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Research on Adjoint Kernelled Quasidifferential
The quasidifferential of a quasidifferentiable function in the sense of Demyanov and Rubinov is not uniquely defined. Xia proposed the notion of the kernelled quasidifferential, which is expected to be a representative for the equivalence class of quasidifferentials. Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure, it is still not very convenient for calculating the kernelled quasidifferentials of and , where and are kernelled quasidifferentiable functions. In this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for and , 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. Under some condition, a formula of the adjoint kernelled quasidifferential is presented.
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., [1–13]).
A function defined on an open set is called quasidifferentiable (q.d.) at a point , in the sense of Demyanov and Rubinov , 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 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 . Suppose that is a quasidifferential of ; then, for any , the pair of sets is still a quasidifferential of . And the set 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 . There were many reports and publications mentioning or dealing with this subject from different points of view (see, for instance, [9–26], etc.).
Pallaschke et al.  introduced the notion of the minimal quasidifferential and proved the existence of equivalent minimal quasidifferential. is called minimal, provided that satisfying and implies and . 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 . Grzybowski  and Scholtes  proved independently the fact that equivalent minimal quasidifferentials, in the two-dimensional case, are uniquely determined up to a translation. For the -dimensional case (), Grzybowski  gave the first example of two equivalent minimal pairs in which are not related by translations, and, as in , 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 . and (defined by (3)) are called sub- and super-kernel, respectively, and is called a quasi-kernel of . 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 . 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 ). In the -dimensional case (), 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.  gave a sufficient condition for a quasi-kernel being a kernelled quasidifferential. In , 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 .
Although the kernelled quasidifferential is known to have good algebraic properties and geometric structure (see ), it is still not very convenient for calculating the kernelled quasidifferentials of and , where and are kernelled quasidifferentiable functions. Hence, in this paper, the notion of adjoint kernelled quasidifferential, which is well-defined for and , 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 are given. In Section 5, under some condition, a formula of the adjoint kernelled quasidifferential is presented.
The support function of a set is defined by It is well known (see, e.g., ) that the mapping called the Minkowski duality is one-to-one correspondence between and the set of all finite sublinear functions is defined on .
Proposition 1. Let ; then
It is true that is convex with particularly, , where denotes the subdifferential in the sense of convex analysis .
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,
Proposition 2. Let , for ; it holds
Proposition 3. Let and . If , then
Let the function defined on be locally Lipschitz continuous and let denote the set where exists. The Clarke subdifferential of at is defined as follows: where “” denotes the convex hull. In the convex case, the Clarke subdifferential coincides with the subdifferential in the sense of convex analysis .
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 . In general, the Demyanov difference is smaller than the Minkowski difference. It is true that According to , the Demyanov difference can be rewritten as
Define the algebraic operations of addition and multiplication by a real number in and the equivalence relation as follows: where , , and . It is easy to check that .
Proposition 4. If , then
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).
Proposition 5. Let denote the set of all functions defined on an open set and quasidifferentiable at a point . Then, the following hold. (1)If , , are real numbers, then , and Note that, in particular, .(2)Let . Then, and(3)If , , then is quasidifferentiable at and(4)Let and Then, , , and where Here, , .
3. Adjoint Kernelled Quasidifferential
The kernelled quasidifferential is known to have good algebraic properties (see ), but it is still not very convenient for calculating the kernelled quasidifferentials of and , 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 Obviously, is nonempty and symmetric. Since having the similar structure to the quasi-kernel of , is called an adjoint quasi-kernel of , 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.
Definition 6. Let . The adjoint quasi-kernel is said to be an adjoint kernelled quasidifferential of at if and only if 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 .
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.
(1) If , , , then
Note that, in particular, .
(2) Let . Then, (3) If , , then
If the adjoint kernelled quasidifferential exists, some operation rules of adjoint kernelled quasidifferential are presented as follows.
Theorem 8. Let denote the set of all functions in and having adjoint kernelled quasidifferential at . Then, the following hold. (1)If , then and (2)If , , then and (3)If , then and (4)If , , then and
Proof. We will prove only Properties (1) and (2). Properties (3) and (4) can be proved in an analogous manner.(1)Since , then From Propositions 5 and 7 and (32), it follows that By the similar way, we can prove that Since , hence, together with (33) and (34), one has that .(2)Since , then, together with Propositions 5 and 7, one has that Similarly, we can prove that Combining (35) with (36) leads to Hence, .
