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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 965931, 12 pages
The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP
1School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China
3School of Information and Calculating Science, North University for Ethnics, Yinchuan 750021, China
Received 22 February 2013; Accepted 21 May 2013
Academic Editor: Ming Mei
Copyright © 2013 Xiaoni Chi et al. 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.
Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.
The second-order cone complementarity problem (SOCCP)  is to find such that where is a continuously differentiable mapping, is the Cartesian product of second-order cones (SOC), that is, with and the -dimensional SOC defined by
Without loss of generality, we may assume that and in the following analysis, since our analysis can be easily extended to the general case.
The SOCCP contains a wide class of problems, such as nonlinear complementarity problems , second-order cone programming [1, 3, 4], and has a variety of engineering and management applications, such as filter design, antenna array weight design, truss design, and grasping force optimization in robotics [5, 6].
Recently, great attention has been paid to smoothing methods, partially due to their superior theoretical and numerical performances [7–10]. Smoothing methods usually reformulate the SOCCP as a system of equations by using smoothing functions of SOC complementarity functions [10, 11]. The smoothing parameter involved in smoothing functions may be treated as a variable  or a parameter with an appropriate parameter control . In the latter case, the Jacobian consistency plays a key role for achieving a rapid convergence of Newton the methods or the Newton-like methods. Hayashi et al.  propose a combined smoothing and regularized method for monotone SOCCP and show its global and quadratic convergence based on the Jacobian consistency of the smoothing natural residual function. Ogasawara and Narushima  show the Jacobian consistency of a smoothed Fischer-Burmeister (FB) function. Chen et al.  present a smoothing function of a generalized FB function in the context of nonlinear complementarity programming and study some of its favorable properties, including the Jacobian consistency property. Based on the results, they  propose a smoothing algorithm for the mixed complementarity problem, which is shown to possess global convergence and local superlinear (or quadratic) convergence.
In this paper, we are concerned with the one-parametric class of SOC complementarity functions defined by  where is an arbitrary but fixed parameter. When , reduces to the vector-valued Fischer-Burmeister function given by and as , it becomes a multiple of the following vector-valued residual function: where denotes the metric projection on the second-order cone . Thus, the one-parametric class of vector-valued functions (3) cover two popular second-order cone complementarity functions.
In this paper, we aim to show the Jacobian consistency of smoothing functions of the one-parametric class of SOC complementarity functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of the smoothing functions, which will help to adjust a parameter appropriately in smoothing methods.
The organization of this paper is as follows. In Section 2, we review some preliminaries including the Euclidean Jordan algebra associated with SOC and subdifferentials. In Section 3, we derive the computable formula for the Jacobian of the one-parametric class of smoothing functions in the SOCCP. In Section 4, we prove the Jacobian consistency of the one-parametric class of smoothing functions and estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions. In Section 5, we study the directional derivative and -subdifferential of the one-parametric class of SOC complementarity functions and then present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions. Finally, we close this paper with some conclusions in Section 6.
In what follows, we denote the nonnegative orthant of by . We use the symbol to denote the Euclidean norm defined by for a vector or the corresponding induced matrix norm. For simplicity, we often use for the column vector . For the SOC , and mean the topological interior and the boundary of , respectively. For a given set , denotes the convex hull of in , and denotes for a matrix .
First, we recall the Euclidean Jordan algebra associated with the SOC and some useful definitions. The Euclidean Jordan algebra for the SOC is the algebra defined by with being its unit element. Given an element , we define where represents the identity matrix. It is easy to verify that for any . Moreover, is symmetric positive definite (and hence invertible) if and only if .
Now, we give the spectral factorization of vectors in associated with the SOC . Let . Then can be decomposed as where , , and , are the spectral values and the associated spectral vectors of given by for , with any such that . By the spectral factorization, a scalar function can be extended to a function for the SOC. For any , we define Since both eigenvalues of any are nonnegative, we define
For any , we define  . Obviously, , , and for any . Moreover, if .
Let be a locally Lipschitzian function. Then, is differentiable almost everywhere by Rademacher’s theorem . The Bouligand- (B-) subdifferential and the Clarke subdifferential of at are defined by respectively, where denotes the set of points at which is differentiable. It is obvious that if is continuously differentiable at .
By using the concepts of subdifferentials, we give the definition of the Jacobian consistency, which was first introduced by Chen et al. , which is a concept relating the generalized Jacobian of a nonsmooth function with the Jacobian of a smoothing function .
Definition 1 (see ). Let be a locally Lipschitzian function. Let be a continuously differentiable function for any such that for any . We say that satisfies the Jacobian consistency property if for any , .
It should be noted that the “inf” appearing in the definition of can be replaced by “min,” since the set is compact at all .
3. Smoothing Function
In this section, we propose a smoothing function of the one-parametric class of SOC complementarity functions and derive the computable formula for its Jacobian.
Since the one-parametric class of SOC complementarity functions defined by (3) is nonsmooth, we consider the function defined by where the smoothing parameter .
Definition 2 (see ). For a nondifferentiable function , one considers a function with a parameter that has the following properties:(i) is differentiable for any ;(ii) for any .
Such a function is called a smoothing function of .
In the following, we will show that the function given by (13) is a smoothing function of . Thus, we can solve a family of smoothing subproblems for and obtain a solution of by letting .
