Volume 2010, Article ID 363012,13pages 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,[email protected]
Received 30 October 2009; Accepted 10 December 2009
Academic Editor: Yeol Je Cho
Copyrightq2010 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 qualification1,2, 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 con-ditions,-saddle point theorems and-duality theorems for several kinds of optimization problems1,8–16.
Recently, Jeyakumar and Glover 11 gave -optimality conditions for convex optimization problems, which hold without any constraint qualification. Yokoyama and Shiraishi16gave 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
-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
i1
xiFi0, x1, x2, . . . , xm∈C,
2.1
where f : Rm → R is a convex function, C is a closed convex cone of Rm, and for i 0,1, . . . , m, Fi ∈ Sn, whereSn is the space of n× nreal symmetric matrices. The space Sn is partially ordered by the L ¨owner order, that is, forM, N ∈ Sn, M Nif and only if M−Nis positive semidefinite. The inner product inSn is defined byM, N TrMN, where Tr·is the trace operation.
LetS:{M∈Sn|M0}. ThenSis self-dual, that is,
S :θ∈S
n|θ, Z0, for any Z∈SS. 2.2
Let Fx : F0 mi1xiFi,Fx :
m
i1xiFi, x x1, . . . , xm ∈ Rm.Then F is a linear operator fromRmtoSnand its dual is defined by
F∗Z TrF
1Z, . . . ,TrFmZ, 2.3 for anyZ∈Sn.Clearly,A:{x∈C|Fx∈S}is the feasible set of SDP.
Definition 2.1. Letg:Rn → R∪ {∞}be a convex function.
1The subdifferential ofg ata ∈ domg,where domg {x ∈ Rn | gx < ∞}, is given by
∂ga v∈Rn|gxga v, x−a , ∀x∈Rn, 2.4
where·,· is the scalar product onRn.
2The-subdifferential ofgata∈domgis given by
∂ga v∈Rn|gxga v, x−a −, ∀x∈Rn. 2.5
Definition 2.2. Let 0.Thenx ∈Ais called an-approximate solution of SDP, if, for any x∈A,
Definition 2.3. The conjugate function of a functiong :Rn → R∪ {∞}is defined by
g∗v supv, x −gx|x∈Rn. 2.7
Definition 2.4. The epigraph of a functiong :Rn → R∪ {∞},epi g, is defined by
epigx, r∈Rn×R|gxr. 2.8
Ifgis sublineari.e., convex and positively homogeneous of degree one, then∂g0 ∂g0, for all 0. Ifgx gx−k,x∈Rn,k ∈R, thenepig∗ epig∗ 0, k. It is worth nothing that ifgis sublinear, then
epig∗∂g0×R
. 2.9
Moreover, ifgis sublinear and ifgx gx−k,x∈Rn, andk∈R, then
epig∗∂g0×k,∞. 2.10
Definition 2.5. LetCbe a closed convex set inRnandx∈C.
1Let NCx {v ∈ Rn | v, y−x 0,for ally ∈ C}. ThenNCx is called the normal cone toCatx.
2Let0. LetNCx {v∈Rn| v, y−x ,for ally∈C}. ThenNCxis called the-normal set toCatx.
3When C is a closed convex cone in Rn, NC0 we denoted byC∗ and called the negative dual cone ofC.
Proposition 2.6see17,18. Letf :Rn → Rbe a convex function and letδC be the indicator function with respect to a closed convex subset C ofRn, that is,δCx 0ifx∈C, andδCx ∞ ifx /∈C. Let0. Then
∂fδC x 00,10
01
∂0fx ∂1δCx.
2.11
Proposition 2.7see7. Letg : Rn → Rbe a continuous convex function and let h : Rn → R∪ {∞}be a proper lower semicontinuous convex function. Then
epigh ∗epig∗epih∗. 2.12
Lemma 2.8. LetFi ∈ Sn, i 0,1, . . . , m. Suppose thatA /∅.Letu ∈ Rm and α ∈ R. Then the following are equivalent:
i
x∈C|F0 m
i1
Fixi 0
⊂ {x∈Rm| u, x α},
ii
u α
∈cl
⎛
⎝
Z,δ∈S×R
F∗Z
−TrZF0−δ
−C∗×R
⎞
⎠.
2.13
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. Letx∈A. Suppose that
Z,δ∈S×R
F∗Z
−TrZF0−δ
−C∗×R
3.1
is closed. Thenxis an-approximate solution of SDP if and only if there existsZ ∈Ssuch that for anyx∈C,
fx−TrZFxfx−. 3.2
Proof. ⇒Letxbe an-approximate solution of SDP. Thenfxfx−, for anyx∈A. Let hx fx−fx . Then hx δAx 0, for anyx ∈ Rn. Thus we have, from
Proposition 2.7,
0∈epihδA∗epih∗epiδA∗
epif∗0, fx− epiδ∗ A,
3.3
and hence,0, −fx∈epif∗epiδ∗A. So there existsu, r∈epif∗such that−u, −fx−r∈ epiδA∗and hence there existsu, r ∈epif∗such that−u, x −fx−r for anyx ∈A. Sincef∗ur,−u, x −fx−f∗ufor anyx∈A; and hence it follows fromLemma 2.8
that
u
−fx f∗u
∈
Z,δ∈S×R
F∗Z
−TrZF0−δ
−C∗×R
. 3.4
Thus there existZ, δ∈S×R, c∗∈C∗, andγ∈Rsuch that
uF∗Z−c∗,
This gives
F∗Z, x− c∗, x −fx u, x −fxf∗u
−TrZF0−δ−γ−fx ,
3.6
for anyx∈Rn. Thus we have
fx−−u, x fx−TrZF0−δ−γ
fx−F∗Z, xc∗, x −TrZF
0−δ−γ
fx−TrZFx c∗, x −δ−γ
fx−TrZFx
3.7
for anyx∈C.
