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Miao et al. Journal of Inequalities and Applications (2016) 2016:291 DOI 10.1186/s13660-016-1243-5

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An alternative approach for a distance inequality associated with the second-order cone and the circular cone Xin-He Miao1 , Yen-chi Roger Lin2 and Jein-Shan Chen2* *

Correspondence: [email protected] 2 Department 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. Let Lθ denote the circular cone in Rn , which is defined by Lθ := x = (x , x ) ∈ R × Rn– | x cos θ ≤ x = x = (x , x ) ∈ R × Rn– | x ≤ x tan θ ,

()

with · denoting the Euclidean norm and θ ∈ (, π ). When θ = π , the circular cone Lθ 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, K is the set of nonnegative reals R+ . It is well known that the second-order cone Kn is a special kind of symmetric cones []. But when θ = π , the circular cone Lθ is 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, provided 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.

Miao et al. Journal of Inequalities and Applications (2016) 2016:291

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In [], Zhou and Chen showed that there is a special relationship between the SOC and the circular cone as follows:

x ∈ Lθ

Ax ∈ K

⇐⇒

n

tan θ with A = 

 In–

,

()

where In– is the (n – ) × (n – ) identity matrix. Based on the relationship () between the SOC and circular cone, Miao et al. [] showed that circular cone complementarity problems can be transformed into the second-order cone complementarity problems. Furthermore from the relationship (), we have x ∈ int Lθ

⇐⇒

Ax ∈ int Kn

and x ∈ bd Lθ

⇐⇒

Ax ∈ bd Kn .

Besides the relationship between second-order cone and circular cone, some topological 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 between 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 vector x = (x , x ) ∈ R × Rn– , the spectral decomposition of x with respect to second-order cone is given by () x = λ (x)u() x + λ (x)ux ,

()

() where λ (x), λ (x), u() x , and ux are expressed as

λi (x) = x + (–) x i

and

u(i) x

 = 

 , (–)i w

i = , ,

()

with w = xx if x = , or any vector in Rn– satisfying w =  if x = . In the setting of the circular cone Lθ , Zhou and Chen [] gave the following spectral decomposition of x ∈ Rn with respect to Lθ : () x = μ (x)v() x + μ (x)vx ,

()


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() where μ (x), μ (x), v() x , and vx are expressed as

μ (x) = x – x cot θ , μ (x) = x + x tan θ ,

and

v() x = v() x =

 +cot θ  +tan θ

 , – cot θ·w  , tan θ·w

()

x if x = , or any vector in Rn– satisfying w =  if x = . Moreover, λ (x), x () λ (x) and u() x , ux are called the spectral values and the spectral vectors of x associated () with Kn , whereas μ (x), μ (x) and v() x , vx are called the spectral values and the spectral

with w =

vectors of x associated with Lθ , respectively. To proceed, we denote x+ (resp. xθ+ ) the projection of x onto Kn (resp. Lθ ); also we set a+ = max{a, } for any real number a ∈ R. According to the spectral decompositions () and () of x, the expressions of x+ and xθ+ can be obtained explicitly, as stated in the following 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 + ux ⎧ n ⎪ x +x ⎨ x, if x ∈ K ,  = , if x ∈ –(Kn )∗ = –Kn , where u = x +x x ; ⎪ ⎩  x u, otherwise, (b)

() xθ+ = x – x cot θ + u() x + x + x tan θ + ux ⎧ ⎪ x +x tan θ ⎨ x, if x ∈ Lθ , θ +tan ∗ . = , if x ∈ –(Lθ ) = –L π –θ , where v = x +x tan θ  ⎪ (  +tan tan θ ) xx ⎩ θ v, otherwise, Based on the expression of the projection xθ+ onto Lθ in Lemma ., it is easy to obtain, for any x = (x , x ) ∈ R × Rn– , the explicit formula of projection of x ∈ Rn onto the dual cone L∗θ (denoted by (xθ )∗+ ): θ ∗

() x + = x – x tan θ + u() x + x + x cot θ + ux ⎧ ∗ ⎪ x +x cot θ ⎨ x, if x ∈ Lθ = L π –θ ,  +cot θ . = , if x ∈ –(L∗θ )∗ = –Lθ , where ω = x +x cot θ  ⎪ (  +cot cot θ ) xx ⎩ θ ω, otherwise,

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 cone Lθ and its dual cone L∗θ , respectively. Let A be the matrix defined as in (). Then the following hold:


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(a) If Ax ∈ Kn , then (Ax)+ = Axθ+ . (b) If Ax ∈ –Kn , then (Ax)+ = A(xθ )∗+ = . θ (c) If Ax ∈/ Kn ∪ (–Kn ), then (Ax)+ = ( +tan )A– (xθ )∗+ , where (xθ )∗+ retains its expression  only in the case of x ∈/ 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 ) =  (x tan θ – x )– +  (x tan θ + x )– , cot θ tan θ   (b) dist(x, Lθ ) = +cot  θ (x tan θ – x )– + +tan θ (x cot θ + x )– , where (a)– = min{a, }.