By we denote the set of all functions in and having kernelled quasidifferential at . Obviously, one has that . The adjoint kernelled quasidifferential is convenient for calculating and can calculate the adjoint kernelled quasidifferential of with kernelled quasidifferential, where , .
Theorem 9. If , then and If , then and
Proof. Since , then , where By Propositions 5 and 7 and (41), we obtain From Propositions 5 and 7 and (40), it follows that Obviously, . This fact, together with (42) and (43), implies that Then, and . Similarly, it can be proved that if , then and . The proof is completed.
Theorem 10. Let and Then, and , where Here, .
Proof. Since and , , then, according to Propositions 5 and 7, we have
Since, for , , where denotes a finite index set, one has that
where , . Hence, together with Proposition 5, it follows that
By Propositions 5 and 7 and (49), we obtain
Based on Propositions 5 and 7 and (47) and (50), one has that hence . The demonstration is completed.
4. 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 under some conditions.
Theorem 11. Suppose that , , and is a minimal quasidifferential of at . Then, the relations below hold Furthermore, ; that is, .
Proof. Let . From the existence of the minimal quasidifferentials, see , it follows that there exists a minimal quasidifferential of at , denoted by , such that , . Consequently,Note that both and are the minimal quasidifferentials of at . According to the translation property of the equivalent minimal quasidifferentials in the one- and two-dimensional case, see [15, 18], there exists , , such that the minimal quasidifferential can be expressed as This leads toIt follows from (53a), (53b), (55a), and (55b) thatTaking the intersection on the right hands of (56a) and of (56b) for all quasidifferentials of at , we have thatOn the other hand, implies thatThe relations (57a), (57b), (58a), and (58b) lead to thatNote that and , . Hence, Equations (59a), (59b), and (60) show that . The proof is completed.
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 . But. by the tool of Demyanov difference of compact convex sets, we get the following interesting result about minimal quasidifferential.
Proposition 12. Suppose that and there exists a quasidifferential such that Then is a minimal quasidifferential of at .
Proof. Let and Obviously, one has By Proposition 4 and (61), we obtain From (13) and (64), it follows that Combining (63) with (65) leads to According to (62) and (66), we conclude that Then, by the definition of the minimal quasidifferential, 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 .
Theorem 13. Suppose that and there exists a quasidifferential such that Then, one hasFurthermore, ; that is, .
Proof. Let . From Proposition 4 and (68), it follows that By the definition of the quasidifferential, it is easy to check that implies . Therefore, we have and . These give By (70) and Proposition 1, we obtain Evidently, (72) is equivalent to the following: Combining (71) with (73) leads to Based on (74) and Proposition 1, one has that Notice that both (70) and (75) hold for any . Taking the intersection on the right-hand sides of (70) and of (75), respectively, for all quasidifferentials of at , it is obtained thatOn the other hand, impliesCombining (76a) with (77b) yields (69a). Likewise, (76b) and (77b) yield (69b). Notice that the relation has been claimed. We thus complete the proof of the theorem.
A decomposition structure of is defined by where and are defined by respectively. Generally, and are positively homogeneous, but not sublinear. It is easy to be seen that It is easy to be seen that, for any , there exists at least one sequence convergent to , where . According to Proposition 2, if and such that there exist sequences and , then and .
The above lines enable us to give the following theorem which provides a sufficient condition for to be an adjoint kernelled quasidifferential.
Let be a shape of that is defined by a similar way according to , such that
Theorem 14. Let and suppose that and are continuous with respect to direction, and, furthermore, there exists a shape of such that, for any and , one has that where denotes the closed convex conical hull. If, for any , , and , there exist sequences such that is one of clusters of ; then , that is, .
Proof. Let be an arbitrary nonzero vector. There exist and such that . According to (82), there exists a sequence , 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 , , , and such that Since each is a convex combination of , , or of , , one has that there are and such that satisfying from (83) and (84), where . Since , , it follows, from the sufficient condition for given before the theorem, that Thus, it follows from (91) that Without loss of generality, assume . Taking the limit to (92), one has that According to the continuity of , (93) becomes Similarly, it can be proved that According to (94) and (95), we conclude Then, by the definition of the quasidifferential, one has , that is, . The demonstration is completed.
5. 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.
Lemma 15. Let ,