Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

Abstract

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

1. Introduction

The weighted second-order cone complementarity problem (WSOCCP) is, for a given weight vector and a continuously differentiable function , to find vectors such thatwhere represents the Jordan product and is the Cartesian product of second-order cone, that is, with . The set is the second-order cone (SOC) of dimension defined byand the interior of the SOC is the set

Here is the Euclidean norm, and

Obviously, if , WSOCCP (1) reduces to second-order cone complementarity problem (SOCCP). In this article, we may assume that and in the following analysis, since it can easily be extended to the general case.

In order to reformulate several equilibrium problems in economics and study highly efficient algorithms to solve these problems, Potra [1] introduced the notion of a weighted complementarity problem (WCP). He showed that the Fisher market equilibrium problem can be modeled as a monotone linear WCP. Moreover, the linear programming and weighted centering (LPWC) problem, which was introduced by Anstreicher [2], can also be formulated as a monotone linear WCP. And Potra [1] analyzed two interior-point methods for solving the monotone linear WCP over the nonnegative orthant. Since then, many scholars are dedicated to investigating the theories and solution methods of WCP. Tang [3] gave a new nonmonotone smoothing-type algorithm to solve the linear WCP. Chi et al. [4] studied the existence and uniqueness of the solution for a class of WCPs.

As is well known, smoothing methods have superior theoretical and numerical performances. For solving the SOCCP by smoothing methods, we usually reformulate the SOCCP as a system of equations based on parametric smoothing functions of SOC complementarity functions [5, 6]. The smoothing parameter involved in smoothing functions may be treated as a variable [7] or a parameter with an appropriate parameter control [8]. In the latter case, the Jacobian consistency is important to achieve a rapid convergence of Newton methods or Newton-like methods. Hayashi et al. [8] proposed a combined smoothing and regularized method for monotone SOCCP, and based on the Jacobian consistency of the smoothing natural residual function, they proved that the method has global and quadratic convergence. Krejić and Rapajić [9] gave a nonmonotone Jacobian smoothing inexact Newton method for nonlinear complementarity problem and proved the global and local superlinear convergence of the method. Chen et al. [10] presented a modified Jacobian smoothing method for the nonsmooth complementarity problem and established the global and fast local convergence for the method.

In this paper, we consider the function for WSOCCPwith a given vector . If , (5) reduces to the SOC complementarity function [6] with :

Since is nonsmooth, we define the following smoothing function :where is a smoothing parameter.

The main contribution of this paper is to show the Jacobian consistency of the smoothing function (7) and estimate the distance between the subgradient of the weighted SOC complementarity function (5) and the gradient of its smoothing function (7). These properties will be critical to solve weighted SOC complementarity problems by smoothing methods.

The paper is organized as follows. In Section 2, we review some concepts and properties. In Section 3, we derive the computable formula for the Jacobian of the smoothing function in WSOCCP. In Section 4, we show the Jacobian consistency of the smoothing function and estimate the distance between the gradient of smoothing function and the subgradient of the weighted SOC complementarity function. Some conclusions are reported in Section 5.

Throughout this paper, denotes the set of nonnegative numbers. and denote the space of -dimensional real column vectors and the space of matrices, respectively. We use to denote the Euclidean norm and define for a vector or the corresponding induced matrix norm. For simplicity, we often use instead of the column vector . and mean the topological interior and the boundary of the SOC , respectively. For a given set , denotes the convex hull of in , and for any matrix , denotes .

2. Preliminaries

In this section, we briefly recall some definitions and results about the Euclidean Jordan algebra [11] associated with the SOC and subdifferentials [12].

For any , their Jordan product is defined as , and is unit element of this algebra. Given an element , we define the symmetric matrixwhere represents the identity matrix. It is easy to verify that for any . Moreover, is positive definite (and hence invertible) if and only if .

For each , let and be the spectral values and the associated spectral vectors of , given byfor , with any such that . Then, admits a spectral factorization associated with SOC in the form of

For any , let [13]. Then, , , and for any . Moreover, if .

Suppose that is a locally Lipschitzian function; then, from Rademacher’s theorem [14], is differentiable almost everywhere. The Bouligand (B-) subdifferential and the Clarke subdifferential of at are defined bywhere denotes the set of points at which is differentiable. Obviously, if is continuously differentiable at .

Definition 1 (see [12]). Let be a locally Lipschitzian function and be a continuously differentiable function for any , and for any , we have . Then, satisfies the Jacobian consistency property if for any , .

3. Smoothing Function

In this section, we study the properties of the smoothing function (7).

Definition 2 (see [8]). For a nondifferentiable function , we consider 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 .