⇐Suppose that there existsZ∈Ssuch that
fx−TrZFxfx−, 3.8
for anyx∈C. Then we have
fxfx−TrZFxfx−, 3.9
for anyx∈A.Thusfxfx−, for anyx∈A. Hencexis an-approximate solution of SDP.
Now we formulate the dual problem SDD of SDP as follows:
SDD maximize fx−TrZFx,
subject to 0∈∂0fx−F∗Z NC1x, Z0,
01∈0, .
3.10
We prove-weak and-strong duality theorems which hold between SDP and SDD.
Theorem 3.2-weak duality. For any feasible solutionxof SDP and any feasible solutiony, Z of SDD,
Proof. Letxandy, Zbe feasible solutions of SDP and SDD respectively. Then TrZFx0 and there existv∈∂0fyandω∈N1Cysuch thatv−ωF∗Z. Thus, we have
fx−fy −TrZFy v, x−y−0Tr
ZFy
−ωF∗Z, x−y− 0Tr
ZFy
F∗Z, x−y−
0−1Tr
ZFy
F∗Z, x−F∗Z, y− 0−1
TrZFy
Tr
Zm i1
xiFi
−Tr
Zm i1
yiFi
−0−1
TrZF0 Tr
Zm i1
yiFi
TrZFx−0−1
−0−1
−.
3.12
Hencefxfy−TrZFy−.
Theorem 3.3-strong duality. Suppose that
Z,δ∈S×R
F∗Z
−TrZF0−δ
−C∗×R
3.13
is closed. Ifx is an-approximate solution of SDP, then there existsZ ∈ S such that x, Z is a 2-approximate solution of SDD.
Proof. Letx∈Abe an-approximate solution of SDP. Thenfxfx−,for anyx∈A.By
Lemma 3.1, there existsZ∈Ssuch that
fx−TrZFxfx−, 3.14
for anyx∈C. Lettingxxin3.14, TrZFx. SinceFx∈SandZ∈S, TrZFx 0.
Thus from3.14,
for anyx∈C. Hencexis an-approximate solution of the following problem:
maximize fx−TrZFx,
subject to x∈C,
3.16
and so, 0 ∈ ∂f−F∗Z δC x, and hence, byProposition 2.6, there exist0,1 ∈ 0, such that01and
0∈∂0fx−F∗
ZN1Cx. 3.17
So,x, Zis a feasible solution of SDD. For any feasible solutiony, Zof SDD,
fx−TrZFx−fy −TrZFy fx−fy −TrZFy
−TrZFx
−−TrZFx
by-weak duality
−−
−2.
3.18
Thusx, Zis a 2-approximate solution to SDD.
Now we characterize the-normal set toRn.
Proposition 3.4. Letx1, . . . , xn∈Rnand0.Then
N
Rn
x1, . . . , xn
i0
n i1i
n
i1
Ai,
3.19
where
Ai
⎧ ⎪ ⎨ ⎪ ⎩
−R ifxi0,
$ −xi
i,0
%
ifxi>0.
Proof. Letx1, . . . , xn∈Rnand0. Then
N
Rn
x1, . . . , xn ∂δRnx1, . . . , xn
∂
n
i1
δR×···×R×R×R×···×R
x1, . . . , xn
i0
n i1i
n
i1
∂iδR×···×R×R×R×···×Rx1, . . . , xn.
3.21
Letv1, . . . , vn∈∂iδR×···×R×R×R×···×Rx1, . . . , xn whereRis at theith position inR× · · · ×R×
R×R× · · · ×R
⇐⇒ for any y1, . . . , yn ∈R× · · · ×R×R×R× · · · ×R,
iv1
y1−x1 · · ·viyi−xi · · ·vnyn−xn ,
⇐⇒ for any yi∈R, iviyi−xi , vj0,
forj ∈ {1, . . . , n} \ {i},
⇐⇒ vi∈
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
−R, ifxi0,
$
−i
xi,0
%
, ifxi>0,
vj0 for j∈ {1, . . . , n} \ {i}.
3.22
Thus, we have
N1Rn
x1, . . . , xn
i0
n i1i
n
i1
{0} × · · · × {0} ×Ai× {0} × · · · × {0}
i0
n i1i
n
i1 Ai.
