Research on Adjoint Kernelled Quasidifferential

and Applied Analysis 3 Proposition 3. Let C ∈ Y n and x ∈ C. If d 1 , d 2 ∈ N C (x), then


Introduction
Quasidifferential calculus, developed by Demyanov and Rubinov, plays an important role in nonsmooth analysis and optimization.The class of quasidifferentiable functions is fairly broad.It contains not only convex, concave, and differentiable functions but also convex-concave, D.C. (i.e., difference of two convex), maximum, and other functions.In addition, it even includes some functions which are not locally Lipschitz continuous.Quasidifferentiability can be employed to study a wide range of theoretical and practical issues in many fields, such as in mechanics, engineering, and economics nonsmooth analysis and fuzzy control theory (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13]).
A function  defined on an open set O ⊂   is called quasidifferentiable (q.d.) at a point  ∈ O, in the sense of Demyanov and Rubinov [5], if it is directionally differentiable at  and there exist two nonempty convex compact sets () and () such that the directional derivative can be represented in the form as   (; ) = max ∈() ⟨, ⟩ + min V∈() ⟨V, ⟩ , ∀ ∈   , (1) where ⟨⋅, ⋅⟩ denotes the usual inner product in   .The pair of sets () = [(), ()] is called a quasidifferential of  at  and () and () are called a subdifferential and a superdifferential, respectively.
It is well known that the quasidifferential is not uniquely defined.Let   be the set of all nonempty convex compact sets in   .Denote  ±  = { ±  |  ∈ ,  ∈ } and  = { |  ∈ }, where ,  ∈   and  ≥ 0. Suppose that [, ] is a quasidifferential of ; then, for any  ∈   , the pair of sets [+, −] is still a quasidifferential of .And the set D() of quasidifferentials of  at  is so large that the whole space   could be covered by the union of subdifferentials or superdifferentials; that is, The quasidifferential uniqueness is an essential problem in quasidifferential calculus, so it is necessary to find a way by which a quasidifferential, particularly a small quasidifferential in some sense, as a representative of the equivalence class of quasidifferentials, can be determined automatically.The problem was for the first time considered in a discussion at IIASA, by Demyanov and Xia in 1984 [4].There were 2 Abstract and Applied Analysis many reports and publications mentioning or dealing with this subject from different points of view (see, for instance, [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], etc.).Pallaschke et al. [18] introduced the notion of the minimal quasidifferential and proved the existence of equivalent minimal quasidifferential.[, ] ∈ D() is called minimal, provided that [ 1 ,  1 ] ∈ D() satisfying  1 ⊂  and  1 ⊂  implies  =  1 and  =  1 .Nevertheless, the minimal quasidifferential is not uniquely defined either.Indeed, any translation of a minimal quasidifferential is still a minimal quasidifferential; in other words, if [, ] is a minimal quasidifferential, then, for any singleton {}, the pair of sets [ + {},  − {}] is still a minimal quasidifferential.For one-dimensional space, equivalent minimal pairs are uniquely determined up to translations, according to [8].Grzybowski [15] and Scholtes [22] proved independently the fact that equivalent minimal quasidifferentials, in the two-dimensional case, are uniquely determined up to a translation.For the -dimensional case ( ≥ 3), Grzybowski [15] gave the first example of two equivalent minimal pairs in  3 which are not related by translations, and, as in [19], Pallaschke and Unbański indicated that a continuum of equivalent pairs are not related by translation for different indices.Some sufficient conditions and both sufficient and necessary conditions for the minimality of pairs of compact convex sets were given and some reduction techniques for the reduction of pairs of compact convex sets via cutting hyperplanes or excision of compact convex subsets were proposed according to Pallaschke and Urbański [20,21].
For the same purpose, Xia [24,25] introduced the notion of the kernelled quasidifferential.It was proved that are nonempty, according to Deng and Gao [14]. and  (defined by ( 3)) are called sub-and super-kernel, respectively, and [, ] is called a quasi-kernel of D().The quasi-kernel is said to be a kernelled quasidifferential of  at  if and only if the quasi-kernel [, ] is a quasidifferential, denoted by If  has a kernelled quasidifferential at  ∈   , then  is said to be a kernelled quasidifferentiable function at .For the case of one-dimensional space, the existence of the kernelled quasidifferential was given by Gao [16].In the two dimensional case, based on the translation of minimal quasidifferentials, it was proved that the kernelled quasidifferential exists for any q.d.function (see [17]).In the -dimensional case ( ≥ 3), whether the pair of sets given in (3) is a quasidifferential of  at  is still an open problem, some progress has been made in the last years.Zhang et al. [26] gave a sufficient condition for a quasi-kernel being a kernelled quasidifferential.In [11], Gao presented a condition in terms of Demyanov difference, called g-condition, in which the kernelled quasidifferential exists.The corresponding subclasses and augmented class of g-q.d.functions on   were defined and some more properties on this class were presented according to Song and Xia [23].
Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure (see [25]), it is still not very convenient for calculating the kernelled quasidifferentials of − and min{  |  ∈ a finite index set }, where  and   are kernelled quasidifferentiable functions.Hence, in this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for − and min{  |  ∈ }, is employed as a representative of the equivalence class of quasidifferentials.Some algebraic properties of the adjoint kernelled quasidifferential are given and the existence of the adjoint kernelled quasidifferential is explored by means of the minimal quasidifferential and the Demyanov difference of convex sets.The rest of the paper is organized as follows.In Section 2, some preliminary definitions and results used in the paper are provided.In Section 3, definitions of adjoint kernelled quasidifferential will be introduced and some operations of adjoint kernelled quasidifferentiable functions are given.In Section 4, we prove that the adjoint kernelled quasidifferential exists in one-and two-dimensional cases and two sufficient conditions for the existence of the adjoint kernelled quasidifferential in   ( ≥ 3) are given.In Section 5, under some condition, a formula of the adjoint kernelled quasidifferential is presented.

