© 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.
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.
Proposition 2.7 (see ).
Following the proof of Lemma in , we can prove the following lemma.
Now we formulate the dual problem SDD of SDP as follows:
Thus from (3.14),
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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