3.23
FromProposition 3.4, we can calculateNR2
Corollary 3.5. Letx1, x2∈R2
and0.Then following hold.
iIfx1, x2 0,0, thenNR2
0,0 −R
2
.
iiIfx1, x2 x1,0andx1>0, thenNR2
x1,0 −/x1,0×−∞,0. iiiIfx1, x2 0, x2andx2>0, thenNR2
0, x2 −∞,0×−/x2,0. ivIfx1, x2 x1, x2,andx1>0andx2>0, then
N
R2
x1, x2
10,20, 12
$
−1
x1 ,0
% ×
$
−2
x2 ,0
%
. 3.24
Now we give an example illustrating our-duality theorems.
Example 3.6. Consider the following convex semidefinite program.
SDP Minimize x1x22,
subject to
0 x1 x1 0
0,
x1, x2∈R2.
3.25
Letfx1, x2 x1x22,
F0
0 0
0 0
, F1
0 1
1 0
, F2
0 0
0 0
, 3.26
and0. Letfx1, x2 x1x22and
Fx1, x2
0 x1 x1 0
. 3.27
ThenA:{0, x2∈R2|x20}is the set of all feasible solutions of SDP and the set of all -approximate solutions of SDP is{x1, x2∈R2|x1 0,0x2 √}. LetF{x1, x2, Z| 0∈∂0fx1, x2−F∗Z NR12
x1, x2, Z 0, 01 ∈0, }. ThenFis the set of all feasible
A:
0,0,
a b b c
|0∈∂0f0,0−F∗
a b b c
NR12
0,0,
a0, c0, b2 ac,
01∈0,
0,0,
a b
b c
|0∈ {1} ×−2√0,2
√
0
−2b,0−R2,
a0, c0, b2 ac,
01∈0,
0,0,
a b b c
|2b,0∈−∞,1×−∞,2√0
,
a0, c0, b2 ac,
01∈0,
0,0,
a b
b c
|a0, c0, b1
2, b 2 ac
,
B:
0, x2,
a b b c
|x2>0, 0∈∂0f0, x2−F∗
a b b c N1 R2
0, x2
a0, c0, b2 ac, 01∈0,
0, x2,
a b
b c
|x2>0, 0∈ {1} ×
2x2−2
√
0,2x22
√
0
−2b,0
−∞,0×
$
−1
x2,0
%
, a0, c0, b2ac,
01∈0,
0, x2,
a b b c
|x2>0, 2b,0∈−∞,1×
$
2x2− 1 x2 −
2√0,2x22
√
0
%
,
a0, c0, b2ac,
01∈0,
0, x2,
a b
b c
|0< x2
√
0&021
2 , a0, c0, b 1 2, b
2ac,
01∈0,
C:
x1,0,
a b b c
|x1>0, 0∈∂0fx1,0−F∗
a b b c
NR12
x1,0,
a0, c0, b2 ac,
01∈0,
x1,0,
a b b c
|x1>0, 0∈ {1} ×
−2√0,2
√
0
−2b,0
$
−1
x1,0
%
×−∞,0, a0, c0, b2ac,
01∈0,
x1,0,
a b b c
|x1>0, 2b,0∈
$
1− 1 x1
,1
%
×−∞,2√0
,
a0, c0, b2 ac,
01∈0,
x1,0,
a b b c
|0< x1, −1−x12bx1, a0, c0, b 1 2, b
2ac,
01∈0,
,
D:
x1, x2,
a b b c
|x1>0, x2 >0, 0∈∂0fx1, x2−F∗
a b b c
NR12
x1, x2, a0, c0, b
2ac,
01∈0,
x1, x2,
a b b c
|x1>0, x2>0,
0∈ {1} ×2x2−2
√
0,2x22
√
0
−2b,0
−11
x1 ,0
×
−21
x2 ,0
,
a0, c0, b2 ac, 1
1121, 01∈0,
x1, x2,
a b b c
|x1>0, x2>0,
2b,0∈
1− 1 1 x1,1
×
2x2− 2
1 x2 −2
√
0,2x22
√
0
,
a0, c0, b2 ac, 1
1121, 01∈0,
x1, x2,
a b b c
|0< x1, −11 −x12bx1,
0< x2
√
0
'
0221
2 , a0, c0, b 1 2, b
2ac, 1
1121, 01∈0,
.
3.28
ThusFA∪B∪C∪D. We can check that for any x1, x2∈Aand anyy1, y2,a bb c ∈F,
fx1, x2f
y1, y2 −Tr
a b b c
Fy1, y2
−, 3.29
that is,-weak duality holds.
Letx1, x2 ∈Abe an-approximate solution of SDP. Thenx1 0 and 0x2
√
. So, we can easily check thatx1, x2,0 0
0 0 ∈F. Since Tr0 0
0 0 Fx1, x2 0, from3.29,
fx1, x2f
y1, y2 −Tr
a b b c
Fy1, y2
−, 3.30
for anyy1, y2,
a b
b c ∈F. Sox1, x2,
a b b c
is an-approximate solution of SDD. Hence -strong duality holds.
Acknowledgment
This work was supported by the Korea Science and Engineering FoundationKOSEFNRL Program grant funded by the Korean governmentMESTno. R0A-2008-000-20010-0.
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