R E S E A R C H Open Access
An alternative approach for a distance
inequality associated with the second-order cone and the circular cone
Xin-He Miao1, Yen-chi Roger Lin2and Jein-Shan Chen2*
*Correspondence:
jschen@math.ntnu.edu.tw
2Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan
Full list of author information is available at the end of the article
Abstract
It is well known that the second-order cone and the circular cone have many analogous properties. In particular, there exists an important distance inequality associated with the second-order cone and the circular cone. The inequality indicates that the distances of arbitrary points to the second-order cone and the circular cone are equivalent, which is crucial in analyzing the tangent cone and normal cone for the circular cone. In this paper, we provide an alternative approach to achieve the aforementioned inequality. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds. Such a clarification is helpful for further study of the relationship between the second-order cone programming problems and the circular cone programming problems.
Keywords: second-order cone; circular cone; projection; distance
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. LetLθ
denote the circular cone inRn, which is defined by Lθ:=
x= (x,x)∈R×Rn–| xcosθ≤x
=
x= (x,x)∈R×Rn–| x ≤xtanθ
, ()
with · denoting the Euclidean norm andθ∈(,π). Whenθ=π, the circular coneLθ
reduces to the well-known second-order cone (SOC)Kn[, ] (also called the Lorentz cone),i.e.,
Kn:=
(x,x)∈R×Rn–| x ≤x .
In particular,Kis the set of nonnegative realsR+. It is well known that the second-order coneKnis a special kind of symmetric cones []. But whenθ=π, the circular coneLθis a non-symmetric cone [, , ].
©The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
In [], Zhou and Chen showed that there is a special relationship between the SOC and the circular cone as follows:
x∈Lθ ⇐⇒ Ax∈Kn withA=
tanθ
In–
, ()
whereIn–is the (n– )×(n– ) identity matrix. Based on the relationship () between the SOC and circular cone, Miaoet al.[] showed that circular cone complementarity prob- lems can be transformed into the second-order cone complementarity problems. Further- more from the relationship (), we have
x∈intLθ ⇐⇒ Ax∈intKn and x∈bdLθ ⇐⇒ Ax∈bdKn.
Besides the relationship between second-order cone and circular cone, some topologi- cal structures play important roles in theoretical analysis for optimization problems. For example, the projection formula onto a cone facilitates designing algorithms for solving conic programming problems [–]; the distance formula from an element to a cone is an important factor in the approximation theory; the tangent cone and normal cone are crucial in analyzing the structure of the solution set for optimization problems [–].
From the above illustrations, an interesting question arises: What is the relationship be- tween second-order cone and circular cone regarding the projection formula, the distance formula, the tangent cone and normal cone, and so on? The issue of the tangent cone and normal cone has been studied in [], Theorem .. In this paper, we focus on the other two issues.
More specifically, we provide an alternative approach to achieve an inequality which was obtained in [], Theorem .. Although the proof is a bit longer than the existing one, the new approach offers a way to clarify when the equality holds, which is helpful for further studying in the relationship between the second-order cone programming problems and the circular cone programming problems.
In order to study the relationship between second-order cone and circular cone, we need to recall some background materials. For any vectorx= (x,x)∈R×Rn–, the spectral decomposition ofxwith respect to second-order cone is given by
x=λ(x)u()x +λ(x)u()x , ()
whereλ(x),λ(x),u()x , andu()x are expressed as
λi(x) =x+ (–)ix and u(i)x =
(–)iw
, i= , , ()
withw=xx
ifx= , or any vector inRn–satisfyingw= ifx= . In the setting of the circular coneLθ, Zhou and Chen [] gave the following spectral decomposition ofx∈Rn with respect toLθ:
x=μ(x)v()x +μ(x)v()x , ()
whereμ(x),μ(x),v()x , andv()x are expressed as
μ(x) =x–xcotθ, μ(x) =x+xtanθ, and
v()x =+cotθ
–cotθ·w
, v()x =+tanθ
tanθ·w
, ()
withw=xx
ifx= , or any vector inRn–satisfyingw= ifx= . Moreover,λ(x), λ(x) andu()x ,u()x are called the spectral values and the spectral vectors ofxassociated withKn, whereasμ(x),μ(x) andv()x ,v()x are called the spectral values and the spectral vectors ofxassociated withLθ, respectively.