Preliminaries
The support function  * (⋅ | ) of a set  ∈   is defined by It is well known (see, e.g., [6]) that the mapping   → particularly,  * (0 | ) = , where  denotes the subdifferential in the sense of convex analysis [27].
For any  ∈   and  ∈   , we denote the max-face of  with respect to  by the formula Obviously, the max-face () coincides with the subdifferential  * ( | ).Denote by   () the normal cone to  at  ∈ ; that is, Let the function  defined on   be locally Lipschitz continuous and let   denote the set where ∇ exists.The Clarke subdifferential  Cl () of  at  is defined as follows: (11) where "co" denotes the convex hull.In the convex case, the Clarke subdifferential coincides with the subdifferential in the sense of convex analysis [28].
A set  ⊂   is called of full measure (with respect to   ), if   \  is a set of measure zero.Let  ∈   and   =   * (⋅|) be the set of all points  ∈   such that ∇ * ( | ) exists.The set   is of full measure in   .Let ,  ∈   and  be a subset of   ∩   of full measure; then the set is called Demyanov difference of  and , where "cl" refers to the closure.This construction was applied implicitly by Demyanov for the study of connections between the Clarke subdifferential and the quasidifferential [3].In general, the Demyanov difference is smaller than the Minkowski difference.It is true that According to [6], the Demyanov difference can be rewritten as Define the algebraic operations of addition and multiplication by a real number in  2  =   ×   and the equivalence relation ∼ as follows: where  ∈ , (, ), and The main formulas of quasidifferential calculus will be stated as Proposition 5. Algebraic operations over quasidifferentials are performed as over elements of the space of compact sets (or what is the same, as over pairs of sets).

