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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 603542, 21 pages
The Vector-Valued Functions Associated with Circular Cones
1Department of Mathematics, School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
2Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan
3Mathematics Division, National Center for Theoretical Sciences, Taipei, Taiwan
Received 6 April 2014; Accepted 15 May 2014; Published 22 June 2014
Academic Editor: Jen-Chih Yao
Copyright © 2014 Jinchuan Zhou and Jein-Shan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 .
In our previous work , 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 , 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 .
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
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
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.
Theorem 6. Let and be defined as in (4). Then, is -differentiable at if and only if is -differentiable at , for .
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].
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