A Nonlinear Projection Neural Network for Solving Interval Quadratic Programming Problems and Its Stability Analysis

Abstract

This paper presents a nonlinear projection neural network for solving interval quadratic programs subject to box-set constraints in engineering applications. Based on the Saddle point theorem, the equilibrium point of the proposed neural network is proved to be equivalent to the optimal solution of the interval quadratic optimization problems. By employing Lyapunov function approach, the global exponential stability of the proposed neural network is analyzed. Two illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

1. Introduction

In lots of engineering applications including regression analysis, image and signal progressing, parameter estimation, filter design and robust control, and so forth [1], it is necessary to solve the following quadratic programming problem:1min2𝑥^𝑇𝑄𝑥+𝑐^𝑇𝑥,subjectto𝑥∈Ω, (1.1) where 𝑄∈𝑅^𝑛×𝑛, 𝑐∈𝑅^𝑛, and Ω is a convex set. When 𝑄 is a positive definite matrix, the problem (1.1) is said to be the convex quadratic program. When 𝑄 is a semipositive definite matrix, the problem (1.1) is said to be the degenerate convex quadratic program. In general, the matrix 𝑄 is not precisely known, but can only be enclosed in intervals, that is, 𝑄≤𝑄≤𝑄. Such quadratic program with interval data is named as interval quadratic program usually. In the recent years, there have been some project neural network approaches for solving the problem (1.1); see, for example, [2–15], and the references therein. In [2], Kennedy and Chua presented a primal network for solving the convex quadratic program. This network contains a finite penalty parameter, so it converges an approximate solution only. To overcome the penalty parameter, in [3, 4], Xia proposed several primal projection neural networks for solving the convex quadratic program and it dual, and analyzed the global asymptotic stability of the proposed neural networks when the constraint set Ω is a box set. In [5, 6], Xia et al. presented a recurrent projection neural network for solving the convex quadratic program and related linear piecewise equation, and gave some conditions of the exponential convergence. In [7, 8], Yang and Cao presented a delayed projection neural network for solving problem (1.1), and analyzed the global asymptotic stability and exponential stability of the proposed neural networks when the constraint set Ω is a unbounded box set. In order to solve the degenerate convex quadratic program, Tao et al. [9] and Xue and Bian [10, 11] proposed two projection neural networks, and proved that the equilibrium point of the proposed neural networks was equivalent to the KT point of the quadratic programming problem. Particularly, in [10], the proposed neural network was shown to have complete convergence and finite-time convergence, and the nonsingular part of the output trajectory with respect to 𝑄 has an exponentially convergent rate. In [12, 13], Hu and Wang designed a general projection neural network for solving monotone linear variational inequalities and extended linear-quadratic programming problems, and proved that the proposed network was exponentially convergent when the constraint set Ω is a polyhedral set.

In order to solve the interval quadratic program, in [14], Ding and Huang presented a new class of interval projection neural networks, and proved the equilibrium point of this neural networks is equivalent to the KT point of a class of interval quadratic program. Furthermore, some sufficient conditions to ensure the existence and global exponential stability for the unique equilibrium point of interval projection neural networks are given. To the best of the authors knowledge, the work in [14] is first to study solving the interval quadratic program by a projection neural network. However, the interval quadratic program discussed in [14] is only a quadratic program without constraints, thus has many limitations in practice. It is well known that the quadratic program with constraints is more popular.

Motivated by the above discussion, in the present paper, a new projection neural network for solving the interval quadratic programming problem with box-set constraints is presented. Based on the Saddle theorem, the equilibrium point of the proposed neural network is proved to be equivalent to the KT point of the interval quadratic program. By using the fixed point theorem, the existence and uniqueness of an equilibrium point of the proposed neural network are analyzed. By constructing a suitable Lyapunov function, a sufficient condition to ensure the existence and global exponential stability for the unique equilibrium point of interval projection neural network is obtained.

This paper is organized as follows. Section 2 describes the system model and gives some necessary preliminaries; Section 3 gives the proof of the existence of equilibrium point of the proposed neural network, and discusses the global exponential stability of the proposed neural network; Section 4 provides two numerical examples to demonstrate the validity of the obtained results. Some conclusions are drawn in Section 5.