Lemma 1. For any and , one has

Proof. We first suppose that . Then,and henceThat is, .
Conversely, suppose that ; then, it follows from (7) thatUpon squaring both sides of it, we obtainLetwhich impliesTherefore,Further, it follows from Proposition 3.4 [15] that

Let , , , and the mapping be defined by

For simplicity, we use to denote when , that is,

By direct calculations, we have

Therefore, . From the definition of spectral factorization, can be decomposed aswhere , and are the spectral values and the associated spectral vectors of given byandfor , whereif ; otherwise, is any vector in such that . For any given and any , it can be verified thatfor any , and

Given and , we defineand when ,

The spectral factorization of and is as follows:

By (29), we can partition as , where

Lemma 2. For any given and any , let and be defined as (5) and (7), respectively. Then, we have(i)The function is continuously differentiable everywhere with any , and its Jacobian is given by Here if ; otherwise, with where(ii)For any , we have . Thus, is a smoothing function of .(iii)For any ,

Proof. (i)For any and any , according to Corollary 5.4 [15] and (28), formula (34) holds. By Proposition 5.2 and its proof [15], we get formula (35).(ii)Given any . For any , we obtain from the spectral factorization of and that where and and are, respectively, given by (25) and (26) for . It is obvious that for . Then, and . Thus, by (i) and Definition 2, is a smoothing function of .(iii)By following the proof of Proposition 5.1 [15], we obtain the desired result.

Next, we study some properties of , which will be used in the subsequent analysis.

Lemma 3. For any , let . Then, we have

Proof. We can obtain the desired result by following the proof of Lemma 2 [16].

Lemma 4. For any , let . Then, one has

where . Moreover, the following equivalence holds:

Proof. Sincefrom Lemma 3, we haveIt follows from these equalities that the results in (45)–(47) hold. Since , we have , i.e.,By the last relation and (45)–(47), we obtain that (49) holds. To prove (48), we only need to verify and by the symmetry of and in . From (45)–(47) and (49),From (51), the equivalence is also true.

4. Jacobian Consistency

In this section, we will show the Jacobian consistency property and estimate the distance between the gradient of the smoothing function (7) and the subgradient of the WSOCCP complementarity function (5). For any , , let . Based on smoothing function (7), we define by

From (1) and (56) and Lemma 1,

Since the function is typically nonsmooth, Newton’s method cannot be applied to the system directly. Thus, we can approximately solve the smooth system at each iteration and make decrease gradually by reducing to zero. First, we show that the function satisfies the Jacobian consistency.

Lemma 5. For any arbitrary but fixed vector , we have for any ,where

Proof. By (34) and the symmetry of and , it suffices to prove

Case 1. If , it follows from (25) thatTherefore,

Case 2. If , it is easy to prove (51), andThus, we obtain the following from (25):For any , we may get from (35) that . We first prove for any ,LetBased on (36), (48), and (64), we haveNext, we prove . From (37), (64), and (65), we haveCombining (68) and (69) yields

Case 3. If , it follows from Lemma 4 that and

Lemma 6. For any arbitrary but fixed vector , we have for any ,whereand is defined by (59).

Proof. By Proposition 5.2 [15] and the chain rule for differentiation, the complementarity function is continuously differentiable at any withThus, it suffices to consider the two cases: and .
For any or , let with sufficiently small , and defineThen, we haveObviously, when , we have for . Then by (7), it suffices to showif is differentiable at .

Case 4. If , we obtain , and from (45), (46), and (48),The last relation together with implies that for sufficiently small , we haveFor sufficiently small , we obtain from (77) and (80),It follows from (81) and (82) that , and hence is differentiable at .
Now we will provewhere , in which and are given by (36) and (37) with and replacing and , respectively. By the expression of and (80),By (45), (46), (48), and (84), we haveThus, from (36) and (81),It follows from (73)–(84) that as ,Then, by following the proof of Case 5 in Lemma 5, we haveTherefore, we obtain from (86) and (88) thatNext we will proveBy (45), (46), (48), (81), and (84), we haveand thenTherefore, we obtain from (88) and (92) that

Case 5. If , it follows from Lemma 4 that . Thus, , , andNow we show the Jacobian consistency of the function (56) and then estimate an upper bound of the parameter for the predicted accuracy of the distance between the gradient of (56) and the subgradient of (55).

Theorem 1. The following results hold. (i) The function defined by (56) with satisfies the Jacobian consistency. (ii) For given and any point , let be any function such thatand let be defined byThen, for any such that , we have

Proof. By (56), it suffices to show the Jacobian consistency of with . Definewherefor , and are defined by (59) and (73). LetIt follows from Lemma 5 and Lemma 6 thatand . Hence,which together with Definition 1 and Lemma 2 implies the Jacobian consistency of with . (ii) For any , it follows from the proof of Theorem 1(i) thatThus, we obtain from (34) and (100) thatThen, similar to the proof of Proposition 4.1 [13], we havewhere is given byHence, by following the proof of Theorem 4.1 [13], the result holds.

5. Conclusions

In this paper, we show the Jacobian consistency of the smoothing function for WSOCCP, which will play a key role in analyzing the rapid convergence of smoothing methods. Moreover, in order to adjust a parameter appropriately in smoothing methods, we estimate the distance between the gradient of the smoothing function and the subgradient of the weighted SOC complementarity function .

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 11861026), Guangxi Key Laboratory of Cryptography and Information Security (no. GCIS201819), and Guangxi Key Laboratory of Automatic Detecting Technology and Instruments, China (no. YQ18112).