Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

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: min𝑓(𝑥)=𝑏^𝑇𝑥³+12𝑥^𝑇s.t.𝑥∈𝐷=_𝑛𝐴𝑥+𝑎^𝑇𝑥,_𝑖=1𝑢_𝑖,𝑣_𝑖,(𝐶𝑃1) where 𝑢_𝑖,𝑣_𝑖∈𝑅, 𝑢_𝑖≤𝑣_𝑖,𝑖=1,2,…,𝑛, and 𝑎=(𝑎₁,…,𝑎_𝑛)^𝑇∈𝑅^𝑛,𝑏=(𝑏₁,…,𝑏_𝑛)^𝑇∈𝑅^𝑛, 𝐴∈𝑆^𝑛, where 𝑆^𝑛 is the set of all symmetric 𝑛×𝑛 matrices. 𝑥³ that means (𝑥³¹,…,𝑥^3𝑛)^𝑇.

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 𝐿-subdifferential and 𝐿-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 𝐿-subdifferentials by quadratic functions [8, 12] to cubic functions. We show how an 𝐿-subdifferential can be explicitly calculated for cubic functions and then develop the global sufficient optimality conditions for (𝐶𝑃1). 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 𝑄 exists. It is hard to identify whether the matrix 𝑄 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 𝐿-subdifferentials and develops the sufficient global optimality condition for (𝐶𝑃1). 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. 𝐿-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 𝑅 and the 𝑛-dimensional Euclidean space is denoted by 𝑅^𝑛. For vectors 𝑥,𝑦∈𝑅^𝑛, 𝑥≥𝑦 means that 𝑥_𝑖≥𝑦_𝑖, for 𝑖=1,…,𝑛. 𝐴≽𝐵 means that the matrix 𝐴−𝐵 is a positive semidefinite. A diagonal matrix with diagonal elements 𝛼₁,…,𝛼_𝑛 is denoted by diag(𝛼₁,…,𝛼_𝑛). Let 𝐿 be a set of real-valued functions defined on 𝑅^𝑛.

Definition 2.1 (𝐿-subdifferentials [13]). Let 𝐿 be a set of real-valued functions. Let 𝑓∶𝑅^𝑛→𝑅. An element 𝑙∈𝐿 is called an 𝐿-subgradient of 𝑓 at a point 𝑥₀∈𝑅^𝑛 if 𝑓(𝑥)≥𝑓𝑥₀+𝑙(𝑥)−𝑙𝑥₀,∀𝑥∈𝑅^𝑛.(2.1)
The set 𝜕_𝐿𝑓(𝑥) of all 𝐿-subgradients of 𝑓 at 𝑥₀ is referred to as 𝐿-subdifferential of 𝑓 at 𝑥₀.
Throughout the rest of the paper, we use the specific choice of 𝐿 defined by 𝐿=𝑏^𝑇𝑥³+12𝑥^𝑇𝑄𝑥+𝛽^𝑇𝑥∣𝑄=diag𝛼₁,…,𝛼_𝑛,𝛼_𝑖∈𝑅,𝛽∈𝑅^𝑛.(2.2)

Proposition 2.2. Let 𝑓(𝑥)=𝑏^𝑇𝑥³+(1/2)𝑥^𝑇𝐴𝑥+𝑎^𝑇𝑥 and 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝑅^𝑛. Then 𝜕_𝐿𝑓𝑥=𝑏^𝑇𝑥³+12𝑥^𝑇𝑄𝑥+𝛽^𝑇𝑥|||^{𝐴−𝑄≽0,𝑄=diag(𝛼1,…,𝛼𝑛)𝛽=(𝐴𝑄)}_{𝑥+𝑎,𝛼𝑖∈𝑅}.(2.3)

