Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 363012, 13 pages
doi:10.1155/2010/363012
Research Article
-Duality Theorems for Convex Semidefinite
Optimization Problems with Conic Constraints
Gue Myung Lee and Jae Hyoung Lee
Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea
Correspondence should be addressed to Gue Myung Lee, gmlee@pknu.ac.kr
Received 30 October 2009; Accepted 10 December 2009
Academic Editor: Yeol Je Cho
Copyright q 2010 G. M. Lee and J. H. Lee. 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.
A convex semidefinite optimization problem with a conic constraint is considered. We formulate
a Wolfe-type dual problem for the problem for its -approximate solutions, and then we prove
-weak duality theorem and -strong duality theorem which hold between the problem and its
Wolfe type dual problem. Moreover, we give an example illustrating the duality theorems.
1. Introduction
Convex semidefinite optimization problem is to optimize an objective convex function over a
linear matrix inequality. When the objective function is linear and the corresponding matrices
are diagonal, this problem becomes a linear optimization problem.
For convex semidefinite optimization problem, Lagrangean duality without constraint
qualification 1, 2, complete dual characterization conditions of solutions 1, 3, 4, saddle
point theorems 5, and characterizations of optimal solution sets 6, 7 have been
investigated.
To get the -approximate solution, many authors have established -optimality conditions, -saddle point theorems and -duality theorems for several kinds of optimization
problems 1, 8–16.
Recently, Jeyakumar and Glover 11 gave -optimality conditions for convex
optimization problems, which hold without any constraint qualification. Yokoyama and
Shiraishi 16 gave a special case of convex optimization problem which satisfies -optimality
conditions. Kim and Lee 12 proved sequential -saddle point theorems and -duality
theorems for convex semidefinite optimization problems which have not conic constraints.
The purpose of this paper is to extend the -duality theorems by Kim and Lee
12 to convex semidefinite optimization problems with conic constraints. We formulate a
Wolfe type dual problem for the problem for its -approximate solutions, and then prove
2
Journal of Inequalities and Applications
-weak duality theorem and -strong duality theorem for the problem and its Wolfe type
dual problem, which hold under a weakened constraint qualification. Moreover, we give an
example illustrating the duality theorems.
2. Preliminaries
Consider the following convex semidefinite optimization problem:
SDP Minimize fx,
subject to F0 m
xi Fi 0,
x1 , x2 , . . . , xm ∈ C,
2.1
i1
where f : Rm → R is a convex function, C is a closed convex cone of Rm , and for
i 0, 1, . . . , m, Fi ∈ Sn , where Sn is the space of n × n real symmetric matrices. The space
Sn is partially ordered by the Löwner order, that is, for M, N ∈ Sn , M N if and only if
M − N is positive semidefinite. The inner product in Sn is defined by M, N TrMN,
where Tr· is the trace operation.
Let S : {M ∈ Sn | M 0}. Then S is self-dual, that is,
S : θ ∈ Sn | θ, Z 0, for any Z ∈ S S.
2.2
m
m
m
Let Fx : F0 i1 xi Fi , Fx :
i1 xi Fi , x x1 , . . . , xm ∈ R . Then F is a linear
m
operator from R to Sn and its dual is defined by
F∗ Z TrF1 Z, . . . , TrFm Z,
2.3
for any Z ∈ Sn . Clearly, A : {x ∈ C | Fx ∈ S} is the feasible set of SDP.
Definition 2.1. Let g : Rn → R ∪ {∞} be a convex function.
1 The subdifferential of g at a ∈ dom g, where dom g {x ∈ Rn | gx < ∞}, is
given by
∂ga v ∈ Rn | gx ga v, x − a, ∀x ∈ Rn ,
2.4
where ·, · is the scalar product on Rn .
2 The -subdifferential of g at a ∈ dom g is given by
∂ ga v ∈ Rn | gx ga v, x − a − , ∀x ∈ Rn .
2.5
Definition 2.2. Let 0. Then x ∈ A is called an -approximate solution of SDP, if, for any
x ∈ A,
fx fx − .