Now, applying Theorem ., we can obtain the relation on the distance formulas associated with the second-order cone and the circular cone. Note that when θ = π , we know Lθ = Kn and Ax = x. Thus, it is obvious that dist(Ax, Kn ) = dist(x, Lθ ). In the following theorem, we only consider the case θ = π . Theorem . For any x = (x , x ) ∈ R × Rn– , according to the expressions of the distance formulas dist(Ax, Kn ) and dist(x, Lθ ), the following hold. (a) For θ ∈ (, π ), we have

dist Ax, Kn ≤ dist(x, Lθ ) ≤ cot θ · dist Ax, Kn . (b) For θ ∈ ( π , π ), we have

dist(x, Lθ ) ≤ dist Ax, Kn ≤ tan θ · dist(x, Lθ ).

3 Proofs of main results 3.1 Proof of Theorem 2.1 (a) If Ax ∈ 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) If Ax ∈ –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) If Ax ∈/ Kn ∪ (–Kn ), from Lemma .(a), we have (Ax)+ =

x tan θ+x  x tan θ+x x  x

x +x cot θ  

tan θ  +cot θ  + cot θ = x +x cot θ x    x +cot θ x +x cot θ  + tan θ θ · A– x +x cot+cot = θ    · cot θ ·  +cot θ

=

x x

∗  + tan θ · A– xθ + . 

The proof is complete.


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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 element x ∈ Rn , i.e., x ∈/ L∗θ ∪ (–Lθ ) or x ∈ L∗θ . When x ∈/ L∗θ ∪ (–Lθ ), the relationship between (Ax)+ and (xθ )∗+ is just as stated in θ )A– (xθ )∗+ . However, when x ∈ L∗θ , we have (xθ )∗+ = x. Theorem .(c), that is, (Ax)+ = ( +tan  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 any x = (x , x ) ∈ R × Rn– , from the spectral decomposition () with respect to the () () () SOC, we have Ax = (x tan θ – x )u() x + (x tan θ + x )ux , where ux and ux are given as () in (). It follows from Lemma .(a) that (Ax)+ = (x tan θ – x )+ u() x + (x tan θ + x )+ ux . Hence, we obtain the distance dist(Ax, Kn ):

dist Ax, Kn = Ax – (Ax)+

() = x tan θ – x – u() x + x tan θ + x – ux

 

  x tan θ – x – + x tan θ + x – . =   (b) For any x = (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+



 tan θ cot θ = x tan θ – x – + x cot θ + x – .    + cot θ  + tan θ

3.3 Proof of Theorem 2.3 (a) For θ ∈ (, π ), we have  < tan θ <  < cot θ and Lθ ⊂ Kn ⊂ L∗θ . We discuss three cases according to x ∈ Lθ , x ∈ –L∗θ , and x ∈/ Lθ ∪ (–L∗θ ). Case . If x ∈ Lθ , then Ax ∈ Kn , which clearly yields dist(Ax, Kn ) = dist(x, Lθ ) = . Case . If x ∈ –L∗θ , then x cot θ ≤ –x and dist(x, Lθ ) = x =

x + x  .

In this case, there are two subcases for the element Ax. If x cot θ ≤ x tan θ ≤ –x , i.e., Ax ∈ –Kn , it follows that

dist Ax, Kn = Ax = x tan θ + x  ≤ x + x  = dist(x, Lθ ) ≤ x + x  cot θ = cot θ · x tan θ + x 

= cot θ · dist Ax, Kn ,


Page 6 of 10

where the first inequality holds since tan θ <  (it becomes an equality only in the case of x = ), and the second inequality holds since cot θ >  (it becomes an equality only in the case of x = ). On the other hand, if x cot θ ≤ –x < x tan θ ≤ , we have


 

  = x tan θ – x – + x tan θ + x –  

  x tan θ – x =  ≤ x tan θ + x  < x + x  = dist(x, Lθ ) ≤ x – x x cot θ    x + x  cot θ – x x cot θ <   