Proof. Suppose that there exists a diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛), such that 𝐴−𝑄≽0. Let 𝑙(𝑥)=𝑏^𝑇𝑥³+12𝑥^𝑇𝑄𝑥+𝛽^𝑇𝑥,𝛽=(𝐴−𝑄)𝑥+𝑎.(2.4) Then it suffices to prove that 𝑙(𝑥)∈𝜕_𝐿𝑓(𝑥). Let 𝜙(𝑥)=𝑓(𝑥)−𝑙(𝑥)=12𝑥^𝑇(𝐴−𝑄)𝑥+(𝑎−𝛽)^𝑇𝑥.(2.5) Since ∇²𝜙(𝑥)=𝐴−𝑄≽0, for all 𝑥∈𝑅^𝑛, we know that 𝜙(𝑥) is a convex function on 𝑅^𝑛. Note that ∇𝜙(𝑥)=(𝐴−𝑄)𝑥+(𝑎−𝛽)=0, and so 𝑥 is a global minimizer of 𝜙(𝑥), that is, 𝜙(𝑥)≥𝜙(𝑥), for all 𝑥∈𝑅^𝑛. This means that 𝑙(𝑥)∈𝜕_𝐿𝑓(𝑥).
Next we prove the converse.
Let 𝑙(𝑥)∈𝜕_𝐿𝑓(𝑥),𝑙(𝑥)=𝑏^𝑇𝑥³+(1/2)𝑥^𝑇𝑄𝑥+𝛽^𝑇𝑥. By definition, 𝑓(𝑥)≥𝑓𝑥+𝑙(𝑥)−𝑙𝑥,∀𝑥∈𝑅^𝑛.(2.6) Hence 𝜙(𝑥)=𝑓(𝑥)−𝑙(𝑥)=12𝑥^𝑇(𝐴−𝑄)𝑥+(𝑎−𝛽)^𝑇𝑥≥𝑓𝑥−𝑙𝑥,∀𝑥∈𝑅^𝑛.(2.7) Thus, 𝑥 is a global minimizer of 𝜙(𝑥). So, ∇𝜙(𝑥)=0 and ∇²𝜙(𝑥)≽0, that is, 𝐴−𝑄≽0,(𝐴−𝑄)𝑥+(𝑎−𝛽)=0,(2.8) hence 𝛽=(𝐴−𝑄)𝑥+𝑎.

For 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝐷, define 𝑥̃_𝑖=⎧⎪⎨⎪⎩−1if1if𝑥_𝑖=𝑢_𝑖,𝐴𝑥_𝑖=𝑣_𝑖,𝑋=𝑥d_𝑖i+ag𝑎_𝑖𝑥̃₁if𝑥_𝑖∈𝑢_𝑖,𝑣_𝑖.,…,𝑥̃_𝑛.(2.9) For 𝑄=diag(𝛼₁,…,𝛼_𝑛),𝛼_𝑖∈𝑅,𝑖=1,…,𝑛, define 𝛼𝑄_𝑖==mdiiang0𝛼,₁𝛼_𝑖,,…,𝛼_𝑛.(2.10)

By Proposition 2.2, we obtain the following sufficient global optimality condition for (𝐶𝑃1).

Theorem 2.3. For (𝐶𝑃1), let 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝐷 and 𝑢=(𝑢₁,…,𝑢_𝑛)^𝑇,𝑣=(𝑣₁,…,𝑣_𝑛)^𝑇. Suppose that there exists a diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛),𝛼_𝑖∈𝑅,𝑖=1,…,𝑛, such that 𝐴−𝑄≽0, and for all 𝑥∈𝐷, 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0,(𝑖=1,…,𝑛). If 𝑋𝐴𝑥+𝑎−12𝑄(𝑣−𝑢)≤𝟎,(2.11) then 𝑥 is a global minimizer of problem (𝐶𝑃1).