2.6
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3
Definition 2.3. The conjugate function of a function g : Rn → R ∪ {∞} is defined by
g ∗ v sup v, x − gx | x ∈ Rn .
2.7
Definition 2.4. The epigraph of a function g : Rn → R ∪ {∞}, epi g, is defined by
epig x, r ∈ Rn × R | gx r .
2.8
If g is sublinear i.e., convex and positively homogeneous of degree one, then ∂ g0 g ∗ epig ∗ 0, k. It is worth
∂g0, for all 0. If g x gx − k, x ∈ Rn , k ∈ R, then epi
nothing that if g is sublinear, then
epig ∗ ∂g0 × R .
2.9
Moreover, if g is sublinear and if g x gx − k, x ∈ Rn , and k ∈ R, then
epi
g ∗ ∂g0 × k, ∞.
2.10
Definition 2.5. Let C be a closed convex set in Rn and x ∈ C.
1 Let NC x {v ∈ Rn | v, y − x 0, for all y ∈ C}. Then NC x is called the
normal cone to C at x.
2 Let 0. Let NC x {v ∈ Rn | v, y − x , for all y ∈ C}. Then NC x is called
the -normal set to C at x.
3 When C is a closed convex cone in Rn , NC 0 we denoted by C∗ and called the
negative dual cone of C.
Proposition 2.6 see 17, 18. Let f : Rn → R be a convex function and let δC be the indicator
function with respect to a closed convex subset C of Rn , that is, δC x 0 if x ∈ C, and δC x ∞
if x /
∈ C. Let 0. Then
∂ f δC x ∂0 fx ∂1 δC x .
0 0,1 0
0 1 2.11
Proposition 2.7 see 7. Let g : Rn → R be a continuous convex function and let h : Rn →
R ∪ {∞} be a proper lower semicontinuous convex function. Then
∗
epi g h epig ∗ epih∗ .
Following the proof of Lemma 2.2 in 1, we can prove the following lemma.
2.12
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∅. Let u ∈ Rm and α ∈ R. Then the
Lemma 2.8. Let Fi ∈ Sn , i 0, 1, . . . , m. Suppose that A /
following are equivalent:
m
x ∈ C | F0 Fi xi 0 ⊂ {x ∈ Rm | u, x α},
i
ii
u
α
i1
⎛
∈ cl⎝
− TrZF0 − δ
Z,δ∈S×R
⎞
F∗ Z
2.13
− C∗ × R ⎠.
3. -Duality Theorem
Now we give -duality theorems for SDP. Using Lemma 2.8, we can obtain the following
lemma which is useful in proving our -strong duality theorems for SDP.
Lemma 3.1. Let x ∈ A. Suppose that
F∗ Z
− TrZF0 − δ
Z,δ∈S×R
− C∗ × R
3.1
is closed. Then x is an -approximate solution of SDP if and only if there exists Z ∈ S such that for
any x ∈ C,
fx − TrZFx fx − .
3.2
Proof. ⇒ Let x be an -approximate solution of SDP. Then fx fx − , for any x ∈ A.
Let hx fx − fx . Then hx δA x 0, for any x ∈ Rn . Thus we have, from
Proposition 2.7,
0 ∈ epih δA ∗ epih∗ epiδA ∗
∗
epif ∗ 0, fx − epiδA
,
3.3
∗
and hence, 0, −fx ∈ epif ∗ epiδA
. So there exists u, r ∈ epif ∗ such that −u, −fx−r ∈
∗
epiδA and hence there exists u, r ∈ epif ∗ such that −u, x − fx − r for any x ∈ A.
Since f ∗ u r, −u, x −fx−f ∗ u for any x ∈ A; and hence it follows from Lemma 2.8
that
u
− fx f ∗ u
∈
Z,δ∈S×R
F∗ Z
− TrZF0 − δ
− C∗ × R .
3.4
Thus there exist Z, δ ∈ S × R , c∗ ∈ C∗ , and γ ∈ R such that
u F∗ Z − c∗ ,
− fx f ∗ u − TrZF0 − δ − γ.
