- Research Article
- Open Access

# -Duality Theorems for Convex Semidefinite Optimization Problems with Conic Constraints

- GueMyung Lee
^{1}Email author and - JaeHyoung Lee
^{1}

**2010**:363012

https://doi.org/10.1155/2010/363012

© G.M. Lee and J.H. Lee 2010

**Received:**30 October 2009**Accepted:**10 December 2009**Published:**12 January 2010

## Abstract

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.

## 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 -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 -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:

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.

Definition 2.1.

Let be a convex function.

where is the scalar product on .

Definition 2.2.

Definition 2.3.

Definition 2.4.

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

Definition 2.5.

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 2.6 (see [17, 18]).

Proposition 2.7 (see [7]).

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

Lemma 2.8.

## 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.

Proof.

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).

Proof.

Hence .

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.

Proof.

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 .

Proposition 3.4.

Proof.

From Proposition 3.4, we can calculate .

Corollary 3.5.

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.

Example 3.6.

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.

## Declarations

### 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).

## Authors’ Affiliations

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