Proof. Suppose that condition (2.11) holds. Let 𝑙(𝑥)=𝑏^𝑇𝑥³+12𝑥^𝑇𝛽=(𝐴−𝑄)𝑄𝑥+𝛽^𝑇𝑥,𝑥+𝑎.(2.12) Then, by Proposition 2.2, 𝑙(𝑥)∈𝜕_𝐿𝑓(𝑥), that is, 𝑓(𝑥)−𝑓𝑥≥𝑙(𝑥)−𝑙𝑥,∀𝑥∈𝑅^𝑛.(2.13) Obviously if 𝑙(𝑥)−𝑙(𝑥)≥0 for each 𝑥∈𝐷, then 𝑥 is a global minimizer of (𝐶𝑃1).
Note that 𝑙(𝑥)−𝑙𝑥=^𝑛_𝑖=1𝛼_𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎^𝑇𝑥−𝑥+𝑏^𝑇𝑥³−𝑥³.(2.14) If each term in the right side of the above equation satisfies 𝛼_𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑥_𝑖+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖≥0,𝑖=1,…,𝑛,𝑥^𝑖∈𝑢_𝑖,𝑣_𝑖,(2.15) then, from (2.14), it holds that 𝑙(𝑥)−𝑙(𝑥)≥0. So 𝑥 is a global minimizer of 𝑙(𝑥) over box constraints.
On the other hand, suppose that 𝑥 is a global minimizer of 𝑙(𝑥),𝑥∈𝐷. Then it holds that 𝑙(𝑥)−𝑙𝑥=^𝑛_𝑖=1𝛼_𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎^𝑇𝑥−𝑥+𝑏^𝑇𝑥³−𝑥³≥0,∀𝑥∈𝐷.(2.16) When 𝑥 is chosen as a special point 𝑥̃∈𝐷 as follows: 𝑥̃=𝑥₁,𝑥₂,…,𝑥_𝑖−1,𝑥_𝑖,𝑥_𝑖+1,…,𝑥_𝑛,𝑥_𝑖∈𝑢_𝑖,𝑣_𝑖,𝑥_𝑖≠𝑥_𝑖,𝑖=1,…,𝑛,(2.17) we still have 𝑙(𝑥̃)−𝑙𝑥=𝛼^𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎^𝑇𝑥−𝑥+𝑏^𝑇𝑥³−𝑥³≥0,𝑖=1,…,𝑛.(2.18) This means that if 𝑥 is a global minimizer of 𝑙(𝑥) over box constraints. Then (2.15) holds.
Combining the above discussion, we can conclude that 𝑥 is a global minimizer of 𝑙(𝑥) over box constraints if and only if (2.15) holds. So next, we just need to prove (2.15) in order to show that 𝑥 is a global minimizer of 𝑙(𝑥).
We first see from (2.11), for each 𝑖=1,…,𝑛, that −𝛼^𝑖2𝑣^𝑖−𝑢_𝑖+𝑥̃_𝑖𝐴𝑥+𝑎_𝑖≤0.(2.19)
Since 𝛼_𝑖≤0, then for each 𝑥_𝑖∈[𝑢_𝑖,𝑣_𝑖],𝑖=1,…,𝑛, −𝛼^𝑖2𝑥^𝑖−𝑢_𝑖+𝑥̃_𝑖𝐴𝑥+𝑎_𝑖≤0,(2.20) and 𝛼_𝑖2𝑥^𝑖−𝑣_𝑖+𝑥̃_𝑖𝐴𝑥+𝑎_𝑖≤0.(2.21)
For each 𝑖=1,…,𝑛, we consider the following three cases.
Case 1. (If 𝑥_𝑖∈(𝑢_𝑖,𝑣_𝑖), then 𝑥̃_𝑖=(𝐴𝑥+𝑎)_𝑖). By (2.20), −𝛼^𝑖2𝑥^𝑖−𝑢_𝑖+𝐴𝑥+𝑎^2𝑖≤0.(2.22)
So, 𝛼_𝑖=0 and (𝐴𝑥+𝑎)_𝑖=0, and then 𝛼_𝑖2𝑥^𝑖≥𝛼−_𝑖𝑥_𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑥_𝑖+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖2𝑥^𝑖=𝑏_𝑖−𝑥_𝑖²+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖≥0.𝑥_𝑖−𝑥_𝑖𝑥^2𝑖+𝑥_𝑖𝑥_𝑖+𝑥^2𝑖(2.23)
Case 2. (If 𝑥_𝑖=𝑢_𝑖, then 𝑥̃_𝑖=−1). By (2.20), 𝛼_𝑖2𝑥^𝑖−𝑢_𝑖+𝐴𝑥+𝑎_𝑖≥0.(2.24)
So we have 𝛼_𝑖2𝑥^𝑖≥−𝛼𝑥_𝑖𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑥_𝑖+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖2𝑥^𝑖=𝛼_𝑖−𝑢_𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑢_𝑖+𝑏_𝑖𝑥^3𝑖−𝑢^3𝑖2𝑥^𝑖−𝑢_𝑖+𝐴𝑥+𝑎_𝑖𝑥^𝑖≥0.−𝑢_𝑖+𝑏_𝑖𝑥^3𝑖−𝑢^3𝑖(2.25)
Case 3. (If 𝑥_𝑖=𝑣_𝑖, then 𝑥̃_𝑖=1). By (2.21), 𝛼_𝑖2𝑥^𝑖−𝑣_𝑖+𝐴𝑥+𝑎_𝑖≤0.(2.26) Then 𝛼_𝑖2𝑥^𝑖≥−𝛼𝑥_𝑖𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑥_𝑖+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖2𝑥^𝑖≥0.−𝑣_𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑣_𝑖+𝑏_𝑖𝑥^3𝑖−𝑣^3𝑖(2.27)
So, if condition (2.11) holds, then (2.15) holds. And, from (2.14), we can conclude that 𝑥 is a global minimizer of (𝐶𝑃1).