3.5
Journal of Inequalities and Applications
5
This gives
F∗ Z, x − c∗ , x − fx u, x − fx f ∗ u
3.6
− TrZF0 − δ − γ − fx ,
for any x ∈ Rn . Thus we have
fx − −u, x fx − TrZF0 − δ − γ
fx − F∗ Z, x c∗ , x − TrZF0 − δ − γ
∗
3.7
fx − TrZFx c , x − δ − γ
fx − TrZFx
for any x ∈ C.
⇐ Suppose that there exists Z ∈ S such that
fx − TrZFx fx − ,
3.8
fx fx − TrZFx fx − ,
3.9
for any x ∈ C. Then we have
for any x ∈ A. Thus fx fx − , for any x ∈ A. Hence x is an -approximate solution of
SDP.
Now we formulate the dual problem SDD of SDP as follows:
SDD
maximize
fx − TrZFx,
subject to 0 ∈ ∂0 fx − F∗ Z NC1 x,
Z 0,
3.10
0 1 ∈ 0, .
We prove -weak and -strong duality theorems which hold between SDP and SDD.
Theorem 3.2 -weak duality. For any feasible solution x of SDP and any feasible solution y, Z
of SDD,
fx f y − Tr ZF y − .
3.11
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Journal of Inequalities and Applications
Proof. Let x and y, Z be feasible solutions of SDP and SDD respectively. Then TrZFx 0
and there exist v ∈ ∂0 fy and ω ∈ NC1 y such that v −ω F∗ Z. Thus, we have
fx − f y − Tr ZF y
v, x − y − 0 Tr ZF y
−ω F∗ Z, x − y − 0 Tr ZF y
F∗ Z, x − y − 0 − 1 Tr ZF y
F∗ Z, x − F∗ Z, y − 0 − 1
Tr ZF y
m
m
Tr Z xi Fi − Tr Z yi Fi − 0 − 1
i1
TrZF0 Tr Z
i1
m
3.12
yi Fi
i1
TrZFx − 0 − 1
−0 − 1
−.
Hence fx fy − TrZFy − .
Theorem 3.3 -strong duality. Suppose that
Z,δ∈S×R
F∗ Z
− TrZF0 − δ
− C∗ × R
3.13
is closed. If x is an -approximate solution of SDP, then there exists Z ∈ S such that x, Z is a
2-approximate solution of SDD.
Proof. Let x ∈ A be an -approximate solution of SDP. Then fx fx−, for any x ∈ A. By
Lemma 3.1, there exists Z ∈ S such that
fx − Tr ZFx fx − ,
3.14
for any x ∈ C. Letting x x in 3.14, TrZFx . Since Fx ∈ S and Z ∈ S, TrZFx 0.
Thus from 3.14,
fx − Tr ZFx fx − Tr ZFx
3.15
Journal of Inequalities and Applications
7
for any x ∈ C. Hence x is an -approximate solution of the following problem:
maximize
fx − Tr ZFx ,
subject to
x ∈ C,
3.16
and so, 0 ∈ ∂ f − F∗ Z δC x, and hence, by Proposition 2.6, there exist 0 , 1 ∈ 0, such that 0 1 and
0 ∈ ∂0 fx − F∗ Z NC1 x.
3.17
So, x, Z is a feasible solution of SDD. For any feasible solution y, Z of SDD,
fx − f y − Tr ZF y
fx − Tr ZFx − f y − Tr ZF y
− Tr ZFx
− − Tr ZFx
by -weak duality
3.18
− − −2.
Thus x, Z is a 2-approximate solution to SDD.
Now we characterize the -normal set to Rn .
Proposition 3.4. Let x1 , . . . , xn ∈ Rn and 0. Then
NR n x1 , . . . , xn n
Ai ,
0 i1
n i
i1 i 3.19
where
⎧
⎪
⎨−R
Ai $ i %
⎪
⎩ − ,0
xi
if xi 0,
if xi > 0.
