The Vector-Valued Functions Associated with Circular Cones

Abstract

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let denote the circular cone in . For a function from to , one can define a corresponding vector-valued function on by applying to the spectral values of the spectral decomposition of with respect to . In this paper, we study properties that this vector-valued function inherits from , including Hölder continuity, -subdifferentiability, -order semismoothness, and positive homogeneity. These results will play crucial role in designing solution methods for optimization problem involved in circular cone constraints.

1. Introduction

The circular cone is a pointed closed convex cone having hyperspherical sections orthogonal to its axis of revolution about which the cone is invariant to rotation, which includes second-order cone as a special case when the rotation angle is 45 degrees. Let denote the circular cone in . Then, the -dimensional circular cone is expressed as The application of lies in engineering field, for example, optimal grasping manipulation for multigingered robots; see [1].

In our previous work [2], we have explored some important features about circular cone, such as characterizing its tangent cone, normal cone, and second-order regularity. In particular, the spectral decomposition associated with was discovered; that is, for any , one has where with if , and being any vector satisfying if . With this spectral decomposition (2), analogous to the so-called SOC-function (see [3–5]) and SDP-function (see [6, 7]), we define a vector-valued function associated with circular cone as below. More specifically, for , we define as It is not hard to see that is well-defined for all . In particular, if , then

Note that when , reduces to the second-order cone (SOC) and the vector-valued function defined as in (4) corresponds to the SOC-function given by where and .

It is well known that the vector-valued function associated with second-order cone and matrix-valued function associated with positive semidefinite cone play crucial role in the theory and numerical algorithm for second-order cone programming and semidefinite programming, respectively. In particular, many properties of and are inherited from , such as continuity, strictly continuity, directional differentiability, Fréchet differentiability, continuous differentiability, and semismoothness. It should be mentioned that, compared with second-order cone and positive semidefinite cone, is a nonsymmetric cone. Hence a natural question arises whether these properties are still true for . In [1], the authors answer the questions from the following aspects:(a) is continuous at if and only if is continuous at for ;(b) is directionally differentiable at if and only if is directionally differentiable at for ;(c) is (Fréchet) differentiable at if and only if is (Fréchet) differentiable at for ;(d) is continuously differentiable at if and only if is continuously continuous at for ;(e) is strictly continuous at if and only if is strictly continuous at for ;(f) is Lipschitz continuous with constant if and only if is Lipschitz continuous with constant ;(g) is semismooth at if and only if is semismooth at for .

In this paper, we further study some other properties associated with , such as Hölder continuity, -order semismoothness, directionally differentiability in the Hadamard sense, the characterization of B-subdifferential, positive homogeneity, and boundedness. Of course, one may wonder whether and always share the same properties. Indeed, they do not. There exists some property that holds for and but fails for and . A counterexample is presented in the final section.

To end the third section, we briefly review our notations and some basic concepts which will be needed for subsequent analysis. First, we denote by the space of -dimensional real column vectors and let . Given , the Euclidean inner product and norm are and . For a linear mapping , its operator norm is . For and , we write (resp., ) to means is uniformly bounded (resp., tends to zero) as . In addition, given a function , we say the following:(a) is Hölder continuous with exponent , if (b) is directionally differentiable at in the Hadamard sense, if the directional derivative exists for all and (c) is -differentiable (Bouligand-differentiable) at , if is Lipschitz continuous near and directionally differentiable at ;(d)if is strictly continuous (locally Lipschitz continuous), the generalized Jacobian is the convex hull of the , where  where denotes the set of all differentiable points of ;(e) is semismooth at , if is strictly continuous near , directionally differentiable at , and for any , (f) is -order semismooth at () if is semismooth at and for any ,  in particular, we say is strongly semismooth if it is -order semismooth;(g) is positively homogeneous with exponent , if (h) is bounded if there exists a positive scalar such that

2. Directional Differentiability, Strict Continuity, Hölder Continuity, and -Differentiability

This section is devoted to study the properties of directional differentiability, strict continuity, and Hölder continuity. The relationship of directional differentiability between and has been given in [1, Theorem 3.2] without giving the exact formula of directional differentiability. Nonetheless, such formulas can be easily obtained from its proof. Here we just list them as follows.

Lemma 1. Let and be defined as in (4). Then, is directionally differentiable at if and only if is directionally differentiable at for . Moreover, for any , we have when and . Consider when and ; otherwise, where

Lemma 2. Let and be defined as in (4). Then, the following hold.(a) is differentiable at if and only if is differentiable at for . Moreover, if , then  otherwise,  where (b) is continuously differentiable (smooth) at if and only if is continuously differentiable (smooth) at for .