Theorem 2.3 shows that the existence of diagonal matrix 𝑄 plays a crucial role because if this diagonal matrix 𝑄 does not exist, then we have no way to use this theorem. If the diagonal matrix 𝑄 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 (𝐶𝑃1), let 𝑥∈𝐷. Assume that, for all 𝑥∈𝐷, it holds that 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,…,𝑛). Then one has the following conclusion.(1) When 𝐴 is a positive semidefinite matrix, if 𝑋𝐴𝑥+𝑎≤𝟎,(2.28)then 𝑥 is a global minimizer of (𝐶𝑃1).(2) When 𝐴 is not a positive semidefinite matrix, if there exists an index 𝑖₀,1≤𝑖₀≤𝑛, such that 𝑥̃_𝑖0𝐴𝑥+𝑎_𝑖0>𝟎,(2.29)then there is no such diagonal matrix Q that meets the requirements of the Theorem 2.3.(3) Let 𝛼_𝑖=2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖𝑄=diag𝛼₁,𝑖=1,…,𝑛,,…,𝛼_𝑛.(2.30)
When 𝐴 is not a positive semidefinite matrix and the condition 𝑋(𝐴𝑥+𝑎)≤𝟎 holds, if 𝐴−𝑄≽0 holds, then 𝑥 is a global minimizer of (𝐶𝑃1). Otherwise, one can conclude that there is no such diagonal matrix 𝑄 that meets the requirements of the Theorem 2.3.

Proof. (1) Suppose that 𝐴≽0 and the condition 𝑋(𝐴𝑥+𝑎)≤𝟎 holds. Choosing 𝑄=𝑄=𝟎, by Theorem 2.3, we can conclude that 𝑥 is a global minimizer of (𝐶𝑃1).
(2) When 𝐴 is not a positive semidefinite matrix, if there exists an index 𝑖₀,1≤𝑖₀≤𝑛, such that 𝑥̃_𝑖0𝐴𝑥+𝑎_𝑖0>0,(2.31) then 2𝑥̃_𝑖0𝐴𝑥+𝑎_𝑖0𝑣_𝑖0−𝑢_𝑖0>0.(2.32)
Suppose there exists a diagonal matrix 𝑄 that meets all conditions in Theorem 2.3. Condition (2.11) can be rewritten in the following form: 𝑥̃_𝑖𝐴𝑥+𝑎_𝑖−12𝛼^𝑖𝑣_𝑖−𝑢_𝑖≤0,𝑖=1,…,𝑛.(2.33) Then it follows that 𝛼_𝑖≥2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖,𝑖=1,…,𝑛.(2.34)
For the index 𝑖₀, we still have the following inequality: 𝛼_𝑖0≥2𝑥̃^𝑖0𝐴𝑥+𝑎_𝑖0𝑣_𝑖0−𝑢_𝑖0>0.(2.35) This conflicts with the fact that 𝛼_𝑖=min{0,𝛼_𝑖}≤0,𝑖=1,…,𝑛.
(3) Next we will consider the case that 𝐴 is not a positive semidefinite matrix, and condition 𝑋(𝐴𝑥+𝑎)≤𝟎 holds.
We construct a diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛) where 𝛼_𝑖=2𝑥̃_𝑖(𝐴𝑥+𝑎)_𝑖/(𝑣_𝑖−𝑢_𝑖),𝑖=1,…,𝑛, and 𝐴−𝑄≽0. Then it suffices to show that condition (2.11) in Theorem 2.3 hold.
Note that 𝑥̃_𝑖(𝐴𝑥+𝑎)_𝑖≤0,𝑖=1,…,𝑛, and then 𝛼_𝑖=2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖≤0,𝑖=1,…,𝑛.(2.36) Since 𝛼_𝑖≤0, we have 𝛼_𝑖=𝛼_𝑖. So 𝛼_𝑖=2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖≤0,𝑖=1,…,𝑛.(2.37) Rewriting the above inequality, we have 𝑥̃_𝑖𝐴𝑥+𝑎_𝑖−12𝛼^𝑖𝑣_𝑖−𝑢_𝑖=0,𝑖=1,…,𝑛.(2.38) Apparently this means that the constructed diagonal matrix 𝑄 also satisfies condition (2.11). According to Theorem 2.3, we can conclude that 𝑥 is a global minimizer of (𝐶𝑃1).
If the constructed diagonal matrix 𝑄 does not meet the condition 𝐴−𝑄≽0, then we can conclude that there is no such diagonal matrix 𝑄 that can meet the requirements of Theorem 2.3.
To show this, suppose that there exists a diagonal matrix 𝑄^∗=diag(𝛼^∗1,…,𝛼^∗𝑛), which satisfies 𝐴−𝑄^∗≽0 and (2.11).
From (2.11), we have 0≥𝛼^∗𝑖≥2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖=𝛼_𝑖,𝑖=1,…,𝑛.(2.39) Obviously if 𝐴−𝑄^∗≽0, then there must exist a diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛) such that 𝐴−𝑄≽0. This conflicts the assumption.