3.20
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Journal of Inequalities and Applications
Proof. Let x1 , . . . , xn ∈ Rn and 0. Then
NR n x1 , . . . , xn ∂ δRn x1 , . . . , xn n
∂
δR×···×R×R ×R×···×R x1 , . . . , xn i1
3.21
n
∂i δR×···×R×R ×R×···×R x1 , . . . , xn .
0 i1
n i
i1 i Let v1 , . . . , vn ∈ ∂i δR×···×R×R ×R×···×R x1 , . . . , xn where R is at the ith position in R × · · · × R ×
R × R × · · · × R
⇐⇒ for any y1 , . . . , yn ∈ R × · · · × R × R × R × · · · × R,
i v1 y1 − x1 · · · vi yi − xi · · · vn yn − xn ,
i vi yi − xi ,
⇐⇒ for any yi ∈ R ,
vj 0,
for j ∈ {1, . . . , n} \ {i},
⎧
⎪
⎪
⎨−R ,
%
⇐⇒ vi ∈ $
⎪
⎪
⎩ − i ,0 ,
xi
3.22
if xi 0,
if xi > 0,
vj 0 for j ∈ {1, . . . , n} \ {i}.
Thus, we have
NR1n x1 , . . . , xn n
{0} × · · · × {0} × Ai × {0} × · · · × {0}
0 i1
n i
i1 i n
3.23
Ai .
0 i1
n i
i1 i From Proposition 3.4, we can calculate NR 2 .
Journal of Inequalities and Applications
9
Corollary 3.5. Let x1 , x2 ∈ R2 and 0. Then following hold.
i If x1 , x2 0, 0, then NR 2 0, 0 −R2 .
ii If x1 , x2 x1 , 0 and x1 > 0, then NR 2 x1 , 0 −/x1 , 0 × −∞, 0.
iii If x1 , x2 0, x2 and x2 > 0, then NR 2 0, x2 −∞, 0 × −/x2 , 0.
iv If x1 , x2 x1 , x2 , and x1 > 0 and x2 > 0, then
NR 2 x1 , x2 $
−
1 0,2 0,
1 2 % $
%
2
1
,0 × − ,0 .
x1
x2
3.24
Now we give an example illustrating our -duality theorems.
Example 3.6. Consider the following convex semidefinite program.
SDP
Minimize
x1 x22 ,
subject to
0 x1
x1 0
0,
3.25
x1 , x2 ∈ R2 .
Let fx1 , x2 x1 x22 ,
F0 0 0
0 0
,
F1 0 1
,
1 0
F2 0 0
0 0
,
3.26
and 0. Let fx1 , x2 x1 x22 and
Fx1 , x2 0 x1
x1 0
.
3.27
Then A : {0, x2 ∈ R2 | x2 0} is the set of all feasible solutions of SDP and the set of all √
approximate solutions of SDP is {x1 , x2 ∈ R2 | x1 0, 0 x2 }. Let F {x1 , x2 , Z |
0 ∈ ∂0 fx1 , x2 − F∗ Z NR12 x1 , x2 , Z 0, 0 1 ∈ 0, }. Then F is the set of all feasible
solution of SDD. Now we calculate the set F.