Adjoint Kernelled Quasidifferential
The kernelled quasidifferential is known to have good algebraic properties (see [25]) From the definition of quasidifferential and Proposition 5, the following proposition can be obtained immediately, which is especially useful in the study of the operation rules of adjoint kernelled quasidifferential.
Theorem 8. Let Δ , * () denote the set of all functions in Δ  () and having adjoint kernelled quasidifferential at .Then, the following hold.

Existence of the Adjoint Kernelled Quasidifferential
In this section, the existence of the adjoint kernelled quasidifferential of a quasidifferentiable function is established.In one-and two-dimensional cases, we prove that the adjoint kernelled quasidifferential exists and give its expression by using of a minimal quasidifferential.We also develop the existence of the adjoint kernelled quasidifferential for a quasidifferentiable function on   ( ≥ 3) under some conditions.
The conclusion of Theorem 11 strongly depends upon the translation of minimal quasidifferentials.Unfortunately, the minimal quasidifferential is not uniquely determined up to a translation in   if  ≥ 3 [15].But. by the tool of Demyanov difference of compact convex sets, we get the following interesting result about minimal quasidifferential.Proposition 12. Suppose that  ∈ Δ  () and there exists a quasidifferential [ 0 (),  0 ()] ∈ D() such that Then [ 0 (),  0 ()] is a minimal quasidifferential of  at .
Inspired by Proposition 12, we present the following theorem, which gives a sufficient condition for the existence of the adjoint kernelled quasidifferential in   ( ≥ 3).
A decomposition structure of   (; ⋅) is defined by where   (; ⋅) and   (; ⋅) are defined by The above lines enable us to give the following theorem which provides a sufficient condition for [, ] to be an adjoint kernelled quasidifferential.
Let F(, −) be a shape of (, −) that is defined by a similar way according to [18], such that and   (; ⋅) are continuous with respect to direction, and, furthermore, there exists a shape F(, −) of (, −) such that, for any  ∈  and V ∈ −, one has that where cone denotes the closed convex conical hull.If, for any  ∈ F(, −),  ∈ (), and V ∈ −(), there exist sequences Proof.Let  ∈   be an arbitrary nonzero vector.There exist  ∈  and V ∈ −  such that  ∈   () ∩  − (V).According to (82), there exists a sequence = 1, 2, . .., convergent to .For each , there are two index sets   and   , with finite indices such that It follows from ( 83)-( 85) and (87) that, for each , there exist Since each   is a convex combination of   ,  ∈   , or of   ,  ∈   , one has that there are   ≥ 0 and   ≥ 0 such that satisfying from ( 83) and ( 84), where    ∈     ()−   () (   ).

Formula of Representative for Quasidifferentials
Theorem 13 only gives the existence of the adjoint kernelled quasidifferential but does not show us how to calculate it.For the practical purpose, we expect to find a way to calculate a representative of the equivalent class of quasidifferentials for a given quasidifferential.The present section is devoted to this topic.
This means that (103) is the kernelled quasidifferential.The proof is concluded.

Proposition 5 .
Let Δ  () denote the set of all functions defined on an open set O ⊂   and quasidifferentiable at a point  ∈ O.Then, the following hold.
but it is still not very convenient for calculating the kernelled quasidifferentials of − and min{  |  ∈ a finite index set }, where  and   are kernelled quasidifferentiable functions.So it is natural and necessary to explore the pair of sets [, ], where  is defined as in (3) and Definition 6.Let  ∈ Δ  ().The adjoint quasi-kernel is said to be an adjoint kernelled quasidifferential of  at  if and only if [, ] ∈ D () .(24) If  has an adjoint kernelled quasidifferential at  ∈   , then  is said to be an adjoint kernelled quasidifferentiable function at .The adjoint kernel [, ] is a quasidifferential, denoted by   * () = [  * (), Obviously,  is nonempty and symmetric.Since having the similar structure to the quasi-kernel of D(), [, ] is called an adjoint quasi-kernel of D(), where  and  are called adjoint sub-kernel and adjoint super-kernel, respectively.Of course  and  are compact convex.This motivates the introduction of the following notions.* ()].