We now consider a special case of (𝐶𝑃1): min𝑓(𝑥)=^𝑛_𝑖=1𝑏_𝑖𝑥^3𝑖+^𝑛_𝑖=1𝑟_𝑖2𝑥^2𝑖+^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖s.t.𝑥∈𝐷=^𝑛_𝑖=1𝑢_𝑖,𝑣_𝑖,(𝐶𝑃2) where 𝑢_𝑖,𝑣_𝑖,𝑎_𝑖,𝑏_𝑖,𝑟_𝑖∈𝑅 and 𝑢_𝑖≤𝑣_𝑖,𝑖=1,2,…,𝑛.

Corollary 2.5. For (𝐶𝑃2), let 𝑥∈𝐷. If, for each 𝑖=1,…,𝑛, 𝑥̃_𝑖𝑟_𝑖𝑥_𝑖+𝑎_𝑖−12𝑟^𝑖̂𝑣_𝑖𝑏−𝑢_𝑖≤0,_𝑖𝑥_𝑖−𝑥_𝑖≥0,(2.40) then 𝑥 is a global minimizer of (𝐶𝑃2), where 𝑟_𝑖̂=min{0,𝑟_𝑖}.

Proof. For (𝐶𝑃2), choose 𝑄=𝐴=diag(𝑟₁,…,𝑟_𝑛). If (2.40) holds, then, by Theorem 2.3, 𝑥 is a global minimizer of (𝐶𝑃2).

3. Sufficient Conditions of Bivalent Programming

In this section, we will consider the following bivalent programming: min𝑓(𝑥)=𝑏^𝑇𝑥³+12𝑥^𝑇s.t.𝑥∈𝐷_𝐵𝐴𝑥+𝑎^𝑇𝑥=^𝑛_𝑖=1𝑢_𝑖,𝑣_𝑖,(𝐶𝑃3) where 𝐴,𝑎,𝑏, and 𝑢_𝑖, 𝑣_𝑖, 𝑖=1,…,𝑛 are the same as in (𝐶𝑃1).

Similar to Theorem 2.3, we will obtain the global sufficient optimality conditions for (𝐶𝑃3).

Theorem 3.1. For (𝐶𝑃3), let 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇∈𝐷_𝐵. Suppose that there exists a diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛),𝛼_𝑖∈𝑅,𝑖=1,…,𝑛 such that 𝐴−𝑄≽0, and for all 𝑥∈𝐷_𝐵, 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,…,𝑛). If 𝑋𝐴𝑥+𝑎−12𝑄(𝑣−𝑢)≤𝟎,(3.1) then 𝑥 is a global minimizer of problem (𝐶𝑃3).

Proof. Suppose that condition (3.1) holds. Let 𝑙(𝑥)=𝑏^𝑇𝑥³+12𝑥^𝑇𝛽=(𝐴−𝑄)𝑄𝑥+𝛽^𝑇𝑥,𝑥+𝑎.(3.2) Then 𝑓(𝑥)−𝑓𝑥≥𝑙(𝑥)−𝑙𝑥,∀𝑥∈𝑅^𝑛.(3.3) Obviously if 𝑙(𝑥)−𝑙(𝑥)≥0 for each 𝑥∈𝐷_𝐵, then 𝑥 is a global minimizer of (𝐶𝑃3).
Note that 𝑙(𝑥)−𝑙𝑥=^𝑛_𝑖=1𝛼_𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎^𝑇𝑥−𝑥+𝑏^𝑇𝑥³−𝑥³.(3.4)
Thus, 𝑥 is a global minimizer of 𝑙(𝑥) with binary constraints if and only if, for each 𝑖=1,…,𝑛,𝑥_𝑖∈{𝑢_𝑖,𝑣_𝑖}, 𝛼_𝑖2𝑥^𝑖−𝑥_𝑖²+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑥_𝑖+𝑏_𝑖𝑥^3𝑖−𝑥^3𝑖≥0.(3.5)
Firstly, we note from (3.1), for each 𝑖=1,…,𝑛, that −𝛼^𝑖2𝑣^𝑖−𝑢_𝑖+𝑥̃_𝑖𝐴𝑥+𝑎_𝑖≤0.(3.6)
Next we only show it from the following two cases.
Case 1. (If 𝑥_𝑖=𝑢_𝑖), then (3.6) is equivalent to 𝛼_𝑖2𝑣^𝑖−𝑢_𝑖+𝐴𝑥+𝑎_𝑖≥0.(3.7) It is obvious that, for each 𝑥_𝑖∈{𝑢_𝑖,𝑣_𝑖}, 𝛼_𝑖2𝑥^𝑖−𝑢_𝑖²+𝑏_𝑖𝑥^3𝑖−𝑢^3𝑖+𝐴𝑥+𝑎_𝑖𝑥_𝑖−𝑢_𝑖≥0.(3.8) So (3.5) holds.
Case 2. (If 𝑥_𝑖=𝑣_𝑖), then (3.6) is equivalent to −𝛼^𝑖2𝑣^𝑖−𝑢_𝑖+𝐴𝑥+𝑎_𝑖≤0.(3.9) It is obvious that, for each 𝑥_𝑖∈{𝑢_𝑖,𝑣_𝑖}, 𝛼_𝑖2𝑣^𝑖−𝑥_𝑖²−𝑏_𝑖𝑣^3𝑖−𝑥^3𝑖−𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑥_𝑖≥0.(3.10) So (3.5) holds.
From (3.5), we can conclude that 𝑥 is a global minimizer of problem (𝐶𝑃3)