Note that the formula of gradient given in [1, Theorem 3.3] and Lemma 2 is the same by using the following facts:

The following result indicating that is Lipschitz continuous on for will be used in proving the Lipschitz continuity between and .

Lemma 3. Let with spectral values , , respectively. Then, we have

Proof. First, we observe that Applying the similar argument to yields Then, the desired result follows from the fact that .

Theorem 4. Let and be defined as in (4). Then, is strictly continuous (local Lipschitz continuity) at if and only if is strictly continuous (local Lipschitz continuity) at for .

Proof. “” Suppose that is strictly continuous at , for ; that is, there exist and , for such that Let and . Define Clearly, is Lipschitz continuous on ; that is, there exists such that , for all . Since is compact, according to [6, Lemma 4.5] or [5, Lemma 3], there exist continuously differentiable functions for converging uniformly to on such that Now, let . Then, from Lemma 3, we know that contains all spectral values of . Therefore, for any , we have for and where we have used the facts that , , and . Since converges uniformly to on , the above equations show that converges uniformly to on . If , then it follows from Lemma 2 that . Hence it follows from (27) that since in this case . If , then where , , are given as in (20) with replaced by for and replaced by . For simplicity of notations, let us denote Note that where the inequality comes from the fact that is continuously differentiable on and (27). Besides, we also note that (i)For , then takes the form of whose eigenvalues are and by [5, Lemma 1]. In other words, in this case, we get from (32) and (33) that (ii)For , since , the eigenvalues of are and 0 with multiplicity . Note that Note that where and In this case the matrix has eigenvalues of and with multiplicity . Hence, where the last step is due to (32), (33), (34), and (36).
Putting (29), (35), and (39) together, we know that Fix any with . Since converges uniformly to on , then for any there exists such that Since is continuously differentiable, is continuously differentiable by Lemma 2. Thus, Because is arbitrary, this ensures that which says is strictly continuous at .
“” Suppose that is strictly continuous at , then there exist and such that Case 1. . Take with . Let Then, and and it follows from (44) that which says is strictly continuous at . The similar argument shows the strict continuity of at .
Case 2. . For any , we have and as well; that is, . It then follows from (44) that This means is strictly continuous at for .

Remark 5. As mentioned in Section 1, the strict continuity between and has been given in [1, Theorem 3.5]. Here we provide an alternative proof, since our analysis technique is different from that in [1, Theorem 3.5]. In particular, we achieve an estimate regarding via its eigenvalues, which may have other applications.

According to Lemma 1 and Theorem 4, we obtain the following result immediately.

Theorem 6. Let and be defined as in (4). Then, is -differentiable at if and only if is -differentiable at , for .

Next, inspired by [8, 9], we further study the Hölder continuity relation between and .

Theorem 7. Let and be defined as in (4). Then, is Hölder continuous with exponent if and only if is Hölder continuous with exponent .

Proof. “” Suppose that is Hölder continuous with exponent . To proceed the proof, we consider the following two cases.
Case 1. and . We assume without loss of generality that . Thus, Let us analyze each term in the above inequality. First, we look into the first term: where the first inequality is due to the Hölder continuity of , the second inequality comes from the fact that (cf. [8, Lemma 2.2]), and the third inequality follows from the fact that (since ). Next, we look into the second term: Similarly, the third term also satisfies Combining (49)–(51) proves that is Hölder continuous with exponent .
Case 2. Either or . In this case, we take , for according to the spectral decomposition. Therefore, we obtain which says is Hölder continuous.
“” Recall that . Hence, for any , which says is Hölder continuous.

3. -Order Semismoothness and -Subdifferential Formula

The property of semismoothness plays an important role in nonsmooth Newton methods [10, 11]. For more information on semismooth functions, see [12–15]. The relationship of semismooth between and has been given in [1, Theorem 4.1]. But the exact formula of the -subdifferential is not presented. Hence the main aim of this section is twofold: one is establishing the exact formula of -subdifferential; another is studing the -order semismoothness for .

Lemma 8. Define and for . Then, and are strongly semismooth at .

Proof. Since , it is clear that and are twice continuously differentiable and hence the gradient is Lipschitz continuous near . Therefore, and are strongly semismooth at , see [16, Proposition 7.4.5].

