x2 :(1)
The smallest constant k for which the previous relation holds is called “the Lipschitzconstant” or “the Lipschitz rank”. If k 1; then f is said to be non-expansive and if k 1;then f is said to be a contraction.Note that if f is Lipschitz on X; then it is (uniformly)continuous on X; but the converse isp3not true: take, e. g. the continuous function f (x) x; x 2 R; with x2 0 we see that thereis no constant k 0 satisfying (1). To understant the meaning of (1), rewrite it as followsjf (x1 )kx1f (x2 )j5 k; 8x1 6 x2 2 X:x2 kHence. a function is Lipschitz on the set Xare bounded.Rn if and only if all its di erence quotientsExample 9.i)The function f (x) kxk ; x 2 Rn ; is Lipschitz on Rn ; with k 1:ii) The function f (x) kxk2 is not Lipschitz on the whole space Rn : Indeed, by choosingx2 0; we havex125 k x1which holds only if kx1 k 5 k:A su cient condition for f to be Lipschitz on a set contained in its domain, is given bythe following proposition.Theorem 6. Let be f : X Rn ! R, with X convex set. If f is di erentiable on X andif all its partial derivatives are bounded on X; then f is Lipschitz on X: Moreover, for everyM 0 such that@f(x) 5 M; 8x 2 X; 8i 1; :::; n;@xipthen relation (1) holds with k nM:De nition 9. Let X Rn be a nonempty open set and f : X ! R. For a point x0 2 X;if there exist a neighborhood N (x0 ) of x0 a nonnegative number k such thatf (x1 )f (x2 ) 5 k x1x2 ; 8x1 ; x2 2 N (x0 );then f is said to be locally Lipschitz at x0 or Lipschitz near x0 or Lipschitz around x0 ; withconstant k:We say that f is locally Lipschitz on X is f is locally Lipschitz at each x 2 X:Thus a function which is locally Lipschitz at a point means that the function satis es theLipschitz condition in a neighborhood of that point. However, it is important to note that the10
value of the Lipschitz constant k in general could change as we change the point. Obviouslywe have the implicationRn (X open) )f Lipschitz on X ) f locally Lipschitz at each point of X;but the converse is not in general true. If however a locally Lipschitz function has a uniformLipschitz constant k at every point x0 2 X; then f is Lipschitz on X in the sense of De nition8. A su cient condition for f to be locally Lipschitz at a point x0 of its domain is given by thefollowing proposition.Theorem 7. If a function f : X Rn ! R is continuously di erentiable (i. e. of C 1 class)in a neighborhood of x0 2 int(X); then f is locally Lipschitz at x0 :Proof. Continuous di erentiability around x0 means that all n partial derivatives of f arecontinuous on a neighborhood of x0 : It follows that there exist constants " 0 and k 0 suchthatkrf (x)k 5 k; for all x 2 N (x0 ; "):Suppose that x1 ; x2 2 N (x0 ; "): Then, by the classical Mean-Value Theorem, there isz 2 (x1 ; x2 ) N (x0 ; ") such thatf (x1 )f (x2 ) rf (z) (x1x2 ):We now havef (x1 )f (x2 ) 5 krf (z)k x1x2 5 k x1x2 ;i. e. f is Lipschitz continuous at x0 :Theorem 7. can be weakened:it can be proved (see, e. g., Nijenhuis (1974)) that if f isstrictly di erentiable at x0 2 int(X); then f is locally Lipschitz at x0 :Other classical directional derivatives used in optimization are the Dini directional derivatives, introduced by U. Dini (1878) for real functions of one variable. For various considerationsand applications of Dini directional derivatives, see, e. g. Bector, Bhatia and Jain (1993),Crouzeix (1981, 1988), Diewert (1981), Giorgi and Komlosi (1993a, b, 1995), Glover (1984),Komlosi (1983, 1995, 2005).De nition 10.Let be f : XRn ! R, X open and x0 2 X: The quantities f (x0 tv)tf (x0 )f (x0 tv)tf (x0 )f D (x0 ; v) lim supt !0 fD (x0 ; v) lim inf t !011;(2);(3)
are, respectively, called the (right-sided) upper Dini directional derivative of f at x0 in thedirection v 2 Rn and the (right-sided) lower Dini directional derivative of f at x0 in thedirection v 2 Rn .Obviously, it is possible to de ne also the left-sided Dini directional derivatives f D (x0 ; v)and fD (x0 ; v):If the limitf (x0 tv) f (x0 )lim D f (x0 ; v)(4)t !0texists, then f D (x0 ; v) fD (x0 ; v) D f (x0 ; v):Note that the limits in (2), (3) always exist ( nite or not), whereas the limit in (4) notalways exists. It is easy to see that, for any v 2 Rn ;f D (x0 ; v) fD (x0 ; v): We must also note that Dini derivatives are positively homogeneous (of degree one) intheir second argument, i. e. with respect to the direction v and for all 0 we have f D (x0 ; v) f D (x0 ; v)andfD (x0 ; v) fD (x0 ; v):Other “classical” directional derivatives were introduced by the French mathematicianJacques Hadamard.De nition 11.Rn ! R, X open and x0 2 X: The quantitiesLet be f : Xf (x0 tv)t!vf (x0 )f (x0 tv)(x ; v) liminft !0 ; v !vtf (x0 ) f H (x0 ; v) lim supt !0 ; vfH 0;are called, respectively, the Hadamard (right-sided) upper directional derivative of f at thepoint x0 in the direction v 2 Rn and the Hadamard (right-sided) lower directional derivative off at the point x0 in the direction v 2 Rn :Some authors call the above quantities “Dini-Hadamard directional derivatives”; Aubinand Cellina (1984) speak of “contingent derivatives”, whereas Penot (1978) uses the term“semiderivatives”. See also further. We are not sure that there are not other denominations inthe literature. Also in this case it is obviously possible to de ne Hadamard left-sided directionalderivatives f H (x0 ; v) and f H (x0 ; v): Note that the limits of De nition 11 always exist butare not nwecessarily nite. Note also that in the one-dimensional case, i. e. Rn R, the12
Hadamard directional derivatives coincide with the corresponding Dini directional derivatives.If we have that the limitlim t !0 ; vf (x0 tv)!vtf (x0 )DH f (x0 ; v)exists, then f is called Hadamard directionally di erentiable at x0 in the direction v 2 Rn(many authors require that the above limit must exist nite). Some authors (e. g. Delfour(2020), Rockafellar and Wets (2009)) speak in this case of the semiderivative of f at x0 in thedirection v 2 Rn and say that f is semidi erentiable at x0 in the direction v 2 Rn : If thisholds for every v 2 Rn ; we say that f is Hadamard directionally di erentiable at x0 or that fis semidi erentiable at x0 : If DH f (x0 ; 0) exists, then it holds DH f (x0 ; 0) 0:It is possible to show (see, e. g., Shapiro (1990)) that the above de nition of Hadamarddirectional di erentiability is equivalent to the following proposition:For any mapping ' : R ! Rn such that'(0) x0 and such thatthe limitlim '(t)'(0)! v for t ! 0 ;tf (x0 )f ('(t))tt !0does exist. Then it holdsDH f (x0 ; v) lim t !0f (x0 )f ('(t))t:The above characterization gives the Hadamard directional derivative DH f (x0 ; v) alonga curve tangential to v: Indeed, some authors speak also, for the case under examination, oftangential directional derivative. Other authors (Craven (1986), Craven and Mond (1979))speak of arcwise directionally di erentiable functions. Obviously the Hadamard directionalderivative DH f (x0 ; v) can also be expressed in terms of sequences:f (x0 tn v n )!1tnDH f (x0 ; v) limnf (x0 );where fv n g Rn and ftn g R are any sequences such that v n ! v and tn ! 0 :Robinson (1987) introduced the concept of Bouligand di erentiability (he called this property “B-di erentiability”). However, for real-valued functions on Rn ; this property is equivalentto Hadamard directional di erentiability, as de ned before (see Rockafellar and Wets (2009),page 294).Note that if DH f (x0 ; v) exists, then also the directional derivative D f (x0 ; v) exists andDH f (x0 ; v) D f (x0 ; v):13
The converse is not necessarily true.Example 10.Consider the function f : R2 ! R de ned by(0; if either x2 (x1 )2 or x2 5 0;f (x1 ; x2 ) 1; in all other cases.Let be x0 (0; 0) : We have D f (x0 ; v) 0; 8v 2 R2 : Moreover, we have DH f (x0 ; v) 0;8v (v1 ; v2 ) 2 R2 ; with v2 6 0: DH f (x0 ; v) does not exist for v (v1 ; 0) ; v1 2 R.An important property of locally Lipschitz functions, which guarantees the converse of theabove result, is contained in the following proposition.Theorem 8.Let be f : XRn ! R, X open and x0 2 X; if f is locally Lipschitzat x0 and D f (x0 ; v) exists, then f is also Hadamard directionally di erentiable at x0 in thedirection v and it holdsD f (x0 ; v) DH f (x0 ; v):Also the Hadamard directional derivatives are positively homogeneous of degree one withrespect to the direction v 2 Rn :De nition 12. Let be f : X Rn ! R, X open and x0 2 X: We say that f is Hadamarddi erentiable at x0 if DH f (x0 ; v) exists ( nite) for all v 2 Rn and this quantity depends linearlyon v 2 Rn :In the above case we haveDH f (x0 ; v) rf (x0 ) v; 8v 2 Rn ;and as a consequence, in nite-dimensional spaces, such as Rn ; we have that Hadamard di erentiability coincides with Fréchet di erentiability.Theorem 9. Let be f : X Rn ! R, X open and x0 2 X: Then f is Fréchet di erentiableat x0 if and only if, for all v 2 Rn ; it holdslim t !0 ; vf (x0 tv)!vtf (x0 ) rf (x0 ) v:We have to note that some authors call “Hadamard di erentiable” what we have called“Hadamard directionally di erentiable”, perhaps because in nite-dimensional spaces Hadamarddi erentiability coincides with Fréchet di erentiability. Also for Hadamard directional derivatives there is not uniformity of notations and de nitions. Often, in the literature, instead ofthe limits written in the formt ! 0 ; v ! v14
(for the Hadamard directional derivatives), the same limits are taken in the form(t; v) ! (0 ; v):The two forms not necessarily coincide. See the paper of F. Giannessi (1995).Theorem 10. If DH f (x0 ; ) exists ( nite) in a neighborhood of v 2 Rn ; then DH f (x0 ; ) iscontinuous at x0 :Proof. By assumption there exists 0 0 such that DH f (x0 ; v) exists for all v 2 B(v; 0 ):Let be given " 0: Then there exists 2 (0; 0 ) such that for all t 2 (0; ) and all v 2 B(v; )it holdsf (x0 tv) f (x0 )DH f (x0 ; v) 5 ":tFor t ! 0 it follows, for all v 2 B(v; );D f (x0 ; v)DH f (x0 ; v) 5 ":Since D f (x0 ; v) DH f (x0 ; v); the thesis follows.We have seen (Theorem 8) that if f is locally Lipschitz at x0 ; then the existence ofD f (x0 ; v) implies the existence of DH f (x0 ; v) and the equality D f (x0 ; v) DH f (x0 ; v): Asa consequence we have the following important result.Theorem 11.Let be f : XRn ! R, X open, x0 2 X and f locally Lipschitz at x0 :Then the following properties are equivalent.(a)f is Gâteaux di erentiable at x0 ;(b)f is Hadamard di erentiable at x0 :We recall that, being X nite-dimensional, under the assumption of Theorem 11 we havealso the equivalence between Gâteaux di erentiability and Fréchet di erentiability. Obviously,for f : R !R Fréchet, Hadamard and Gâteaux di erentiability at x0 are equivalent conceptsand coincide with the usual classical derivative of f at x0 : It is worth noting, furthermore, thatif f : X Rn ! R, X open and x0 2 X; has a ( nite) directional Hadamard derivative at x0in all directions v 2 Rn ; i. e. DH f (x0 ; v) exists ( nite) for all v 2 Rn ; then f is continuous atx0 ; but not necessarily locally Lipschitz continuous at x0 (see, e. g., Demyanov and Rubinov(1995), Delfour (2020)).We recall that a Gâteaux di erentiable function is not necessarily continuous (Example 5).Obviously, if f : XRn ! R is Hadamard di erentiable at x0 2 X; X open, i. e. it isFréchet di erentiable at x0 ; then it is continuous at x0 :For directionally di erentiable functions and Hadamard directionally di erentiable functions there exist calculus rules (sum, di erence, product and quotient): if f1 and f2 are, forexample, Hadamard directionally di erentiable at a point x; then their sum, di erence, product15
and quotient (if f2 (x) 6 0) are also Hadamard directionally di erentiable at x and the followingformulas hold.DH (f1 f2 )(x; v) DH f1 (x; v) DH f2 (x; v);DH (f1 f2 )(x; v) f1 (x)DH f2 (x; v) f2 (x)DH f1 (x; v);DHf1f2(x; v) 1f1 (x)DH f2 (x; v)2(f2 (x))f2 (x)DH f1 (x; v) :Unfortunately, formulas similar to these ones are no longer valid for Dini and Hadamardupper and lower directional derivatives.3. Directional Derivatives in Convex and Generalized ConvexFunctionsDirectional derivatives play an important role in Convex Analysis and OptimizationTheory. In the present section we give an overview of the main properties of convex and generalized convex functions with regard to directional derivatives. We begin with convex functions;for the related proofs, see, e. g., the fundamental book of Rockafellar (1970) and the books ofBagirov, Karmitsa and Mäkelä (2014), Bertsekas (2009), Bertsekas, Nedic and Ozdaglar (2003),Borwein and Lewis (2000), Dhara and Dutta (2012), Durea and Strugariu (2014), Giorgi, Guerraggio and Thierfelder (2004), Hiriar
If f: XˆRn!Radmits at x0 all npartial derivatives, then the vector @f @x 1 (x0);:::; @f @x n (x0) is called the gradient of fat x0 and denoted as rf(x0): Note that f may have directional derivatives in all nonzero directions at x0;yet not be continuous at x0:Note, moreover, that we may not be able to express the directional derivatives of a given function at a point x0 as a linear function of .
Directional derivatives and gradient vectors (Sect. 14.5). I Directional derivative of functions of two variables. I Partial derivatives and directional derivatives. I Directional derivative of functions of three variables. I The gradient vector and directional derivatives. I Properties of the the gradient vector.
4.2.1 directional derivatives and the gradient in R2 Now that we have a little experience in partial differentiation let's return to the problem of the directional derivative. We saw that . This is the formula I advocate for calculation of directional derivatives. This formula most elegantly summarizes how the directional derivative works.
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Abstract. Directional derivatives can be thought of as general-izations of partial derivatives which describe how a multivariable function changes as you change the inputs to the function by walk-ing along a line in the xy-plane. One important application of directional derivatives appears in an algorithm called the simplex
Derivatives of Trig Functions – We’ll give the derivatives of the trig functions in this section. Derivatives of Exponential and Logarithm Functions – In this section we will get the derivatives of the exponential and logarithm functions. Derivatives of Inverse Trig Functions – Here we will look at the derivatives of inverse trig functions.
Ch21 Carboxylic acid Derivatives(landscape).docx Page 1 Carboxylic acid Derivatives Carboxylic acid derivatives are described as compounds that can be converted to carboxylic acids via simple acidic or basic hydrolysis. The most important acid derivatives are esters, amides and nitriles, although acid halides and anhydrides are also derivatives (really activated forms of a File Size: 2MB
Partial and Directional derivatives, Di erentiability Directional derivatives of f : Rn!R Let f : Rn!R and a 2Rn:Also let u 2Rn with kuk 1: Then the limit, when exists, D uf(a) : lim t!0 f(a tu) f(a) t d dt f(a tu) jt 0; rate of change of f at a in the direction u; is calleddirectional derivativeof f at a in the direction u: D uf(a .
Scrum for Video Game Development Mike Cohn - background Mountain Goat Software, LLC 1 2 Wednesday, January 23, 2008