10
Journal of Inequalities and Applications
:
A
a b
0, 0,
| 0 ∈ ∂0 f0, 0 − F∗
b c
a b
b c
NR12 0, 0,
a 0, c 0, b ac, 0 1 ∈ 0, 2
0, 0,
a b
b c
√ √
| 0 ∈ {1} × −2 0 , 2 0 − 2b, 0 − R2 ,
a 0, c 0, b ac, 0 1 ∈ 0, 2
0, 0,
a b
b c
√ | 2b, 0 ∈ −∞, 1 × −∞, 2 0 ,
a 0, c 0, b ac, 0 1 ∈ 0, 2
0, 0,
b c
B :
a b
0, x2 ,
a b
1 2
| a 0, c 0, b , b ac ,
2
∗
| x2 > 0, 0 ∈ ∂0 f0, x2 − F
b c
a b
b c
NR12 0, x2 a 0, c 0, b ac, 0 1 ∈ 0, 2
a b
0, x2 ,
b c
√
√ | x2 > 0, 0 ∈ {1} × 2x2 − 2 0 , 2x2 2 0 − 2b, 0
%
1
2
−∞, 0 × − , 0 , a 0, c 0, b ac, 0 1 ∈ 0, x2
$
a b
0, x2 ,
b c
$
%
√
√
1
− 2 0 , 2x2 2 0 ,
| x2 > 0, 2b, 0 ∈ −∞, 1 × 2x2 −
x2
a 0, c 0, b ac, 0 1 ∈ 0, 2
0, x2 ,
a b
√
| 0 < x2 b c
0 1 ∈ 0, ,
0 &
0 21
1
, a 0, c 0, b , b2 ac,
2
2
Journal of Inequalities and Applications
a b
∗ a b
NR12 x1 , 0,
| x1 > 0, 0 ∈ ∂0 fx1 , 0 − F
C :
x1 , 0,
b c
b c
a 0, c 0, b ac, 0 1 ∈ 0, 2
x1 , 0,
a b
b c
√ √
| x1 > 0, 0 ∈ {1} × −2 0 , 2 0 − 2b, 0
%
1
2
− , 0 × −∞, 0, a 0, c 0, b ac, 0 1 ∈ 0, x1
$
x1 , 0,
a b
b c
$
%
√ 1
| x1 > 0, 2b, 0 ∈ 1 − , 1 × −∞, 2 0 ,
x1
a 0, c 0, b ac, 0 1 ∈ 0, 2
x1 , 0,
a b
b c
| 0 < x1 , −1 −x1 2bx1 , a 0, c 0, b 1 2
, b ac,
2
0 1 ∈ 0, ,
:
D
x1 , x2 ,
a b
b c
∗
| x1 > 0, x2 > 0, 0 ∈ ∂0 fx1 , x2 − F
a b
b c
NR12 x1 , x2 ,
a b
b c
x1 , x2 ,
a 0, c 0, b ac, 0 1 ∈ 0, 2
| x1 > 0, x2 > 0,
1 2 √
√ 0 ∈ {1} × 2x2 − 2 0 , 2x2 2 0 − 2b, 0 − 1 , 0 × − 1 , 0 ,
x1
x2
a 0, c 0, b2 ac, 11 12 1 , 0 1 ∈ 0, a b
x1 , x2 ,
b c
| x1 > 0, x2 > 0,
11
12
√
√
− 2 0 , 2x2 2 0 ,
2b, 0 ∈ 1 − , 1 × 2x2 −
x1
x2
a 0, c 0, b2 ac, 11 12 1 , 0 1 ∈ 0, 11
12
Journal of Inequalities and Applications
x1 , x2 ,
0 < x2 a b
b c
| 0 < x1 , −11 −x1 2bx1 ,
'
√
0 0 212
2
, a 0, c 0, b 1 2
, b ac, 11 12 1 ,
2
0 1 ∈ 0, .
3.28
∪ B ∪ C
∪ D.
We can check that for any x1 , x2 ∈ A and any y1 , y2 ,
Thus F A
fx1 , x2 f y1 , y2
a b
− Tr
F y1 , y2
− ,
b c
a b
b c
∈ F,
3.29
that is, -weak duality holds.
√
Let x1 , x2 ∈ A be an -approximate
0 0 solution of SDP. Then x1 0 and 0 x2 .
So, we can easily
that x1 , x2 , 0 0 ∈ F.
check
Since Tr 00 00 Fx1 , x2 0, from 3.29,
a b
− ,
fx1 , x2 f y1 , y2 − Tr
F y1 , y2
b c
3.30
for any y1 , y2 , ab bc ∈ F. So x1 , x2 , a b is an -approximate solution of SDD. Hence
b c
-strong duality holds.
Acknowledgment
This work was supported by the Korea Science and Engineering Foundation KOSEF NRL
Program grant funded by the Korean government MESTno. R0A-2008-000-20010-0.
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