to appear in Optimization, 2016 Further relationship between second-order cone and positive semidefinite cone Jinchuan Zhou 1 Department of Mathematics School of Science Shandong University of Technology Zibo 255049, P.R. China E-mail: [email protected]Jingyong Tang 2 College of Mathematics and Information Science Xinyang Normal University Xinyang 464000, Henan, P.R.China E-mail: [email protected]Jein-Shan Chen 3 Department of Mathematics National Taiwan Normal University Taipei 11677, Taiwan E-mail: [email protected]February 2, 2016 (1st revised on May 2, 2016) (2nd revised on July 21, 2016) Abstract. It is well known that second-order cone programming can be regarded as a special case of positive semidefinite programming by using the arrow matrix. This paper further studies the relationship between second-order cones and positive semi- definite matrix cones. In particular, we explore the relationship to expressions regarding 1 The author’s work is supported by National Natural Science Foundation of China (11101248, 11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026). 2 The author’s work is supported by Basic and Frontier Technology Research Project of Henan Province (162300410071). 3 Corresponding author. The author’s work is supported by Ministry of Science and Technology, Taiwan. 1
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to appear in Optimization, 2016
Further relationship between second-order cone and positivesemidefinite cone
Abstract. It is well known that second-order cone programming can be regarded as
a special case of positive semidefinite programming by using the arrow matrix. This
paper further studies the relationship between second-order cones and positive semi-
definite matrix cones. In particular, we explore the relationship to expressions regarding
1The author’s work is supported by National Natural Science Foundation of China (11101248,
11271233) and Shandong Province Natural Science Foundation (ZR2010AQ026).2The author’s work is supported by Basic and Frontier Technology Research Project of Henan
Province (162300410071).3Corresponding author. The author’s work is supported by Ministry of Science and Technology,
Taiwan.
1
distance, projection, tangent cone, normal cone, and the KKT system. Understanding
these relationships will help us to see the connection and difference between the SOC
For y 6= 0, this implies 〈x, y〉 = 0⇐⇒ 〈Lx, L−y〉 = 0 if and only if n = 2. This completes
the proof. 2
19
We point out that the relationship between LTnK(x) and TSn+(Lx) has been described
in Theorem 4.2. Although the normal cone is the polar cone of the tangent cone for a
given convex set, it fails to achieve the relationship between LNKn (x) and NSn+(Lx) by
taking polar on both sides of (16), because the operator L is not invariant under polar
operator. More precisely, for x, y,
〈x, y〉 ≤ 0 ; 〈Lx, Ly〉 ≤ 0.
In fact, if 〈x, y〉 ≤ 0, i.e., x1y1 + xT2 y2 ≤ 0, whereas 〈Lx, Ly〉 = nx1y1 + 2xT2 y2. It is clear
that for n ≥ 3,
x1y1 + xT2 y2 ≤ 0 ; nx1y1 + 2xT2 y2 ≤ 0.
For example, taking x = (2, 1, 1) and y = (2,−3,−2) gives 〈x, y〉 = −1 < 0 and
〈Lx, Ly〉 = 2 > 0. All the above explains why we need a different approach to prove
Theorem 5.4.
6 Relation on KKT Systems
In this section, we turn our attention to the relation on KKT systems. First, we know
that the following second-order cone programming problem (SOCP)
min f(x)
s.t. g(x) ∈ Kn (24)
can be rewritten as a positive semidefinite programming problem (SDP)
min f(x)
s.t. Lg(x) ∈ Sn+,(25)
where g : IRn → IRn is expressed as g = (g1, g2, · · · gn). The KKT systems of the
above SOCP (24) and SDP (25) are respectively denoted by K(x) and K(Lx), which are
expressed as
K(x) :=
{λ ∈ IRn
∣∣∣∣ 0 = ∇f(x) +n∑i=1
λi∇gi(x), λ ∈ NKn(g(x))
},
K(Lx) :=
{Γ ∈ IRn×n
∣∣∣∣ 0 = ∇f(x) +
(n∑i=1
Γii
)∇g1(x) + 2
n∑i=2
Γ1i∇gi(x), Γ ∈ NSn+(Lg(x))
}.
In order to describe the relation between K(x) and K(Lx), we define the following two
mappings: Given x ∈ IRn and X ∈ Sn, we define
M(X) :=
(n∑i=1
Xii 2X12 · · · 2X1n
)(26)
20
and
M(x) :=
{Γ ∈ Sn−
∣∣∣∣ n∑i=1
Γii = x1, Γ1i =1
2xi, i = 2, · · · , n
}. (27)
Then, the relation between KKT system of the above two problems is given as below.
Theorem 6.1. Let x = (x1, x2) ∈ IR × IRn−1 belong to Kn, i.e., x ∈ Kn. Suppose that
the mappings M and M are defined as in (26) and (27), respectively. Then, the following
statements hold:
(a) M(NKn(x)) = NSn+(Lx) and M(NSn+(Lx)) = NKn(x);
(b) M(K(x)) = K(Lx) and M(K(Lx)) = K(x).
