Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 871741, 16 pages

doi:10.1155/2012/871741

Research Article

Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

Xiaomei Zhang,¹ Yanjun Wang,¹ and Weimin Ma²

¹Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China

²School of Economics and Management, Tongji University, Shanghai 200092, China

Received 9 February 2012; Accepted 11 June 2012

Academic Editor: Jianming Shi

Copyright © 2012 Xiaomei Zhang et al. 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.

Abstract.

We present some sufficient global optimality conditions for a special cubic minimization problem with box constraints or binary constraints by extending the global subdifferential approach proposed by V. Jeyakumar et al. (2006). The present conditions generalize the results developed in the work of V. Jeyakumar et al. where a quadratic minimization problem with box constraints or binary constraints was considered. In addition, a special diagonal matrix is constructed, which is used to provide a convenient method for justifying the proposed sufficient conditions. Then, the reformulation of the sufficient conditions follows. It is worth noting that this reformulation is also applicable to the quadratic minimization problem with box or binary constraints considered in the works of V. Jeyakumar et al. (2006) and Y. Wang et al. (2010). Finally some examples demonstrate that our optimality conditions can effectively be used for identifying global minimizers of the certain nonconvex cubic minimization problem.

1. Introduction

Consider the following cubic minimization problem with box constraints:

where , , and , , where is the set of all symmetric matrices. that means .

The cubic optimization problem has spawned a variety of applications, especially in cubic polynomial approximation optimization [1], convex optimization [2], engineering design, and structural optimization [3]. Moreover, research results about cubic optimization problem can be applied to quadratic programming problems, which have been widely studied because of their broad applications, to enrich quadratic programming theory.

Several general approaches can be used to establish optimality conditions for solutions to optimization problems. These approaches can be broadly classified into three groups: convex duality theory [4], local subdifferentials by linear functions [5–7], and global -subdifferential and -normal cone by quadratic functions [8–11]. The third approach, which we extend in this paper, is often adopted to develop optimality conditions for special optimization forms: quadratic minimizations with box or binary constraints, quadratic minimization with quadratic constraints, bivalent quadratic minimization with inequality constraints, and so forth.

In this paper, we consider the cubic minimization problem, which generalizes the quadratic functions frequently considered in the mentioned papers. The proof method is based on extending the global -subdifferentials by quadratic functions [8, 12] to cubic functions. We show how an -subdifferential can be explicitly calculated for cubic functions and then develop the global sufficient optimality conditions for . We also derive the global optimality conditions for special cubic minimization problems with binary constraints. But when we use the sufficient conditions, we have to determine whether a diagonal matrix exists. It is hard to identify whether the matrix exists. So we rewrite the sufficient conditions in an other way through constructing a certain diagonal matrix. This method is applicable to the quadratic minimization problem with box or binary constraints considered in [8, 12].

This paper is organized as follows. Section 2 presents the notions of -subdifferentials and develops the sufficient global optimality condition for . The global optimality condition for special cubic minimization with binary constraints is presented in Section 3. In Section 4, numerical examples is given to illustrate the effectiveness of the proposed global optimality conditions.

2. -Subdifferentials and Sufficient Conditions

In this section, basic definitions and notations that will be used throughout the paper are given. The real line is denoted by and the -dimensional Euclidean space is denoted by . For vectors , means that , for . means that the matrix is a positive semidefinite. A diagonal matrix with diagonal elements is denoted by . Let be a set of real-valued functions defined on .

Definition 2.1 (-subdifferentials [13]). Let be a set of real-valued functions. Let . An element is called an -subgradient of at a point if

The set of all -subgradients of at is referred to as -subdifferential of at .

Throughout the rest of the paper, we use the specific choice of defined by

Proposition 2.2. Let and . Then

Proof. Suppose that there exists a diagonal matrix , such that . Let

Then it suffices to prove that . Let

Since , for all , we know that is a convex function on . Note that , and so is a global minimizer of , that is, , for all . This means that .

Next we prove the converse.

Let . By definition,

Hence

Thus, is a global minimizer of . So, and , that is,

hence .

For , define

For , define

By Proposition 2.2, we obtain the following sufficient global optimality condition for .

Theorem 2.3. For , let and . Suppose that there exists a diagonal matrix , such that , and for all , . If

then is a global minimizer of problem .

Proof. Suppose that condition (2.11) holds. Let

Then, by Proposition 2.2, , that is,

Obviously if for each , then is a global minimizer of .

Note that

If each term in the right side of the above equation satisfies

then, from (2.14), it holds that . So is a global minimizer of over box constraints.

On the other hand, suppose that is a global minimizer of . Then it holds that

When is chosen as a special point as follows:

we still have

This means that if is a global minimizer of over box constraints. Then (2.15) holds.

Combining the above discussion, we can conclude that is a global minimizer of over box constraints if and only if (2.15) holds. So next, we just need to prove (2.15) in order to show that is a global minimizer of .

We first see from (2.11), for each , that

Since , then for each ,

and

For each , we consider the following three cases.

Case 1. (If , then ). By (2.20),

So, and , and then

Case 2. (If , then ). By (2.20),

So we have

Case 3. (If , then ). By (2.21),

Then

So, if condition (2.11) holds, then (2.15) holds. And, from (2.14), we can conclude that is a global minimizer of .

Theorem 2.3 shows that the existence of diagonal matrix plays a crucial role because if this diagonal matrix does not exist, then we have no way to use this theorem. If the diagonal matrix exists, then the key problem is how to find it. These questions also exist in [8, 12].