2. A Projection Neural Network Model

Consider the following interval quadratic programming problem:1min2𝑥^𝑇𝑄𝑥+𝑐^𝑇𝑥𝑄subjectto𝑔≤𝐷𝑥≤ℎ,≤𝑄≤𝑄, (2.1) where 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛, 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛, 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛∈𝑅^𝑛×𝑛; 𝑐,𝑔,ℎ∈𝑅^𝑛, and 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛) is a positive definite diagonal matrix. 𝑄≤𝑄≤𝑄 means 𝑞_𝑖𝑗≤𝑞_𝑖𝑗≤𝑞_𝑖𝑗, 𝑖,𝑗=1,…,𝑛. The Lagrangian function of the problem (2.1) is 1𝐿(𝑥,𝑢,𝜂)=2𝑥^𝑇𝑄𝑥+𝑐^𝑇𝑥−𝑢^𝑇(𝐷𝑥−𝜂),𝑄≤𝑄≤𝑄, (2.2) where 𝑢∈𝑅^𝑛 is referred to as the Lagrange multiplier and 𝜂∈𝑋={𝑢∈𝑅^𝑛∣𝑔≤𝑢≤ℎ}. Based on the well-known Saddle point theorem [1], 𝑥^∗ is an optimal solution of (2.1) if and only if there exist 𝑢^∗ and 𝜂^∗, satisfying 𝐿(𝑥^∗,𝑢,𝜂^∗)≤𝐿(𝑥^∗,𝑢^∗,𝜂^∗)≤𝐿(𝑥,𝑢^∗,𝜂), that is,12𝑥^∗𝑇𝑄𝑥^∗+𝑐^𝑇𝑥^∗−𝑢^𝑇(𝐷𝑥^∗−𝜂^∗)≤12𝑥^∗𝑇𝑄𝑥^∗+𝑐^𝑇𝑥^∗−𝑢^∗𝑇(𝐷𝑥^∗−𝜂^∗)≤12𝑥^𝑇𝑄𝑥+𝑐^𝑇𝑥−𝑢^∗𝑇(𝐷𝑥−𝜂). (2.3) By the first inequality in (2.3), (𝑢−𝑢^∗)^𝑇(𝐷𝑥^∗−𝜂^∗)≥0, for all 𝑢∈𝑅^𝑛, hence 𝐷𝑥^∗=𝜂^∗. Let 𝑓(𝑥)=(1/2)𝑥^𝑇𝑄𝑥+𝑐^𝑇𝑥−𝑢^∗𝑇𝐷𝑥. By the second inequality in (2.3), 𝑓(𝑥^∗)−𝑓(𝑥)≤𝑢^∗𝑇(𝜂−𝜂^∗), for all 𝑥∈𝑅^𝑛, 𝜂∈𝑋. If there exists 𝑥∈𝑅^𝑛 such that 𝑓(𝑥^∗)−𝑓(𝑥)>0, then 0<𝑢^∗𝑇(𝜂−𝜂^∗), for all 𝜂∈𝑋, which is contradictive when 𝜂=𝜂^∗. Thus, for any 𝑥∈𝑅^𝑛, it follows that 𝑓(𝑥^∗)−𝑓(𝑥)≤0 and 𝑢^∗𝑇(𝜂−𝜂^∗)≥0, for all 𝜂∈𝑋.

By using the project formulation [16], the above inequality can be equivalently represented as 𝜂^∗=𝑃_𝑋(𝜂^∗−𝑢^∗), where 𝑃_𝑋(𝑢)=[𝑃_𝑋(𝑢₁),𝑃_𝑋(𝑢₂),…,𝑃_𝑋(𝑢_𝑛)]^𝑇 is a project function, and, for 𝑖=1,2,…,𝑛, 𝑃_𝑋(𝑢_𝑖𝑔)=^𝑖,𝑢_𝑖<𝑔_𝑖,𝑢_𝑖,𝑔_𝑖≤𝑢_𝑖≤ℎ_𝑖,ℎ_𝑖,𝑢_𝑖>ℎ_𝑖. (2.4) On the other hand, 𝑓(𝑥^∗)≤𝑓(𝑥), for all 𝑥∈𝑅^𝑛. This implies that ∇𝑓(𝑥^∗)=𝑄𝑥^∗+𝑐−𝐷𝑢^∗=0. (2.5) Thus, 𝑥^∗ is an optimal solution of (2.1) if and only if there exist 𝑢^∗ and 𝜂^∗, such that (𝑥^∗,𝑢^∗,𝜂^∗) satisfies𝐷𝑥=𝜂,𝑄𝑥+𝑐−𝐷𝑢=0,𝜂=𝑃_𝑋(𝜂−𝑢). (2.6) From (2.6), it follows that 𝐷𝑥=𝑃_𝑋(𝐷𝑥−𝑢). Hence, 𝑥^∗ is an optimal solution of (2.1) if and only if there exists 𝑢^∗ such that (𝑥^∗,𝑢^∗) satisfies 𝑄𝑥+𝑐−𝐷𝑢=0,𝐷𝑥=𝑃_𝑋(𝐷𝑥−𝑢). (2.7) Substituting 𝑢=𝐷⁻¹(𝑄𝑥+𝑐) into the equation 𝐷𝑥=𝑃_𝑋(𝐷𝑥−𝑢), we have 𝐷𝑥=𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐), (2.8) where 𝐷⁻¹=1𝑑₁010⋯0𝑑₂1⋯0⋮⋮⋱⋮00⋯𝑑_𝑛,𝐷⁻¹1𝑄=𝑑₁𝑞₁₁1𝑑₁𝑞₁₂⋯1𝑑₁𝑞_1𝑛1𝑑₂𝑞₂₁1𝑑₂𝑞₂₂⋯1𝑑₂𝑞_2𝑛1⋮⋮⋱⋮𝑑_𝑛𝑞_𝑛11𝑑_𝑛𝑞_𝑛2⋯1𝑑_𝑛𝑞_𝑛𝑛. (2.9) By the above discussion, we can obtain the following proposition.

Proposition 2.1. Let 𝑥^∗ be a solution of the project equation 𝐷𝑥=𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐), (2.10) then, 𝑥^∗ is an optimal solution of the problem (2.1).

In the following, we propose a neural network, which is said to be the interval projection neural network, for solving (2.1) and (2.10), whose dynamical equation is defined as follows: 𝑑𝑥(𝑡)𝑑𝑡=𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝐷𝑥,𝑡>𝑡₀,𝑥(𝑡₀)=𝑥₀,𝑄≤𝑄≤𝑄. (2.11) The neural networks (2.11) can be equivalently written as𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑃_