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On Clarke-Mititelu subdifferential.

1. INTRODUCTION AND PRELIMINARIES

The first model of generalized gradient is the subdifferential of convex functions, introduced by Moreau [15], and founded by Rockafellar [19].

Generalized gradient models were developed taking into account practical reasons, see [18] by Pshenichnyi, [3] by Clarke, [20] by Rockafellar, [6] by Demyanov and Dixon, [8] by Hiriart-Urruty, [11] by Mititelu.

The first model called quasidifferential is introduced by Pshenichnyi [18] in the framework given in the following.

Let E be a topological linear space and [E.sup.*] its dual. The function f: E [right arrow] R is called quasidifferentiable at the point [x.sup.0] [member of] E if the direction derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

exists for all direction v [member of] E. The quasidifferential (in the sense of Pshenichnyi) of the function f at the point [x.sup.0] is a weakly compact set [M.sub.f]([x.sup.0]) of [E.sup.*], defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or, in equivalent form,

Mf([x.sup.0]) = {[xi] [member of] [E.sup.*] | f'([x.sup.0]; v) [greater than or equal to] <[xi], v>, [for all]v [member of] E}.

The following two remarks are in order:

1) if the function f is Gateaux differentiable at the point [x.sup.0], then the function f is quasidifferentiable at [x.sup.0] and [M.sub.f]([x.sup.0]) = {[nabla]f([x.sup.0])};

2) the quasidifferential [M.sub.f] is defined by analogy with the subdifferential of convex functions [11], [15], [19].

One of the most important model of generalized gradients is called subdifferential and is introduced by Clarke [3], for Lipschitz functions.

Using various types of direction derivatives, many subsequent theories have developed the concept of subdifferential for nonsmooth functions. For an overall view, the reader is referred to the next monographs: [3] by Clarke, [6] by Demyanov and Dixon, [9] by Kusraev and Kutateladze, [11] by Mititelu, [17] by Penot, and [20] by Rockafellar, as well as to the following research papers: [7] by Demyanov and Rubinov, [12]/[14] by Mititelu, [16] by Morzhin, [21] by Tolstonogov.

2. CLARKE STRUCTURE

Let be given f : [R.sup.n] [right arrow] R a locally Lipschitz function around a point [x.sup.0]. The Clarke derivative of the function f at the point [x.sup.0] on the direction of v in [R.sup.n] is the direction derivative

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The set

[partial derivative]f([x.sup.0]) = {[xi] [member of] [R.sup.n] | [f.sup.0]([x.sup.0]; v) [greater than or equal to] <[xi], v>, [for all]v [member of] [R.sup.n]}

is called Clarke subdifferential or generalized gradient of the function f at the point [x.sup.0].

Since any locally Lipschitz is almost everywhere differentiable in the sense of Lebesgue measure, the subdifferential admits the equivalent representation [4]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where D is the set in [R.sup.n] where f is differentiable and [nabla]f is its derivative.

If the function is lower semicontinuous, then [3], [4]

[partial derivative]f(x) = {[xi] [member of] [R.sup.n] | ([xi], -1) [member of] [N.sub.epif] (x, f (x))},

where [N.sub.epif] is the Clarke normal cone to the epigraph of the function f at the point (x, f(x)) [member of] epif.

According to Clarke [5], some specific properties of a locally K-Lipschitz function in a neighborhood of a point x are:

1) [absolute value of [f.sup.0](x; v)] [less than or equal to] K[parallel]v[parallel];

2) [absolute value of [f.sup.0](x; [v.sub.1]) - [f.sup.0](x; [v.sub.2])] [less than or equal to] K[parallel][v.sub.1] - - [v.sub.2][parallel];

3) [parallel][xi] [parallel] [less than or equal to] K, [for all][xi] [member of ] [partial derivative]f(x).

This theory allowed the development of nonsmooth analysis, with wide applications in Mathematical Programming, Variational Problems, Numerical Analysis. For some details, see [5], [9], [22], [23].

However, the subdifferential of Clarke may be generalized to a wider class of functions. That is why, this theory was extended to the class of non-Lipschitzian functions, a new structure being introduced by Stefan Mititelu in his works [12], [13] and [14].

3. MITITELU STRUCTURE

The structure proposed by Stefan Mititelu is an extension of the subdifferential in the sense of Clarke to general nonsmooth functions.

Let be given f: A [right arrow] R an arbitrary function, defined on the open set A in [R.sup.n]. The following properties are remarkable. For a complete study of subdifferentials of the functions obtained by algebraic operations with almost everywhere Frechet differentiable functions, we address the reader to the monograph [11], as well as to the research works [12], [13] and [14].

Proposition 3.1. Suppose the function [f.sup.0](x; v) is finite.

1) The function [f.sup.0](*; v) is upper semicontinuous at the point x.

2) If f is continuous around x, the function [f.sup.0] is upper semicontinuous at the point (x; v).

Proposition 3.2. The function [f.sup.0](x; *) is sublinear, that is subadditive and positive homogeneous on [R.sup.n].

Proposition 3.3. Suppose the point x belongs to A and [f.sup.0](x; *) is finite. Then, there exists f'(x; v), that is the direction derivative of the function f at the point x, on the direction of v. Moreover, f'(x; v) = [f.sup.0](x; v), for all v [member of] [R.sup.n].

In [12], Mititelu introduced

Definition 3.1. Let be given f: A [right arrow] R an arbitrary function, where A is open in [R.sup.n], and the point x in A. The set

[partial derivative]f(x) = {[xi] [member of] [R.sup.n] | [f.sup.0](x; v) [greater than or equal to] <[xi], v>, [for all]v [member of] [R.sup.n]}

is called subdifferential or generalized gradient of the function f at the point x.

If [partial derivative]f(x) [not equal to] [empty set], then the function f is called subdifferentiable at the point x. The elements of [partial derivative]f(x) are called subgradients.

Denote by D([f.sup.0.sub.x])the effective domain of the function [f.sup.0](x; *), that is the set

D([f.sup.0.sub.x]) = {v [member of] [R.sup.n] | [f.sup.0]x; v) < [infinity]}.

When [f.sup.0](x; *) [not equal to] -[infinity], then f is proper convex. It follows that the domain D([f.sup.0.sub.x]) is a nonempty convex set.

We underline that if [f.sup.0](x; v)= -[infinity], for all v [member of] [R.sup.n], then [partial derivative]f(x) = [empty set]. That is why, to give a complete characterization of the model proposed by Mititelu, in the following we shall suppose [partial derivative]f(x) [not equal to] [empty set].

In [12], St. Mititelu shows that the infinite values of the function [f.sup.0](x; *) are not essential in the determination of the subdifferential [partial derivative]f(x), essential being its finite values only. More exactly, we have the following two results.

Theorem 3.1. If the function f is subdifferentiable at the point x, then [partial derivative]f(x) admits the representation

[partial derivative]f (x) = {[xi] [member of] [R.sup.n] | [f.sup.0](x; v) [greater than or equal to] <[xi], v>, [for all]v [member of] D([f.sup.0.sub.x])}.

Theorem 3.2. The following statements hold true:

If [f.sup.0](x; v) = -[infinity], for all v [member of] [R.sup.n], then f is subdifferentiable at x [member of] A, the subdifferential [partial derivative]f (x) is a nonempty closed convex set, and

[f.sup.0](x; v) = sup{<[xi], v> | [xi] [member of] [partial derivative]f (x)}, [for all]v [member of] [R.sup.n].

Moreover, if [f.sup.0](x; *) is finite, then [partial derivative]f(x) is compact and

[f.sup.0](x; v) = max{<[xi], v> | [xi] [member of] [partial derivative]f(x)}, [for all]v [member of] D([f.sup.0.sub.x]).

The subdifferential [partial derivative]f(x) of an almost everywhere Frechet differentiable function f: A [right arrow] R, around a point x [member of] A, admits an equivalent representation as convex hull of the limit normals of f at the point x. This is proved by the following two results.

Proposition 3.4. Let x be given a point in A and the sequences ([x.sub.k]) and ([[xi].sub.k]) in A and [R.sup.n] respectively. If [x.sub.k] [right arrow] x in A and [[xi].sub.k] [right arrow] [xi] in [R.sup.n] such that [[xi].sub.k] [member of] [partial derivative]f(xk),then [xi] [member of] [partial derivative]f(x).

Denote by [D.sub.f] the set of all x where f is almost everywhere Frechet differentiable around the point x.

Theorem 3.3 (Mititelu, [12]). Suppose the function f is almost everywhere Frechet differentiable around the point x and [f.sup.0](x; *) is finite. Then f is subdifferentiable around the point x and the subdifferential [partial derivative]f (x) admits the representation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Mititelu structure allowed to obtain new versions of some classical results in smooth analysis, [13]. To emphasize these results, we need

Definition 3.2 (Mititelu, [13]). Let be given f: A [right arrow] R an arbitrary function, defined on the open set A in [R.sup.n]. The point x [member of] A is called Clarke critical point of the function f if [f.sup.0](x; v) [greater than or equal to] 0, for all v [member of] [R.sup.n].

Remark that the inequality in Definition 3.2 is equivalent to 0 [member of] [partial derivative]f(x).

The following results are versions of Fermat Theorem and Rolle Theorem, respectively, in nonsmooth framework.

Theorem 3.4 (Mititelu, [13]). If x [member of] A is an extremum point of the function f, then x is a Clarke critical point off.

Consider a and b be two distinct points in the set A. Introduce the set [a, b] (closed interval in A)by [a, b] = {[lambda]a + (1 - [lambda])b | 0 [less than or equal to] [lambda] [less than or equal to] 1}.

Theorem 3.5 (Mititelu, [13]). If the function f satisfies the conditions:

1) f(a) = f(b); 2) f is continuous on [a, b],

there exists a point c [member of] (a, b) such that either [f.sup.0](c; b - a) [greater than or equal to] 0 or [f.sup.0](c; a - b) [greater than or equal to] 0.

Consider again f: A [right arrow] R an arbitrary function, defined on the open set A in [R.sup.n] and the compound function [phi]:[0, 1] [right arrow] R, [phi](t) = f([x.sub.t]), where [x.sub.t] = a + t(b - a). In this setting, we have

Lemma 3.1. If the function f is subdifferentiable on (a, b) and [phi] is subdifferentiable on (0, 1), then for all t [member of] (0, 1),

[partial derivative][phi](t) [subset or equal to] [partial derivative]f([x.sub.t]), b - a > = {< [xi], b - a > | [xi] [member of] [partial derivative]f([x.sub.t])}.

Using Lemma 3.1 and Theorem 3.5, we obtain a generalization of the well known mean theorem of Lagrange. This is given in

Theorem 3.6 (Mititelu, [13]). Suppose the function f is continuous on [a,b]. Then there exist a point c [member of] (a, b), where f is subdifferentiable, and a subgradient [xi] [member of] [partial derivative]f(c) such that

f(b) - f(a) = <[xi], b - a> [member of] <[partial derivative]f(c), b - a> .

The following result is a version of Cauchy Theorem in nonsmooth framework.

Theorem 3.7 (Mititelu, [13]). Suppose the functions f and g satisfy the following two conditions:

1) f and g are continuous on [a, b], 2) [g.sup.0](x; b - a) < 0, for all x [member of] (a, b).

Then, there exist a point c [member of] (a, b), where f and g are subdifferentiable, and the subgradients [xi] [member of] [partial derivative]f(c) and [eta] [member of] [partial derivative]g(c) such that

[f(b) - f(a)/g(b) - g(a)] = [<[xi], b - a>/<[eta], b - a>.

Proof. For a proof, see [13]. []

Looking at the result in Theorem 3.7, we can easily conclude why l'Hospital rules from smooth real analysis cannot be applied to the functions of several variables.

We continue our considerations by pointing out a result on the subdifferential of composite functions. In this respect, consider L([R.sup.n], [R.sup.m]) the space of linear functionals defined on [R.sup.n] with values on [R.sup.m]. Denote by [F.sup.0](x; v) the Clarke directional derivative of a vector function F [member of] L([R.sup.n], [R.sup.m]), at a point x in A, where A [member of] [R.sup.n], on the direction of v [member of] Rn. This one is defined by analogy to [f.sup.0](x; v). Moreover, if F = ([F.sub.1],..., [F.sub.m]), then [F.sup.0](x; v) = ([F.sup.0.sub.1](x; v),...,[F.sup.0.sub.m](x; v)) and

[partial derivative]F(x) = ([partial derivative][F.sub.1](x),...,[partial derivative][F.sub.m](x)) = [partial derivative][F.sub.1](x) x ... x [partial derivative][F.sub.m](x).

is the subdifferential of the vector function F at the point x.

Using this background, we are now in position to introduce

Definition 3.3. The vector function F: A [right arrow] [R.sup.n] is upper differentiable at the point x [member of] A if the Clarke directional derivative [F.sup.0](x; *) is linear on [R.sup.n], that is there exists [D.sup.s]F(x) [member of] L([R.sup.n], [R.sup.m]) such that

[F.sup.0](x; v) = <[D.sup.s]F(x), v>, [for all]v [member of] [R.sup.n].

The element [D.sup.s]F(x) in Definition 3.3 is called the upper derivative of the function F at the point x.

Theorem 3.8 (Mititelu, [14]). Given the functions F: A [right arrow] [R.sup.m] and g: [R.sup.m] [right arrow] R, consider the composite function g [omicron] F: A [right arrow] R such that the following conditions are fulfilled:

1) The function F is continuous around the point x in A and admits upper derivative [D.sup.s]F(x);

2) The function g is continuous around the point F(x) in [R.sup.m].

On these conditions, the subdifferential [partial derivative](g [omicron] F)(x) is not empty and satisfies the inclusion

[partial derivative](g [omicron] F)(x) [subset or equal to] [partial derivative]g(F(x)) [omicron] [D.sup.s]F(x) = {z [omicron] [D.sup.s]F(x) | z [member of] [partial derivative]g(F(x))}.

Moreover, if F is one to one function, the above inclusion becomes equality.

Proof. For a proof in detail, see the monograph [11] and the research work [14]. []

The following two results are direct consequences of Theorem 3.8.

The first one is an Euler type theorem. This is given in

Corollary 3.1. Let C be an open cone in [R.sup.n] and f: C [right arrow] R a subdifferentiable function. Suppose that the function f is homogeneous of degree m. Then

mf(x) [member of] <[partial derivative]f(x), x>, [for all]x [member of] C.

The second one is a local inversion type theorem. This is given in

Corollary 3.2. Let be given the open sets A and I in [R.sup.n] and R respectively. Consider the function f: A [right arrow] I, its inverse [f.sup.-1]: I [right arrow] A, and the points x [member of] A and y = f(x) [member of] I.

1) If the function f is continuous around x and its inverse [f.sup.-1] admits upper derivative [D.sup.s][f.sup.-1](y), then 1 [member of] < [partial derivative]f(x), [D.sup.s][f.sup.-1](y) >.

2) If the function f is continuous around the point x and admits upper derivative [D.sup.s]f(x) and its inverse [f.sup.-1] is subdifferentiable at y, then [I.sub.n] [member of] < [partial derivative][f.sup.- 1](y), [D.sup.s]f(x) >, where [I.sub.n] is the unit matrix.

A remarkable particular case of Corollary 3.2 is obtained for n = 1. In this case, if the function f is differentiable at the point x, with f'(x) [not equal to] 0, then its inverse is differentiable at the point y and the following formula holds:

([f.sup.-1])'(y) = 1/f'(x),

We underline that the result listed above is the well-known local inversion formula from smooth analysis.

For a geometrical interpretation of the subdifferential calculus, we address the reader to the monograph [11]. Here it is shown that the geometry of tangent and normal cones at a point, proper to Lipschitzian calculus, is preserved for arbitrary nonsmooth functions too.

4. CONCLUSION

The generalized subdifferential introduced by Mititelu set up conditions for developing a complete nonsmooth analysis. According to Mititelu, this concept is called Clarke-extended subdifferential. However, to the best of our knowledge, we think it can be called Clarke-Mititelu subdifferential.

This work emphasizes the role of certain scientists in designing algorithms research based on generalized gradients, more accurately to the development of non-smooth analysis [3]/[20]. Our note suggests various links between this topic and Geometrical Methods in tatistics [1], Mathematical Modeling in Ecology [2], Economics [9], Optimal Control [10], [16], [21], Generalized Convexity [11], as well as Optimization Methods on Manifolds [22], [23].

Acknowledgement. We like to acknowledge the continued support and encouragements provided by Prof. Dr. Stefan Mititelu while this work was in preparation. We are also thankful to the unknown referees for their constructive comments helping to give the present form to the paper.

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[3] F.H. Clarke: Necessary Conditions for Nonsmooth Problems in Optimal Control and the Calculus of Variations, Ph. D. Thesis, Washington Univ., 1973.

[4] F. H. Clarke: Generalized gradients and applications, Trans.Am. Math.Soc., 205(1975), 247-262.

[5] F. H. Clarke: Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983.

[6] V. F. Demyanov and L. C. W. Dixon (Eds.): Quasidifferential Calculus, Math. Programming Study 29, North-Holland, Amsterdam, 1986.

[7] V. F. Demyanov and A. M. Rubinov: On quasidifferentiable functionals, Sov. Math. Dokl., 21(1980), 14-17.

[8] J. B. Hiriart-Urruty: New concepts in nondifferentiable programming, Bull. Soc. Math. Fr., Supl. Mem., 60(1979), 57-85.

[9] A. G. Kusraev and S. S. Kutateladze: Subdifferentials: Theory and Applications, Mathematics and Its Applications, 323, Kluwer, 1995.

[10] P. D. Loewen and R. T. Rockafellar: Optimal control of unbounded differential inclusions, SIAM J. Control Optim., 32(1994), No. 2, 442-470.

[11] St. Mititelu: Generalized Convexities, Monographs and Textbooks 9, Geometry Balkan Press, Bucharest, 2009 (in Romanian).

[12] St. Mititelu: Generalized subdifferential calculus, An. Stiint. Univ. "Ovidius" Constanta, Ser. Mat., 1(1993), No. 1, 13-22.

[13] St. Mititelu: Nonconvex subdifferential calculus, An. Stiint. Univ. "Ovidius" Constanta, Ser. Mat., 3(1995), No. 1, 118-126.

[14] St. Mititelu: Chain rules and applications in nonsmooth analysis, An. Stiint. Univ. "Ovidius" Constanta, Ser. Mat., 6(1998), No. 1, 131-140.

[15] J. J. Moreau: Functionelles sous-differentiables, C. R. Acad. Sci. Paris, 257(1963), 4117-4119.

[16] O. V. Morzhin: On approximation of the subdifferential of the nonsmooth penalty functional in the problems of optimal control, Autom. Remote Control, 70(2009), No. 5, 761-771.

[17] J. Penot: Calcul sous-differentiel et optimisation, Publications Mathematiques de l'Universite de Pau, 1974.

[18] B. N. Pshenichnyi: Necessary Conditions for an Extremum, Marcel Dekker, New York, 1971.

[19] R. T. Rockafellar: Convex Analysis, Princeton University Press, 1970.

[20] R. T. Rockafellar: La theorie de sous-gradients et ses applications a l'optimisation, Les Presses de l'Universite de Montreal, 1979.

[21] A. A. Tolstonogov: Relaxation in nonconvex optimal control problems with subdifferential operators, J. Math. Sci, 140(2007), No. 6, 850-872.

[22] C. Udriste, G. Bercu and M. Postolache: 2_D Hessian Riemannian manifolds, J. Adv. Math. Stud., 1(2008), No. 1-2, 135-142.

[23] C. Udriste: Convex Functions and Optimization Methods on Riemannian Manifolds, Mathematics and Its Applications, 297, Kluwer, 1994.

Dedicated to Professor Stefan Mititelu on the occasion of his seventieth birthday

MIHAI POSTOLACHE AND OLTIN DOGARU

University "Politehnica" of Bucharest

Faculty of Applied Sciences

Splaiul Independentei, No. 313, 060042 Bucharest, Romania

E-mail address: mihai@mathem.pub.ro

University "Politehnica" of Bucharest

Faculty of Applied Sciences

Splaiul Independenttei, No. 313, 060042 Bucharest, Romania

E-mail address: oltin.horia@yahoo.com

Received: October 23, 2009. Revised: May 12, 2010.

2010 Mathematics Subject Classification: 26A51.
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Author:Postolache, Mihai; Dogaru, Oltin
Publication:Journal of Advanced Mathematical Studies
Date:Jul 1, 2010
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