To proceed, we denotex+(resp.xθ+) the projection ofxontoKn(resp.Lθ); also we set a+=max{a, }for any real numbera∈R. According to the spectral decompositions () and () ofx, the expressions ofx+andxθ+can be obtained explicitly, as stated in the fol- lowing lemma.
Lemma .([, ]) Let x= (x,x)∈R×Rn–have the spectral decompositions given as ()and()with respect to SOC and circular cone,respectively.Then the following hold:
(a)
x+= x–x
+u()x + x+x
+u()x
=
⎧⎪
⎨
⎪⎩
x, ifx∈Kn,
, ifx∈–(Kn)∗= –Kn, u, otherwise,
whereu=
x+x x+x
x x
;
(b)
xθ+= x–xcotθ
+u()x + x+xtanθ
+u()x
=
⎧⎪
⎨
⎪⎩
x, ifx∈Lθ,
, ifx∈–(Lθ)∗= –Lπ–θ, v, otherwise,
wherev=
x
+xtanθ
+tanθ
(x+x+tantanθθ tanθ)xx
.
Based on the expression of the projectionxθ+ontoLθin Lemma ., it is easy to obtain, for anyx= (x,x)∈R×Rn–, the explicit formula of projection ofx∈Rnonto the dual coneL∗θ(denoted by (xθ)∗+):
xθ∗
+= x–xtanθ
+u()x + x+xcotθ
+u()x
=
⎧⎪
⎨
⎪⎩
x, ifx∈L∗θ=Lπ–θ,
, ifx∈–(L∗θ)∗= –Lθ, ω, otherwise,
whereω=
x
+xcotθ
+cotθ
(x+x+cotcotθθ cotθ)xx
.
2 Main results
In this section, we give the main results of this paper.
Theorem . For any x∈Rn,let xθ+ and(xθ)∗+ be the projections of x onto the circular coneLθand its dual coneL∗θ,respectively.Let A be the matrix defined as in().Then the following hold:
(a) IfAx∈Kn,then(Ax)+=Axθ+. (b) IfAx∈–Kn,then(Ax)+=A(xθ)∗+= .
(c) IfAx∈/Kn∪(–Kn),then(Ax)+= (+tanθ)A–(xθ)∗+,where(xθ)∗+retains its expression only in the case ofx∈/L∗θ∪(–Lθ).
Theorem . Let x= (x,x)∈R×Rn–have the spectral decompositions with respect to the SOC and the circular cone given as in()and(),respectively.Then the following hold:
(a) dist(Ax,Kn) =
(xtanθ–x)–+(xtanθ+x)–, (b) dist(x,Lθ) =
cotθ
+cotθ(xtanθ–x)–++tantanθθ(xcotθ+x)–, where(a)–=min{a, }.
Now, applying Theorem ., we can obtain the relation on the distance formulas asso- ciated with the second-order cone and the circular cone. Note that whenθ=π, we know Lθ=Kn andAx=x. Thus, it is obvious thatdist(Ax,Kn) =dist(x,Lθ). In the following theorem, we only consider the caseθ=π.
Theorem . For any x= (x,x)∈R×Rn–,according to the expressions of the distance formulasdist(Ax,Kn)anddist(x,Lθ),the following hold.
(a) Forθ∈(,π),we have dist Ax,Kn
≤dist(x,Lθ)≤cotθ·dist Ax,Kn . (b) Forθ∈(π,π),we have
dist(x,Lθ)≤dist Ax,Kn
≤tanθ·dist(x,Lθ).
3 Proofs of main results 3.1 Proof of Theorem 2.1
(a) IfAx∈Kn, by the relationship () between the SOC and the circular cone, we have x∈Lθ. Thus, it is easy to see that (Ax)+=Ax=Axθ+.
(b) IfAx∈–Kn, we know that –Ax∈Kn, which implies (Ax)+= . Besides, combining with (), we have –x∈Lθ, which leads to (xθ)∗+= . Hence, we have (Ax)+=A(xθ)∗+= .
(c) IfAx∈/Kn∪(–Kn), from Lemma .(a), we have
(Ax)+= x
tanθ+x
xtanθ+x
x x
=tanθ
+cotθ x
+xcotθ
+cotθ x+xcotθ
+cotθ x x
= +tanθ
·A–
x
+xcotθ
+cotθ x+xcotθ
+cotθ ·cotθ·xx
= +tanθ
·A– xθ∗
+.