The following corollary will answer the questions above.

Corollary 2.4. For , let . Assume that, for all , it holds that . Then one has the following conclusion.

(1) When is a positive semidefinite matrix, if

then is a global minimizer of .

(2) When is not a positive semidefinite matrix, if there exists an index , such that

then there is no such diagonal matrix Q that meets the requirements of the Theorem 2.3.

(3) Let

When is not a positive semidefinite matrix and the condition holds, if holds, then is a global minimizer of . Otherwise, one can conclude that there is no such diagonal matrix that meets the requirements of the Theorem 2.3.

Proof. (1) Suppose that and the condition holds. Choosing , by Theorem 2.3, we can conclude that is a global minimizer of .

(2) When is not a positive semidefinite matrix, if there exists an index , such that

then

Suppose there exists a diagonal matrix that meets all conditions in Theorem 2.3. Condition (2.11) can be rewritten in the following form:

Then it follows that

For the index , we still have the following inequality:

This conflicts with the fact that .

(3) Next we will consider the case that is not a positive semidefinite matrix, and condition holds.

We construct a diagonal matrix where , and . Then it suffices to show that condition (2.11) in Theorem 2.3 hold.

Note that , and then

Since , we have . So

Rewriting the above inequality, we have

Apparently this means that the constructed diagonal matrix also satisfies condition (2.11). According to Theorem 2.3, we can conclude that is a global minimizer of .

If the constructed diagonal matrix does not meet the condition , then we can conclude that there is no such diagonal matrix that can meet the requirements of Theorem 2.3.

To show this, suppose that there exists a diagonal matrix , which satisfies and (2.11).

From (2.11), we have

Obviously if , then there must exist a diagonal matrix such that . This conflicts the assumption.

We now consider a special case of :

where and .

Corollary 2.5. For , let . If, for each ,

then is a global minimizer of , where .

Proof. For , choose . If (2.40) holds, then, by Theorem 2.3, is a global minimizer of .

3. Sufficient Conditions of Bivalent Programming

In this section, we will consider the following bivalent programming:

where , and , , are the same as in .

Similar to Theorem 2.3, we will obtain the global sufficient optimality conditions for .

Theorem 3.1. For , let . Suppose that there exists a diagonal matrix such that , and for all , . If

then is a global minimizer of problem .

Proof. Suppose that condition (3.1) holds. Let

Then

Obviously if for each , then is a global minimizer of .

Note that

Thus, is a global minimizer of with binary constraints if and only if, for each ,

Firstly, we note from (3.1), for each , that

Next we only show it from the following two cases.

Case 1. (If ), then (3.6) is equivalent to

It is obvious that, for each ,

So (3.5) holds.

Case 2. (If ), then (3.6) is equivalent to

It is obvious that, for each ,

So (3.5) holds.

From (3.5), we can conclude that is a global minimizer of problem

Similar to Corollary 2.4, we have the following corollary.

Corollary 3.2. For , let . Suppose that, for all , .

(1) When is a positive semidefinite matrix, if

then is a global minimizer of .

(2) Let

When is not a positive semidefinite matrix, if holds, then is a global minimizer of . Otherwise, one can conclude that there is no such diagonal matrix that meets the requirements of Theorem 3.1.

We just show the proof of (2).

Proof. We construct the diagonal matrix , where , . If , we just need to test condition (3.1).

Because

then rewriting the above equations, we have

It obviously means that the diagonal matrix also satisfies condition (3.1). According to Theorem 3.1, is a global minimizer of .

Note that there is difference between formula (3.1) and formula (2.11). In formula (3.1), the diagonal elements of a diagonal matrix are allowed to be positive or nonpositive. But in formula (2.11), the diagonal elements of a diagonal matrix must meet the conditions . So we have to discuss the sign of the terms in Corollary 3.2.

We now consider a special case of :

where and .

Corollary 3.3. For , let . If, for each ,

then is a global minimizer of .

Proof. For , choose . If conditions (3.15) hold, then, by Theorem 3.1, is a global minimizer of .

4. Numerical Examples

In this section, six examples are given to test the proposed global sufficient optimality condition.

Example 4.1. Consider the following problem:

Let

and . Obviously is a positive semidefinite matrix.

Considering , obviously we have, for each ,  . Note that and , and so

According to Corollary 2.4(1), is a global minimizer.

Example 4.2. Consider the following problem:

Let

and . Obviously not is a positive semidefinite matrix.

Considering , obviously we have, for each , .

Note that . Then letting , we have . Let , and . Then , which satisfies . According to Corollary 2.4(3), is a global minimizer.

Example 4.3. Consider the following problem:

Let and .

Consider .

Let . Then

According to Corollary 2.5, is a global minimizer.

Example 4.4. Consider the following problem:

Let

and . Obviously not is a positive semidefinite matrix.

Considering , it follows that, for each , . Note that and . Let . Similarly we have, and . Then , which satisfies . According to Theorem 3.1(2), is a global minimizer.

Example 4.5. Consider the following problem:

Let , and .

Consider .

Let . Then

According to Corollary 3.3, is a global minimizer.

Example 4.6. Consider the following problem:

Let and .

Consider .

Let . Then

We can see that the conditions are not true in in Corollary 3.3. But is a global minimizer. This fact exactly shows that the conditions are just sufficient.

Acknowledgment

This research was supported by Innovation Program of Shanghai Municipal Education Commission (12ZZ071), Shanghai Pujiang Program (11PJC059), and the National Natural Science Foundation of China (71071113).

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