The relationship of -order semismoothness between and is given below. Recall from [7] that in the definition of -order semismooth, we can restrict in (11) belonging to differentiable points.

Theorem 9. Let and be defined as in (4). Given , then the following statements hold. (a)If is -order semismooth at for , then is -order semismooth at .(b)If is -order semismooth at , then is -semismooth at for .(c)For , is -semismooth at if and only if is -order semismooth at for .

Proof. (a) Take satisfying . We consider the following two cases to complete the proof.
Case 1. For , as is sufficiently close to . Since , we know that for by Lemma 2. Then, according to Lemma 1, the first component of is expressed as Because is continuously differentiable over , it is strongly semismooth at by Lemma 8. Therefore, Combining this and the -semismoothness of at , we have where the second equation is due to Lemma 3 and the last equality comes from the boundedness of , since is strictly continuous at . Similar argument holds for . Hence the first component of (54) is .
Next, let us look into the second component of (54), which involved . By Lemma 1 again, it can be expressed as Note that is continuous differentiable (and hence is semismooth) with and . Thus, expression (58) can be rewritten as The second equation comes from (57), strongly semismoothness of at , and since is Lipschitz at (which is ensured by the -order semismoothness of ). Analogous arguments apply for the second component of (54) involving . From all the above, we may conclude that which says is -order semismooth at under this case.
Case 2. For , if , then the proof is trivial. If , then the first component of (54) satisfies because is -order semismooth at . The second component of (54), by letting and , takes the form where the last step is due to the -order semismoothness of .
(b) Suppose that is -order semismooth at . Let such that is differentiable at . We discuss the following two cases.
Case 1. For , from being Lipschitz at (and hence the differentiable points are dense near ), there exists such that and is differentiable at and as is sufficiently small (since by ). Denote . Then, which implies and . Since is differentiable at and , is also differentiable at by Lemma 2. Notice that Hence, which follows from the fact that and can be arbitrarily small (hence ). Thus, it is clear that In addition, it can be verified that since as is sufficiently small. Similarly, Therefore, we obtain which further implies . Then, by the hypothesis being -order semismooth at , that is, we have In fact, the left-hand side of (71) takes the form of where the last step is due to the fact that is bounded and since is Lipschitz at . Hence (71) means which says is -order semismooth at . Applying similar arguments show that is -order semismooth at .
Case 2. For , letting . Since is differentiable at and , for , is differentiable at by Lemma 2; that is, . Because is -order semismooth at , we have which, together with the fact , is equivalent to This clearly proves that is -order semismooth at , for .
(c) The necessity comes from part (b), and the sufficiency follows from (62) and (63).

Corollary 10. Let and be defined as in (4). For , is -order semismooth at if and only if is -order semismooth at , for .

Remark 11. In the framework of second-order cone and positive semidefinite cone, the corresponding result to part (a) has been established; see [5, Proposition 7] and [6, Proposition 4.10]. In [17], the author study the -order semismoothness of the spectral operator for . Here we further show that if is -semismooth, then is -semismooth for all . In addition, if , then the -order semismoothness of and coincide with each other for all .

Inspired by [5, Lemma 4], we also obtain the following result.

Theorem 12. Let be strictly continuous. Then, for any , the -differential is well defined and nonempty. Moreover,(i)if , then (ii)if , then

Proof. Denote by the set in the right side of (77).
Case 1. . For any sequence with . Then, we compute Since is strictly continuous, we know that and are bounded and hence have cluster points. We assume, without loss of generality, that and . Note that is differentiable at for by Lemma 2. Besides, from and the fact that any cluster point of is in by definition, we have This means that any cluster points of are element of ; that is, .
Conversely, for any and satisfying and , there exist and with being differentiable at and and and . Since , it implies that is large enough. Now, let For points , it is easy to see that for all by using the following facts: Hence, and are differentiable at by Lemma 2 (since is differentiable at for ). Then, we compute where Since the limit of is an element of , we obtain .
Case 2. . Consider any sequence with being differentiable at for all . By passing to a subsequence, we can assume that either for all or for all . If , then by Lemma 2 we know that is differentiable at for and . Hence, the cluster point of is an element of (78) with and . If , by passing to a subsequence we can assume without loss of generality that for some with . Note that Moreover, from and being bounded (due to the strictly continuous of ), we can assume that and . Using (80) and any cluster point of in , we have In addition, where and hence converges to , since for , due to . Using the outer semicontinuity of we get that the cluster point of belongs to . Hence any cluster of belongs to an element of the set of the right side in (78).

