- Research Article
- Open Access
-Duality Theorems for Convex Semidefinite Optimization Problems with Conic Constraints
© G.M. Lee and J.H. Lee 2010
- Received: 30 October 2009
- Accepted: 10 December 2009
- Published: 12 January 2010
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.
- Approximate Solution
- Feasible Solution
- Convex Function
- Linear Matrix Inequality
- Constraint Qualification
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 , and characterizations of optimal solution sets [6, 7] have been investigated.
Recently, Jeyakumar and Glover  gave -optimality conditions for convex optimization problems, which hold without any constraint qualification. Yokoyama and Shiraishi  gave a special case of convex optimization problem which satisfies -optimality conditions. Kim and Lee  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  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 -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.
Consider the following convex semidefinite optimization problem:
where is a convex function, is a closed convex cone of , and for , where is the space of real symmetric matrices. The space is partially ordered by the L wner order, that is, for if and only if is positive semidefinite. The inner product in is defined by , where is the trace operation.
for any Clearly, is the feasible set of SDP.
Let be a convex function.
where is the scalar product on .
If is sublinear (i.e., convex and positively homogeneous of degree one), then , for all . If , , , then . It is worth nothing that if is sublinear, then
Moreover, if is sublinear and if , , and , then
Let be a closed convex set in and .
(1)Let . Then is called the normal cone to at .
(2)Let . Let . Then is called the -normal set to at .
(3)When is a closed convex cone in , we denoted by and called the negative dual cone of .
Proposition 2.7 (see ).
Following the proof of Lemma in , we can prove the following lemma.
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.
for any .
( ) Suppose that there exists such that
for any Thus , for any . Hence is an -approximate solution of SDP.
Now we formulate the dual problem SDD of SDP as follows:
We prove -weak and -strong duality theorems which hold between SDP and SDD.
Theorem 3.2 ( -weak duality).
Theorem 3.3 ( -strong duality).
is closed. If is an -approximate solution of SDP, then there exists such that is a -approximate solution of SDD.
for any . Letting in (3.14), . Since and , .
Thus from (3.14),
Thus is a 2 -approximate solution to SDD.
Now we characterize the -normal set to .
From Proposition 3.4, we can calculate .
Let and Then following hold.
(i)If , then
(ii)If and , then
(iii)If and , then
Now we give an example illustrating our -duality theorems.
that is, -weak duality holds.
Let be an -approximate solution of SDP. Then and . So, we can easily check that .
Since , from (3.29),
for any . So is an -approximate solution of SDD. Hence -strong duality holds.
This work was supported by the Korea Science and Engineering Foundation (KOSEF) NRL Program grant funded by the Korean government (MEST)(no. R0A-2008-000-20010-0).
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