Similar to Corollary 2.4, we have the following corollary.

Corollary 3.2. For (𝐶𝑃3), let 𝑥∈𝐷_𝐵. Suppose that, for all 𝑥∈𝐷_𝐵, 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,…,𝑛).
(1) When 𝐴 is a positive semidefinite matrix, if 𝑋𝐴𝑥+𝑎≤𝟎,(3.11) then 𝑥 is a global minimizer of (𝐶𝑃3).
(2) Let 𝛼_𝑖=2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖𝑄=diag𝛼₁,𝑖=1,…,𝑛,,…,𝛼_𝑛.(3.12) When 𝐴 is not a positive semidefinite matrix, if 𝐴−𝑄≽0 holds, then 𝑥 is a global minimizer of (𝐶𝑃3). Otherwise, one can conclude that there is no such diagonal matrix 𝑄 that meets the requirements of Theorem 3.1.

We just show the proof of (2).

Proof. We construct the diagonal matrix 𝑄=diag(𝛼₁,…,𝛼_𝑛), where 𝛼_𝑖=2𝑥̃_𝑖(𝐴𝑥+𝑎)_𝑖/(𝑣_𝑖−𝑢_𝑖), 𝑖=1,…,𝑛. If 𝐴−𝑄≽0, we just need to test condition (3.1).
Because 𝛼_𝑖=2𝑥̃^𝑖𝐴𝑥+𝑎_𝑖𝑣_𝑖−𝑢_𝑖,𝑖=1,…,𝑛,(3.13) then rewriting the above equations, we have 𝑥̃_𝑖𝐴𝑥+𝑎_𝑖−12𝛼^𝑖𝑣_𝑖−𝑢_𝑖=0,𝑖=1,…,𝑛.(3.14) It obviously means that the diagonal matrix 𝑄 also satisfies condition (3.1). According to Theorem 3.1, 𝑥 is a global minimizer of (𝐶𝑃3).

Note that there is difference between formula (3.1) and formula (2.11). In formula (3.1), the diagonal elements 𝛼_𝑖 of a diagonal matrix 𝑄 are allowed to be positive or nonpositive. But in formula (2.11), the diagonal elements 𝛼_𝑖 of a diagonal matrix 𝑄 must meet the conditions 𝛼_𝑖≤0. So we have to discuss the sign of the terms 𝑥̃_𝑖(𝐴𝑥+𝑎)_𝑖(𝑖=1,…,𝑛) in Corollary 3.2.

We now consider a special case of (𝐶𝑃3): min𝑓(𝑥)=^𝑛_𝑖=1𝑏_𝑖𝑥^3𝑖+^𝑛_𝑖=1𝑟_𝑖2𝑥^2𝑖+^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖s.t.𝑥∈𝐷_𝐵=^𝑛_𝑖=1𝑢_𝑖,𝑣_𝑖,(𝐶𝑃4) where 𝑏_𝑖,𝑟_𝑖,𝑎_𝑖,𝑢_𝑖,𝑣_𝑖∈𝑅 and 𝑢_𝑖≤𝑣_𝑖,𝑖=1,…,𝑛.

Corollary 3.3. For (𝐶𝑃4), let 𝑥∈𝐷_𝐵. If, for each 𝑖=1,…,𝑛, 𝑥̃_𝑖𝑟_𝑖𝑥_𝑖+𝑎_𝑖−𝑟^𝑖2𝑣^𝑖𝑏−𝑢_𝑖≤0,_𝑖𝑥_𝑖−𝑥_𝑖≥0,(3.15) then 𝑥 is a global minimizer of (𝐶𝑃4).