The proof is complete.
Remark . Here, we say a few more words as regards part (c) in Theorem .. Indeed, if Ax∈/ Kn∪(–Kn), there are two cases for the elementx∈Rn, i.e.,x∈/ L∗θ∪(–Lθ) or x∈L∗θ. Whenx∈/L∗θ∪(–Lθ), the relationship between (Ax)+and (xθ)∗+is just as stated in Theorem .(c), that is, (Ax)+= (+tanθ)A–(xθ)∗+. However, whenx∈L∗θ, we have (xθ)∗+=x.
This implies that the relationship between (Ax)+and (xθ)∗+ is not very clear. Hence, the relation between (Ax)+and (xθ)∗+in Theorem .(c) is a bit limited.
3.2 Proof of Theorem 2.2
(a) For anyx= (x,x)∈R×Rn–, from the spectral decomposition () with respect to the SOC, we haveAx= (xtanθ–x)u()x + (xtanθ+x)u()x , whereu()x andu()x are given as in (). It follows from Lemma .(a) that (Ax)+= (xtanθ–x)+u()x + (xtanθ+x)+u()x . Hence, we obtain the distancedist(Ax,Kn):
dist Ax,Kn
=Ax– (Ax)+
= xtanθ–x
–u()x + xtanθ+x
–u()x
=
xtanθ–x –+
xtanθ+x –.
(b) For anyx= (x,x)∈R×Rn–, from the spectral decomposition () with respect to circular cone and Lemma .(b), with the same argument, it is easy to see that
dist(x,Lθ) =x–xθ+
=
cotθ
+cotθ xtanθ–x
–+ tanθ
+tanθ xcotθ+x –.
3.3 Proof of Theorem 2.3
(a) Forθ∈(,π), we have <tanθ< <cotθ andLθ⊂Kn⊂L∗θ. We discuss three cases according tox∈Lθ,x∈–L∗θ, andx∈/Lθ∪(–L∗θ).
Case .Ifx∈Lθ, thenAx∈Kn, which clearly yieldsdist(Ax,Kn) =dist(x,Lθ) = .
Case .Ifx∈–L∗θ, thenxcotθ≤–xand dist(x,Lθ) =x=
x+x.
In this case, there are two subcases for the elementAx. Ifxcotθ≤xtanθ ≤–x,i.e., Ax∈–Kn, it follows that
dist Ax,Kn
=Ax=
xtanθ+x≤
x+x
=dist(x,Lθ)≤
x+xcotθ
=cotθ·
xtanθ+x
=cotθ·dist Ax,Kn ,
where the first inequality holds sincetanθ < (it becomes an equality only in the case of x= ), and the second inequality holds sincecotθ> (it becomes an equality only in the case ofx= ). On the other hand, ifxcotθ≤–x<xtanθ≤, we have
dist Ax,Kn
=Ax– (Ax)+
=
xtanθ–x –+
xtanθ+x –
=
xtanθ–x
≤
xtanθ+x<
x+x
=dist(x,Lθ)≤
x–xxcotθ
<
x+
xcotθ–xxcotθ
=cotθ·dist Ax,Kn ,
where the third inequality holds becausex ≤–xcotθ, and the fourth inequality holds sincex> –xtanθ≥. Therefore, for the subcases ofx∈–L∗θ, we can conclude that
dist Ax,Kn
≤dist(x,Lθ)≤cotθ·dist Ax,Kn ,
anddist(x,Lθ) =cotθ·dist(Ax,Kn) holds only in the case ofx= .
Case .Ifx∈/ Lθ∪(–L∗θ), then –xtanθ<x<xcotθ, which yieldsxtanθ<x andxcotθ> –x. Thus, we have
dist(x,Lθ) =x–xθ+
=
cotθ
+cotθ xtanθ–x
–+ tanθ
+tanθ xcotθ+x –
=
cotθ
+cotθ xtanθ–x
.
On the other hand, it follows from –xtanθ<x<xcotθ andθ∈(,π) that –x< –xtanθ<xtanθ<x.
This implies that dist Ax,Kn
=Ax– (Ax)+
=
xtanθ–x –+
xtanθ+x –
=
xtanθ–x
.