= cot θ · dist Ax, Kn , where the third inequality holds because x ≤ –x cot θ , and the fourth inequality holds since x > –x tan θ ≥ . Therefore, for the subcases of x ∈ –L∗θ , we can conclude that

dist Ax, Kn ≤ dist(x, Lθ ) ≤ cot θ · dist Ax, Kn , and dist(x, Lθ ) = cot θ · dist(Ax, Kn ) holds only in the case of x = . Case . If x ∈/ Lθ ∪ (–L∗θ ), then –x tan θ < x < x cot θ , which yields x tan θ < x and x cot θ > –x . Thus, we have dist(x, Lθ ) = x – xθ+



 cot θ tan θ tan θ – x + x x cot θ + x – =   –    + cot θ  + tan θ 

 cot θ x tan θ – x . =  + cot θ On the other hand, it follows from –x tan θ < x < x cot θ and θ ∈ (, π ) that –x < –x tan θ < x tan θ < x . This implies that


 

  = x tan θ – x – + x tan θ + x –  

  x tan θ – x . = 


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From this and θ ∈ (, π ), we see that

dist Ax, Kn =

  x tan θ – x 

 cot θ x tan θ – x  + cot θ

   x tan θ – x = dist(x, Lθ ) =   + tan θ 

 = dist Ax, Kn .   + tan θ
dist(x, Lθ ). Moreover, by the expressions of dist(Ax, Kn ) and dist(x, Lθ ), it is easy to verify

dist Ax, Kn =

 dist(x, Lθ ).  + tan θ

On the other hand, if –x tan θ < x tan θ ≤ –x < x , then we have Ax ∈ –Kn , which implies

dist Ax, Kn = Ax = x tan θ + x  . Therefore, it follows that dist(x, Lθ ) = < ≤

 cot θ x tan θ – x   + cot θ

  x tan θ – x 

x tan θ + x  = dist Ax, Kn .

Since



tan θ x tan θ – x –  + tan θ x tan θ + x  = –x x tan θ – x tan θ – x 

= –x tan θ · x tan θ + x tan θ + x –x tan θ – x


Page 9 of 10

≥ x –x tan θ – x ≥ , where the first inequality holds due to –x tan θ < x tan θ , and the second inequality holds due to x tan θ < –x and θ ∈ ( π , π ), we have

dist Ax, Kn ≤ tan θ · dist(x, Lθ ). From all the above analyses and the fact that max{tan θ , can conclude that

 } = tan θ +tan θ

for θ ∈ ( π , π ), we

dist(x, Lθ ) ≤ dist Ax, Kn ≤ tan θ · dist(x, Lθ ). Thus, the proof is complete.

Remark . We point out that Theorem . is equivalent to the results in [], Theorem ., that is, for any x, z ∈ Rn , we have

A– dist Az, Kn ≤ dist(z, Lθ ) ≤ A– dist Az, Kn

()

– –

A dist A– x, Lθ ≤ dist x, Kn ≤ A dist A– x, Lθ .

()

and

However, the above inequalities depends on the factors A and A– . 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 programming problems and the circular cone programming problems. In particular, from the proof of Theorem ., it is clear that dist(Ax, Kn ) = tan θ · dist(x, Lθ ) holds only under the cases of x = (x , x ) ∈ Lθ or x = ; otherwise we would have the strict inequality dist(Ax, Kn ) < tan θ · dist(x, Lθ ). In contrast, it takes tedious algebraic manipulations to obtain such situations by using () and (). The following example elaborates more why dist(Ax, Kn ) = tan θ · dist(x, Lθ ) holds only in the cases of x = (x , x ) ∈ Lθ or x = . Example . Let x = (x , x ) ∈ R × Rn– and

tan θ A= 

 In–

.

When x ∈ Lθ , we have Ax ∈ Kn . It is clear to see that dist(Ax, Kn ) = tan θ · dist(x, Lθ ) = . When x = , i.e., x = (x , ) ∈ R × Rn– , it follows that Ax = (x tan θ , ). If x ≥ , we have x ∈ Lθ and Ax ∈ Kn , which implies that dist(Ax, Kn ) = tan θ · dist(x, Lθ ) = . In the other case, if x < , we see that x ∈ –L∗θ and Ax ∈ –Kn . All the above gives dist(Ax, Kn ) = Ax = |x | tan θ = tan θ · dist(x, Lθ ).


Page 10 of 10

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 1 Department of Mathematics, Tianjin University, Tianjin, 300072, China. 2 Department 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)