Proof. (a) We first show
M(NKn(x)) ⊆ NSn+(Lx) and M(NSn+(Lx)) ⊆ NKn(x). (28)
Let y ∈ NKn(x). Take Γ ∈ M(y). Then Γ ∈ Sn− satisfiesn∑i=1
Γii = y1 and Γ1i = 12yi for
i = 2, · · · , n. Hence
〈Γ, Lx〉 =n∑i=1
x1Γii + 2n∑i=2
xiΓ1i =n∑i=1
xiyi = xTy = 0,
where the last step is due to y ∈ NKn(x). This says Γ ∈ NSn+(Lx), i.e., M(NKn(x)) ⊆NSn+(Lx).
For the other part, taking Γ ∈ NSn+(Lx), then −Γ ∈ Sn+, which implies −M(Γ) =
M(−Γ) ∈ Kn by [13, Theorem 1], i.e., M(Γ) ∈ −Kn. Note that
〈M(Γ), x〉 =n∑i=1
Γiix1 +n∑i=2
2xiΓ1i = 〈Γ, Lx〉 = 0,
where the last step is obtained by Γ ∈ NSn+(Lx). In summary, we have M(Γ) ∈ NKn(x).
This shows M(NSn+(Lx)) ⊆ NKn(x).
Conversely, Let Γ ∈ NSn+(Lx). Then, Γ ∈ Sn−. It follows from (28) that M(Γ) ∈NKn(x); hence Γ ∈ M(M(Γ)) ⊆ M(NKn(x)). For y ∈ NKn(x), we see M(y) ∈ NSn+(Lx)
by (28). Thus, y = M(M(y)) ⊆M(NSn+(Lx)).
(b) It follows from part(a) immediately. 2
Clearly, M is a singleton mapping whereas M is a set-valued mapping. In particular,
in the proof of Theorem 6.1, we need to choose an element in M . Below we present a
way to pick an element in M .
21
Theorem 6.2. For x ∈ Kn and y ∈ NKn(x), let
Γ :=
[αy1
12yT2
12y2 β 1
y1y2y
T2
]if y 6= 0,
O if y = 0,
where
α =1
2
(1±
√1− ‖y2‖2
y21
)and β =
1
2
(1∓
√1− ‖y2‖2
y21
)y2
1.
Then Γ ∈ M(y).
Proof. Note first that α, β ≥ 0 due to y ∈ −Kn. Since the case in which y = 0 is trivial,
it suffices to prove the case where y 6= 0. Consider the following two subcases.
Case 1: For y2 6= 0, by a simple calculation, we can reach
αβ =‖y2‖2
4and β = (1− α)y2
1.
Using this, we have
n∑i=1
Γii = αy1 +β
y1
= y1 and Γ1i =1
2yi, ∀i = 2, 3, · · · , n.
Then, it remains to verify Γ � O, i.e.,
(u vT ) Γ
(u
v
)≤ 0 ∀ (u vT ) ∈ IRn.
This can be seen by verifying the following:
(u vT )
[αy1
12yT2
12y2 β 1
y1y2y
T2
](u
v
)= αy1u
2 + yT2 vu+ β1
y1‖y2‖2(yT2 v)2
= −(√−αy1u)2 + yT2 vu−
(√− βy1
1
‖y2‖yT2 v
)2
= −
(√−αy1u−
√− βy1
1
‖y2‖yT2 v
)2
≤ 0.
Case 2: For y2 = 0, we have
Γ =
[αy1 0
0 β 1y1y2y
T2
]22
where α = 0 and β = y21 or α = 1 and β = 0. Then, it is clear to see
αy1 + β1
y1
= y1,
which indicates
n∑i=1
Γii = y1 and Γ1i = 0 =1
2yi, i = 2, 3, · · · , n.
Moreover, in this case we also have Γ � O because
(u vT ) Γ
(u
v
)= (u vT )
[αy1 0
0 β 1y1y2y
T2
](u
v
)= αy1u
2 + β1
y1
(yT2 v)2
≤ 0,
where the last step follows from α, β ≥ 0 and y1 < 0. 2
7 Conclusion
In this paper, we have explored the relation between the SOC and its PSD counterpart in
term of distances, projections, tangent cones, normal cones, and the KKT systems. It is
known that SOCP and SDP are closely related; for example, SOCP can be regarded as a
special case of SDP, and SOCP relaxation provides a nice approach to SDP as mentioned
in [10]. The results obtained in this paper help us to understand the differences between
the SOC and its PSD reformulation better.
Acknowledgements. The authors are gratefully indebted to the anonymous referees
for their valuable suggestions that allowed us to improve the original presentation of the
paper.
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