From this andθ∈(,π), we see that
dist Ax,Kn
=
xtanθ–x
<
cotθ
+cotθ xtanθ–x
=dist(x,Lθ) =
+tanθ
xtanθ–x
=
+tanθ dist Ax,Kn .
Therefore, in these cases ofx∈/Lθ∪(–L∗θ), we can conclude
dist Ax,Kn
<dist(x,Lθ) =
+tanθdist Ax,Kn .
To sum up, from all the above and the fact thatmax{cotθ,
+tanθ}=cotθ forθ∈(,π), we obtain
dist Ax,Kn
≤dist(x,Lθ)≤cotθ·dist Ax,Kn .
(b) Forθ∈(π,π), we have <cotθ< <tanθ andL∗θ⊂Kn⊂Lθ. Again we discuss the following three cases.
Case .Ifx∈Lθ, thenAx∈Kn, which implies thatdist(Ax,Kn) =dist(x,Lθ) = .
Case .Ifx∈–L∗θ, thenxcotθ≤–xand dist(x,Lθ) =x=
x+x.
It follows fromxcotθ≤–xandθ∈(π,π) thatxtanθ≤xcotθ≤–x, which leads toAx∈–Kn. Hence, we have
dist Ax,Kn
=Ax=
xtanθ+x.
With this, it is easy to verify thatdist(Ax,Kn)≥dist(x,Lθ) forθ ∈(π,π). Moreover, we note that
tanθ·dist(x,Lθ) =
tanθ x+x
≥
xtanθ+x=dist Ax,Kn . Thus, it follows that
dist(x,Lθ)≤dist Ax,Kn
≤tanθ·dist(x,Lθ),
anddist(Ax,Kn) =tanθ·dist(x,Lθ) holds only in the case ofx= .
Case .Ifx∈/Lθ∪(–L∗θ), then we have –xtanθ<x<xcotθ and dist(x,Lθ) =x–xθ+
=
cotθ
+cotθ xtanθ–x
–+ tanθ
+tanθ xcotθ+x –
=
cotθ
+cotθ xtanθ–x
.
Since –xtanθ <x <xcotθ, it follows immediately that –xtanθ <xtanθ <
x. Again, there are two subcases for the elementAx. If –xtanθ< –x<xtanθ<
x, then we haveAx∈/Kn∪(–Kn). Thus, it follows that dist Ax,Kn
=Ax– (Ax)+
=
xtanθ–x –+
xtanθ+x –
=
xtanθ–x
. This together withθ∈(π,π) yields
dist Ax,Kn
>dist(x,Lθ).
Moreover, by the expressions ofdist(Ax,Kn) anddist(x,Lθ), it is easy to verify
dist Ax,Kn
=
+tanθ dist(x,Lθ).
On the other hand, if –xtanθ<xtanθ≤–x<x, then we haveAx∈–Kn, which implies
dist Ax,Kn
=Ax=
xtanθ+x. Therefore, it follows that
dist(x,Lθ) =
cotθ
+cotθ xtanθ–x
<
xtanθ–x
≤
xtanθ+x=dist Ax,Kn . Since
tanθ xtanθ–x
– +tanθ xtanθ+x
= –xxtanθ–xtanθ–x
= –xtanθ· xtanθ+xtanθ
+x –xtanθ–x
≥ x –xtanθ–x
≥,
where the first inequality holds due to –xtanθ<xtanθ, and the second inequality holds due toxtanθ< –xandθ∈(π,π), we have
dist Ax,Kn
≤tanθ·dist(x,Lθ).
From all the above analyses and the fact thatmax{tanθ,
+tanθ}=tanθforθ∈(π,π), we can conclude that
dist(x,Lθ)≤dist Ax,Kn
≤tanθ·dist(x,Lθ).
Thus, the proof is complete.
