Open image in new window -Duality Theorems for Convex Semidefinite Optimization Problems with Conic Constraints

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Abstract

A convex semidefinite optimization problem with a conic constraint is considered. We formulate a Wolfe-type dual problem for the problem for its Open image in new window -approximate solutions, and then we prove Open image in new window -weak duality theorem and Open image in new window -strong duality theorem which hold between the problem and its Wolfe type dual problem. Moreover, we give an example illustrating the duality theorems.

Keywords

Approximate Solution Feasible Solution Convex Function Linear Matrix Inequality Constraint Qualification 

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 Open image in new window -approximate solution, many authors have established Open image in new window -optimality conditions, Open image in new window -saddle point theorems and Open image in new window -duality theorems for several kinds of optimization problems [1, 8, 9, 10, 11, 12, 13, 14, 15, 16].

Recently, Jeyakumar and Glover [11] gave Open image in new window -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 Open image in new window -optimality conditions. Kim and Lee [12] proved sequential Open image in new window -saddle point theorems and Open image in new window -duality theorems for convex semidefinite optimization problems which have not conic constraints.

The purpose of this paper is to extend the Open image in new window -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 Open image in new window -approximate solutions, and then prove Open image in new window -weak duality theorem and Open image in new window -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:

where Open image in new window is a convex function, Open image in new window is a closed convex cone of Open image in new window , and for Open image in new window , where Open image in new window is the space of Open image in new window real symmetric matrices. The space Open image in new window is partially ordered by the L Open image in new window wner order, that is, for Open image in new window if and only if Open image in new window is positive semidefinite. The inner product in Open image in new window is defined by Open image in new window , where Open image in new window is the trace operation.

for any Open image in new window Clearly, Open image in new window is the feasible set of SDP.

Definition 2.1.

Let Open image in new window be a convex function.

where Open image in new window is the scalar product on Open image in new window .

Definition 2.2.

Definition 2.3.

The conjugate function of a function Open image in new window is defined by

Definition 2.4.

The epigraph of a function Open image in new window , Open image in new window , is defined by

If Open image in new window is sublinear (i.e., convex and positively homogeneous of degree one), then Open image in new window , for all Open image in new window . If Open image in new window , Open image in new window , Open image in new window , then Open image in new window . It is worth nothing that if Open image in new window is sublinear, then

Moreover, if Open image in new window is sublinear and if Open image in new window , Open image in new window , and Open image in new window , then

Definition 2.5.

Let Open image in new window be a closed convex set in Open image in new window and Open image in new window .

(1)Let Open image in new window . Then Open image in new window is called the normal cone to Open image in new window at Open image in new window .

(2)Let Open image in new window . Let Open image in new window . Then Open image in new window is called the Open image in new window -normal set to Open image in new window at Open image in new window .

(3)When Open image in new window is a closed convex cone in Open image in new window , Open image in new window we denoted by Open image in new window and called the negative dual cone of Open image in new window .

Proposition 2.6 (see [17, 18]).

Let Open image in new window be a convex function and let Open image in new window be the indicator function with respect to a closed convex subset C of Open image in new window , that is, Open image in new window if Open image in new window , and Open image in new window if Open image in new window . Let Open image in new window . Then

Proposition 2.7 (see [7]).

Let Open image in new window be a continuous convex function and let Open image in new window be a proper lower semicontinuous convex function. Then

Following the proof of Lemma Open image in new window in [1], we can prove the following lemma.

Lemma 2.8.

3. Open image in new window-Duality Theorem

Now we give Open image in new window -duality theorems for SDP. Using Lemma 2.8, we can obtain the following lemma which is useful in proving our Open image in new window -strong duality theorems for SDP.

Lemma 3.1.

is closed. Then Open image in new window is an Open image in new window -approximate solution of SDP if and only if there exists Open image in new window such that for any Open image in new window ,

Proof.

This gives

for any Open image in new window .

( Open image in new window ) Suppose that there exists Open image in new window such that

for any Open image in new window Thus Open image in new window , for any Open image in new window . Hence Open image in new window is an Open image in new window -approximate solution of SDP.

Now we formulate the dual problem SDD of SDP as follows:

We prove Open image in new window -weak and Open image in new window -strong duality theorems which hold between SDP and SDD.

Theorem 3.2 ( Open image in new window -weak duality).

For any feasible solution Open image in new window of SDP and any feasible solution Open image in new window of SDD,

Proof.

Let Open image in new window and Open image in new window be feasible solutions of SDP and SDD respectively. Then Open image in new window and there exist Open image in new window and Open image in new window such that Open image in new window . Thus, we have

Hence Open image in new window .

Theorem 3.3 ( Open image in new window -strong duality).

Suppose that

is closed. If Open image in new window is an Open image in new window -approximate solution of SDP, then there exists Open image in new window such that Open image in new window is a Open image in new window -approximate solution of SDD.

Proof.

Let Open image in new window be an Open image in new window -approximate solution of SDP. Then Open image in new window for any Open image in new window By Lemma 3.1, there exists Open image in new window such that

for any Open image in new window . Letting Open image in new window in (3.14), Open image in new window . Since Open image in new window and Open image in new window , Open image in new window .

Thus from (3.14),

for any Open image in new window . Hence Open image in new window is an Open image in new window -approximate solution of the following problem:
and so, Open image in new window , and hence, by Proposition 2.6, there exist Open image in new window , Open image in new window such that Open image in new window and
So, Open image in new window is a feasible solution of SDD. For any feasible solution Open image in new window of SDD,

Thus Open image in new window is a 2 Open image in new window -approximate solution to SDD.

Now we characterize the Open image in new window -normal set to Open image in new window .

Proposition 3.4.

Proof.

Thus, we have

From Proposition 3.4, we can calculate Open image in new window .

Corollary 3.5.

Let Open image in new window and Open image in new window Then following hold.

(i)If Open image in new window , then Open image in new window

(ii)If Open image in new window and Open image in new window , then Open image in new window

(iii)If Open image in new window and Open image in new window , then Open image in new window

Now we give an example illustrating our Open image in new window -duality theorems.

Example 3.6.

Consider the following convex semidefinite program.
Then Open image in new window is the set of all feasible solutions of SDP and the set of all Open image in new window -approximate solutions of SDP is Open image in new window . Let Open image in new window . Then Open image in new window is the set of all feasible solution of SDD. Now we calculate the set Open image in new window .

that is, Open image in new window -weak duality holds.

Let Open image in new window be an Open image in new window -approximate solution of SDP. Then Open image in new window and Open image in new window . So, we can easily check that Open image in new window Open image in new window .

Since Open image in new window , from (3.29),

for any Open image in new window Open image in new window . So Open image in new window is an Open image in new window -approximate solution of SDD. Hence Open image in new window -strong duality holds.

Notes

Acknowledgment

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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Copyright information

© G.M. Lee and J.H. Lee 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Department of Applied MathematicsPukyong National UniversityPusanSouth Korea

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