Proof. For (𝐶𝑃4), choose 𝑄=𝐴=diag(𝑟₁,…,𝑟_𝑛). If conditions (3.15) hold, then, by Theorem 3.1, 𝑥 is a global minimizer of (𝐶𝑃4).

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: min73𝑥³¹+5𝑥³²+2𝑥³³+32𝑥²¹+𝑥²²+12𝑥²³+2𝑥₁𝑥₂+𝑥₁𝑥₃+𝑥₂𝑥₃+32𝑥¹+5𝑥₂+3𝑥₃s.t.𝑥∈𝐷=³_𝑖=1[1,2].(4.1)
Let 𝐴=⎡⎢⎢⎢⎢⎣321221111⎤⎥⎥⎥⎥⎦(4.2) and 𝑎=(3/2,5,3)^𝑇,𝑏=(7/3,5,2)^𝑇. Obviously 𝐴 is a positive semidefinite matrix.
Considering 𝑥=(1,1,1)^𝑇, obviously we have, for each 𝑥_𝑖∈[1,2],  𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,2,3). Note that 𝐴𝑥+𝑎=(15/2,10,6)^𝑇 and 𝑋=diag(−1,−1,−1), and so 𝑋𝐴𝑥+𝑎=⎡⎢⎢⎢⎢⎣−15−−2160⎤⎥⎥⎥⎥⎦<𝟎.(4.3) According to Corollary 2.4(1), 𝑥=(1,1,1) is a global minimizer.

Example 4.2. Consider the following problem: min−𝑥³¹−3𝑥³²+12𝑥³⁴−𝑥²¹−𝑥²²+32𝑥²³−12𝑥²⁴−2𝑥₁𝑥₂+𝑥₁𝑥₄+2𝑥₂𝑥₄−𝑥₁−4𝑥₂+5𝑥₄s.t.𝑥∈𝐷=⁴_𝑖=1[−1,1].(4.4)
Let 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−−0122−−02223000−1201⎤⎥⎥⎥⎥⎥⎥⎦(4.5) and 𝑎=(−1,−4,0,5)^𝑇,𝑏=(−1,−3,0,1/2)^𝑇. Obviously 𝐴 not is a positive semidefinite matrix.
Considering 𝑥=(1,1,0,−1)^𝑇, obviously we have, for each 𝑥_𝑖∈[−1,1], 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,2,3,4).
Note that 𝐴𝑥+𝑎=(−6,−10,0,9)^𝑇. Then letting 𝑋=diag(1,1,0,−1), we have 𝑋(𝐴𝑥+𝑎)=(−6,−10,0,−9)^𝑇≤𝟎. Let 𝛼₁=2𝑥̃₁(𝐴𝑥+𝑎)₁/(𝑣₁−𝑢₁)=−6, 𝛼₂=−10,𝛼₃=0 and 𝛼₄=−9. Then 𝑄=diag(−6,−10,0,−9), which satisfies 𝐴−𝑄≻0. According to Corollary 2.4(3), 𝑥=(1,1,0,−1) is a global minimizer.

Example 4.3. Consider the following problem: min−𝑥³¹+23𝑥³²−3𝑥³³−2𝑥²¹−35𝑥²²−4𝑥²³+12𝑥²⁴−2𝑥₁+5𝑥₂−3𝑥₃s.t.𝑥∈𝐷=⁴_𝑖=1[−1,1].(4.6)
Let 𝐴=diag(−4,−6/5,−8,1) and 𝑎=(−2,5,−3,0)^𝑇,𝑏=(−1,2/3,−3,0)^𝑇,𝑟=(−4,−6/5,−8,1)^𝑇.
Consider 𝑥=(1,−1,1,0)^𝑇.
Let 𝑋=diag(1,−1,1,0),𝑟̂=(−4,−6/5,−8,0)^𝑇. Then 𝑥̃₁𝑟₁𝑥₁+𝑎₁−12𝑟¹̂𝑣₁𝑥̃₂−𝑢₁=−2<0,𝑥̃₃𝑟₂𝑥₂+𝑎₂−12𝑟²̂𝑣₂−𝑢₂=−5<0,𝑥𝑟̃₃₄𝑥₃+𝑎₃−12𝑟³̂𝑣₃−𝑢₃=−3<0,𝑥𝑟₄𝑥₄+𝑎₄−12𝑟⁴̂𝑣₄−𝑢₄=0,_𝑖∈[−1,1],𝑏_𝑖𝑥_𝑖−𝑥_𝑖≥0,(𝑖=1,2,3,4).(4.7) According to Corollary 2.5, 𝑥=(1,−1,1,0) is a global minimizer.