Remark . We point out that Theorem . is equivalent to the results in [], Theo- rem ., that is, for anyx,z∈Rn, we have
A–dist Az,Kn
≤dist(z,Lθ)≤A–dist Az,Kn
() and
A––dist A–x,Lθ
≤dist x,Kn
≤ Adist A–x,Lθ
. ()
However, the above inequalities depends on the factorsAandA–. Here, we provide a more concrete and simple expression for the inequality. What is the benefit of such a new expression? Indeed, the new approach provides the situation where the equality holds, which is helpful for further study of the relationship between the second-order cone pro- gramming problems and the circular cone programming problems. In particular, from the proof of Theorem ., it is clear thatdist(Ax,Kn) =tanθ ·dist(x,Lθ) holds only un- der the cases ofx= (x,x)∈Lθ orx= ; otherwise we would have the strict inequality dist(Ax,Kn) <tanθ·dist(x,Lθ). In contrast, it takes tedious algebraic manipulations to ob- tain such situations by using () and ().
The following example elaborates more whydist(Ax,Kn) =tanθ·dist(x,Lθ) holds only in the cases ofx= (x,x)∈Lθorx= .
Example . Letx= (x,x)∈R×Rn–and
A=
tanθ
In–
.
Whenx∈Lθ, we haveAx∈Kn. It is clear to see thatdist(Ax,Kn) =tanθ·dist(x,Lθ) = .
Whenx= ,i.e.,x= (x, )∈R×Rn–, it follows thatAx= (xtanθ, ). Ifx≥, we have x∈Lθ andAx∈Kn, which implies thatdist(Ax,Kn) =tanθ·dist(x,Lθ) = . In the other case, ifx< , we see thatx∈–L∗θandAx∈–Kn. All the above givesdist(Ax,Kn) =Ax=
|x|tanθ=tanθ·dist(x,Lθ).
Competing interests
The authors declare that none of the authors have any competing interests in the manuscript.
Authors’ contributions
All authors participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Tianjin University, Tianjin, 300072, China.2Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan.
Acknowledgements
We would like to thank the editor and the anonymous referees for their careful reading and constructive comments which have helped us to significantly improve the presentation of the paper. The first author’s work is supported by National Natural Science Foundation of China (No. 11471241). The third author’s work is supported by Ministry of Science and Technology, Taiwan.
Received: 4 May 2016 Accepted: 14 November 2016
References
1. Pinto Da Costa, A, Seeger, A: Numerical resolution of cone-constrained eigenvalue problems. Comput. Appl. Math.
28, 37-61 (2009)
2. Zhou, JC, Chen, JS: Properties of circular cone and spectral factorization associated with circular cone. J. Nonlinear Convex Anal.14, 807-816 (2013)
3. Chen, JS, Chen, X, Tseng, P: Analysis of nonsmooth vector-valued functions associated with second-order cones.
Math. Program.101, 95-117 (2004)
4. Facchinei, F, Pang, J: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
5. Faraut, U, Korányi, A: Analysis on Symmetric Cones. Oxford Mathematical Monographs. Oxford University Press, New York (1994)
6. Nesterov, Y: Towards nonsymmetric conic optimization. Discussion paper 2006/28, Center for Operations Research and Econometrics (2006)
7. Zhou, JC, Chen, JS, Hung, HF: Circular cone convexity and some inequalities associated with circular cones. J. Inequal.
Appl.2013, Article ID 571 (2013)
8. Miao, XH, Guo, SJ, Qi, N, Chen, JS: Constructions of complementarity functions and merit functions for circular cone complementarity problem. Comput. Optim. Appl.63, 495-522 (2016)
9. Chen, JS: A new merit function and its related properties for the second-order cone complementarity problem. Pac.
J. Optim.2, 167-179 (2006)
10. Fukushima, M, Luo, ZQ, Tseng, P: Smoothing functions for second-order cone complementarity problems. SIAM J.
Optim.12, 436-460 (2001)
11. Hayashi, S, Yamashita, N, Fukushima, M: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim.15, 593-615 (2005)
12. Kong, LC, Sun, J, Xiu, NH: A regularized smoothing Newton method for symmetric cone complementarity problems.
SIAM J. Optim.19, 1028-1047 (2008)
13. Zhang, Y, Huang, ZH: A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities.
J. Comput. Appl. Math.233, 2312-2321 (2010)
14. Jeyakumar, V, Yang, XQ: Characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl.87, 747-755 (1995)
15. Mangasarian, OL: A simple characterization of solution sets of convex programs. Oper. Res. Lett.7, 21-26 (1988) 16. Miao, XH, Chen, JS: Characterizations of solution sets of cone-constrained convex programming problems. Optim.
Lett.9, 1433-1445 (2015)