For convenience, we give some notations. For any , , and any , we define the mapping by and drop the subscript for simplicity for , and thus, By direct calculations, we obtain and therefore . Then, the spectral factorization of is where , , and , are the spectral values and the associated spectral vectors of given by for . Here, if , and otherwise, is any vector in such that .
For any , it is not difficult to verify that for any , and Therefore, for any , and any , we can also define and for , The spectral factorization of and is given by, respectively, By (22), we can partition as , where
Theorem 3. For any and , let and be, respectively, defined by (3) and (13). Then, the following results hold.(i) The function is continuously differentiable everywhere, and its Jacobian is given by where if , and otherwise, with (ii) For any , . Thus, is a smoothing function of .
Proof. (i) For any and any , it follows from Corollary 5.4 in , the chain rule for differentiation, and (21) that formula (27) holds. Formula (28) is due to Proposition 5.2 and its proof in .
(ii) Fix any , . For any , it follows from the spectral factorization of and that where and and are, respectively, given by (18) and (19) for . It is obvious that for . Then, and hence, . Therefore, it follows from (i) and Definition 2 that is a smoothing function of .
Next, we give some properties of , which will be used in the subsequent analysis.
Lemma 4. For any , , let . Then one has Moreover, the following equivalence holds:
4. The Jacobian Consistency
In this section, we show the Jacobian consistency of the smoothing function , which will play an important role for establishing the rapid convergence of smoothing methods. Moreover, we estimate the distance between the gradient of the smoothing functions and the subgradient of the one-parametric class of the SOC complementarity functions, which will help to adjust a parameter appropriately in smoothing methods.
It has been shown in Proposition 3.1 in  that the function with any satisfies Let and define It is easy to see that is the perturbation of the system of equations . On account of (1), (42), and (43), we have Since is typically nonsmooth, we can solve approximately the smooth system by using Newton’s method at each iteration, and then obtain a solution of by reducing the parameter to zero.
First, we show that the function satisfies the Jacobian consistency.
Lemma 5. For any , one has where
Proof. By (27) and the symmetry of and , it suffices to show that Case (i). If , we have from (18) that for , and then, Therefore, Case (ii). If , we obtain , and (41) holds, and thus, Then we have from (18) that For any , it follows from (28) that . We first show that for any . Let By (35), we have and therefore, We next show that . In fact, we obtain from (52) Combining (55) and (56) yields Case (iii). If , then , and This completes the proof.
Lemma 6. For any , one has where
Proof. It follows from Proposition 5.2  and the chain rule for differentiation that the one-parametric class of SOC complementarity functions is continuously differentiable at any with
Therefore, it suffices to consider the two cases: and .
For any or , let with sufficiently small , and define Direct calculations yield Therefore, we obtain It is obvious that as , we have , , and for .
By the definition of -subdifferential and (3), it suffices to show that if is differentiable at .
Case (i). If , it is easy to see that , , and is differentiable at . Then, we have Case (ii). If , we obtain , and thus, from (38) and (39), Since and is sufficiently small, we obtain By (66) and (70), we have Relations (71) and (72) imply that , and thus, is differentiable at .
Now we will show that By Proposition 5.2 in , we have , where and are given by (29) with and replacing and , respectively. By (70), According to (35), (39), (71), and (74), we have and thus, It follows from (70), (71), (72), and (74) that , , and as . Then, by following the proof of Case (ii) in Lemma 5, we obtain Hence, we have from (76) and (77) that
Next, we will show that By (38), (39), and (74), we have and then, Hence, we have from (77) and (81) that This completes the proof.
Theorem 7. The function defined by (43) with satisfies the Jacobian consistency.
Proof. By (43), it suffices to show the Jacobian consistency of the function with . Let where for , and and are given by (61). Define On the one hand, we obtain from Lemma 5 that . On the other hand, we have by Lemma 6 that , and therefore, . This together with Theorem 3 and Definition 1 implies the Jacobian consistency of with .
Now, we are in a position to estimate an upper bound of the parameter for the predicted accuracy of the distance between the gradient of and the subgradient of .
Theorem 8. Let , be given, and be any point. Let be any function such that and let be defined by Then, for any such that , we have
Proof. Since it follows from the proof of Theorem 7 that for any , we obtain Therefore, we have from (27) and (85) that Then, by following the proof of Proposition 4.1 in , we obtain where is defined as Hence, similar to the proof of Theorem 4.1 in , we have the desired result.
5. An Alternative Proof
In this section, we study the directional derivative and -subdifferential of the one-parametric class of SOC complementarity functions . Based on these results, we present an alternative way to prove the Jacobian consistency of the one-parametric class of smoothing functions .
By Corollary 3.3 in , it is not difficult to see that the function given as (3) is directionally differentiable everywhere. However, as far as we know, the expression of its directional derivative is not given in the available literature. In this section, we derive its expression and prove that the -subdifferential of at a general point coincides with that of its directional derivative function at the origin.
In light of the -subdifferential of [4, 10, 19], we obtain the following four results, which can be shown by following the proofs of Proposition 9, Lemma 11, Lemma 12, and Proposition 13 in , respectively.
Proposition 9. For any given , , the directional derivative of at has the following form.(i) If , then .(ii) If , then (iii) If , then where , , is given as (20), and the function is defined by with
Lemma 10. For any given , let be defined by (95). Then, the B-subdifferential of the function at is
Lemma 11. For any given , let . Then,
Proposition 12. For any given , we have