Example 4.4. Consider the following problem: min4𝑥³¹+52𝑥³²−3𝑥³³−83𝑥³⁴−32𝑥²¹−32𝑥²²+3𝑥²³−𝑥²⁴+2𝑥₁𝑥₂+𝑥₁𝑥₄+𝑥₂𝑥₃−𝑥₃𝑥₄+5𝑥₁+92𝑥²−2𝑥₃−2𝑥₄s.t.𝑥∈𝐷_𝐵=⁴_𝑖=1{−1,1}.(4.8)
Let 𝐴=⎡⎢⎢⎢⎢⎢⎢⎣−2013−2103−0161−−1012⎤⎥⎥⎥⎥⎥⎥⎦(4.9) and 𝑎=(5,9/2,−2,−2)^𝑇,𝑏=(4,5/2,−3,−8/3)^𝑇. Obviously 𝐴 not is a positive semidefinite matrix.
Considering 𝑥=(−1,−1,1,1)^𝑇, it follows that, for each 𝑥∈𝐷_𝐵, 𝑏_𝑖(𝑥_𝑖−𝑥_𝑖)≥0(𝑖=1,2,3,4). Note that 𝐴𝑥+𝑎=(7,13/2,2,−6)^𝑇 and 𝑋=diag(−1,−1,1,1). Let 𝛼₁=(2𝑥̃₁(𝐴𝑥+𝑎)₁)/(𝑣₁−𝑢₁)=−7. Similarly we have, 𝛼₂=−13/2,𝛼₃=2 and 𝛼₄=−6. Then 𝑄=diag(−7,−13/2,2,−6), which satisfies 𝐴−𝑄≻0. According to Theorem 3.1(2), 𝑥=(−1,−1,1,1) is a global minimizer.

Example 4.5. Consider the following problem: min3𝑥³¹−8𝑥³²+5𝑥³³+12𝑥³⁴−𝑥²¹+4𝑥²²−53𝑥²³−3𝑥²⁴+𝑥₁−2𝑥₂+3𝑥₃+4𝑥₄s.t.𝑥∈𝐷_𝐵=⁴_𝑖=1{−1,0}.(4.10)
Let 𝑎=(1,−2,3,4)^𝑇,𝑏=(3,−8,5,1/2)^𝑇, and 𝑟=(−2,8,−10/3,−6)^𝑇.
Consider 𝑥=(−1,0,−1,−1)^𝑇.
Let 𝑋=diag(−1,1,−1,−1). Then 𝑥̃₁𝑟₁𝑥₁+𝑎₁−12𝑟¹𝑣₁𝑥̃₂−𝑢₁=−2<0,𝑥̃₃𝑟₂𝑥₂+𝑎₂−12𝑟²𝑣₂−𝑢₂=−6<0,𝑥̃₄𝑟₃𝑥₃+𝑎₃−12𝑟³𝑣₃−𝑢₃=−134<0,𝑥∈𝐷𝑟_4𝐵𝑥₄+𝑎₄−12𝑟⁴𝑣₄−𝑢₄=−7<0,,𝑏_𝑖𝑥_𝑖−𝑥_𝑖≥0,(𝑖=1,2,3,4).(4.11) According to Corollary 3.3, 𝑥=(−1,0,−1,−1) is a global minimizer.

Example 4.6. Consider the following problem: min3𝑥³¹−8𝑥³²+5𝑥³³+12𝑥³⁴−𝑥²¹−4𝑥²²+53𝑥²³−3𝑥²⁴−𝑥₁+2𝑥₂+3𝑥₃+4𝑥₄s.t.𝑥∈𝐷_𝐵=⁴_𝑖=1{−1,0}.(4.12) Let 𝑎=(−1,2,3,4)^𝑇,𝑏=(3,−8,5,1/2)^𝑇 and 𝑟=(−2,−8,10/3,−6)^𝑇.
Consider 𝑥=(−1,0,−1,−1)^𝑇.
Let 𝑋=diag(−1,1,−1,−1). Then 𝑥̃₁𝑟₁𝑥₁+𝑎₁−12𝑟¹𝑣₁𝑥̃₂−𝑢₁=0,𝑥̃₃𝑟₂𝑥₂+𝑎₂−21𝑟²𝑣₂−𝑢₂=6>0,𝑥̃₄𝑟₃𝑥₃+𝑎₃−21𝑟³𝑣₃−𝑢₃=−43<0,𝑟₄𝑥₄+𝑎₄−21𝑟⁴𝑣₄−𝑢₄=−7<0.(4.13)
We can see that the conditions are not true in 𝑥=(−1,0,−1,−1)^𝑇 in Corollary 3.3. But 𝑥=(−1,0,−1,−1) 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).