We point out one thing for Theorem 12(ii). In the set of the right side in (78), cannot be replaced by because is usually a smaller subset of . For example, letting and , , we have while which is the main reason causing what we just pointed out.

At present, we roughly describe for . In other words, how to get the exact formula on . Toward this end, we need to introduce the following definition. Given , define where “limsup” is the outer limits in the sense of set-valued mapping; see [18, 19] for more details.

First, for , according to Lemma 2 let us write the gradient of as

The exact formula of -subdifferential is given below.

Theorem 13. Given , the following statements hold. (a)If , then (b)If , then

Proof. (a) Denote by the set in the right side of (93). Take . By definition, there exists a sequence with satisfying . Since , then for sufficiently large. Note that for by Lemma 2 and from (92) Note also that , , for , and Hence . This establishes .
Conversely, take ; that is, there exists such that By definition of , there exists such that , , , and Let Note that , it is easy to see the , and for . Hence, and are differentiable at by Lemma 2 (since is differentiable at for ). Hence according to the formula of gradients , (95), (97), and (98), together with the fact , we have , which in turn implies .
(b) Take ; then by definition there exists with such that . By passing to a subsequence, we can assume that either for all or for all . If , then by Lemma 2  . Hence the cluster point of is an element of. If , by passing to a subsequence we can assume that for some with . Note that (since by Lemma 2), , for , and ; it follows from (95) that belongs to an element of (94) with Conversely, take belonging to the left side of (94); that is, there exists such that or exist and with such that If for some , then there exists with such that . Let , then , for , , and by Lemma 2. Thus, , which further implies ; that is, . The remaining case can be proved by using the same argument following (97) by replacing by . The proof is complete.

Due to the important role played by , we present the estimate of as below.

Lemma 14. Given , the following statements hold. (a)If , then (b)If , then

Proof. The case of is clear, while the case of can be proved by using the similar argument following (88).

The exact estimate of at can be obtained provided that additional assumption is imposed on .

Lemma 15. Suppose that is strictly continuous and directionally differentiable function satisfying Then, for , we have

Proof. It follows from the definition of via (91) that According to (104), it is easy to see that These are the elements of with , since . Now we claim that It only needs to show First, we observe that for some . Conversely, taking and using yield for some . Due to being dense in , for any , we define Take and ; then for , we have which imply Thus, Similarly, we have In summary, This completes the proof.

Corollary 16. Suppose is strictly continuous and directionally differentiable function satisfying Then, for any ,

Proof. This result follows by combining Theorem 13 and Lemma 15.

We point out that if , then Corollary 16 reduces to [5, Lemma 5].

4. Directional Differentiability in Hadamard Sense, Positive Homogeneity, and Boundedness

In this section, we study some other important properties between and , such as directional differentiability in the Hadamard sense, positive homogeneity, and boundedness. These are new discoveries and are not studied in [1, 5, 6] or other settings. First, we note that if a function is directionally differentiable in the Hadamard senses, then it must be directionally differentiable. The converse statement holds true if this function is from to , since in this case the direction just has ±1. More precisely, for a real-valued function , is directionally differentiable if and only if is directionally differentiable in the Hadamard sense. Indeed, in the proof of [5, Proposition 3], the authors already employ the property of directionally differentiable in the Hadamard sense and even the assumption is directionally differentiable. However, for general mappings to or , these two concepts are not equivalent. In other words, if is directionally differentiable in the Hadamard sense, then is directionally differentiable. But, the converse is invalid in general, unless some additional assumption is imposed; for example, local Lipschitz continuity [20]. Nonetheless, we will show for the special function , these two concepts are still equivalent.

Theorem 17. Let and be defined as in (4). Then, is directionally differentiable at if and only if is directionally differentiable at in the Hadamard sense.

Proof. “” This direction is clear.
“” Suppose that is directionally differentiable at . Then, from Lemma 1, is directionally differentiable, and hence is directionally differentiable in the Hadamard sense, since is a function from to . To proceed the proof, we consider the following three cases.
Case 1. and . Let . If , then the proof is trivial. If , then where is taken to be . Since is directionally differentiable in the Hadamard sense, then Therefore, because the term in big brace approaches to zero and is bounded. These two limits imply which says is directionally differentiable at in the Hadamard sense.
Case 2. and . Note that For the first component, we compute that where the last step is due to the fact that is directionally differentiable in the Hadamard sense.
For the second component, we have The above two limits show that
Case 3. . Then Then, the first component of behaves as follows (when and ): where in the last step we have used the fact that is directionally differentiable in the Hadamard sense. Recall that is continuously differentiable at . Then, the second component of behaves as follows (when and ): The above two limits show that The proof is complete.

Theorem 18. Let and be defined as in (4). Then, the following statements are equivalent.(a) is directionally differentiable at ;(b) is directionally differentiable at in the Hadamard sense;(c) is directionally differentiable at , for ;(d) is directionally differentiable at , for in the Hadamard sense.

Proof. The equivalence comes from Theorem 17; is due to the fact that ; follows from Lemma 1.

Below we study the relationship of positive homogeneity and boundedness between and .

Theorem 19. Let and be defined as in (4). Then, is positively homogeneous at with exponent if and only if is positively homogeneous at for with exponent .

Proof. “” Suppose that is positively homogeneous at for with exponent . For any with , we observe that , for , whenever and , for . Hence, when , we have When , we know that for . Thus, All the above shows that is positively homogeneous at with exponent .
“” Suppose that is positively homogeneous at with exponent ; that is, . Then, we have which in turn implies , since . Hence, is positively homogenous at for with exponent .

Theorem 20. Let and be defined as in (4). Then, is bounded if and only if is bounded.

Proof. “” Suppose that is bounded by . The proof for this direction follows from the following inequality: “” Suppose that is bounded by . This direction is trivial because for any , one has , and hence

5. Final Remarks

So far, we have shown that many properties holding for can be extended to the setting for . One may wonder whether and always share the same properties. The answer is no! Here, we present a simple property that holds for but fails for . Some more different properties between and are discovered in [21].

To see the counterexample, we recall that a function is said to be an odd (even, resp.) function if (, resp.) for all .

Proposition 21. Let and be given as in (6). Then is an odd (even, resp.) function on if and only if is an odd (even, resp.) function on .

Proof. “” In the setting of second-order cone, we observe that
which implies “” For , we have

The below example illustrates that the above relationship fails to hold for and .

Example 22. Let be given as in (4) with (). Then,(a) is an odd function, but is not an odd function at ;(b) is an even function, but is not an even function at .

For , it is clear that is an odd function. Nonetheless, we verify that which says . Thus, is not an odd function at .

Similarly, for which is an even function, we compute Hence, which says is not an even function at .

Finally, let us discuss the relationship between circular cone and the (nonsymmetric) matrix cone introduced in [17, 22], where the authors study the epigraph of six different matrix norms, such as the Frobenius norm, the norm, norm, the spectral or the operator norm, the nuclear norm, and the Ky Fan -norm. If we regard a matrix as a high-dimensional vector, then the circular cone is equivalent to the matrix cone with Frobenius norm. More precisely, denote where denotes the Frobenius norm of ; that is, . Notice that the circular cone can be equivalently written as Let , and where diag denotes the diagonal matrix; then reduces to . Conversely, the matrix cone can be also regarded as a circular cone with Therefore, is a -dimensional circular cone.

In addition, we know for a vector (regarding as a matrix in ) the singular value decomposition is where , , and with and are arbitrary orthonormal vectors orthogonal to . It indicates that the singular value of is and 0 with multiplicity . Hence the spectral/operator norm (largest singular value), the Ky Fan -norm (the sum of -largest singular value), or the nuclear norm (the sum of the singular values) are all . This means that is a special case of the matrix cone studied in [17, 22], where the properties of spectral operator are studied, such as well-definiteness, the directional differentiability, the Fréchet differentiability, the locally Lipschitz continuity, the -order -differentiability (), the -order -semismooth (), and the characterization of Clarke’s generalized Jacobian. In this paper, by using the special structure of circular cone, we mainly establish the -subdifferential (the approach we considered here is more directly and depended on the special structure of circular cone), the directional differentiability in the Hadamard sense, and the -order semismooth for .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are gratefully indebted to Professor Defeng Sun for drawing their attention to the relationship between circular cone and matrix cone introduced in [17, 22]. In particular, the characterization of -subdifferential given in Theorem 13 is inspired by the analysis technique used in [17, Chapter 3]. Jinchuan Zhou’s work is supported by the National Natural Science Foundation of China (11101248, 11271233), Shandong Province Natural Science Foundation (ZR2010AQ026, ZR2012AM016), and Young Teacher Support Program of Shandong University of Technology. Jein-Shan Chen’s work is supported by Ministry of Science and Technology, Taiwan.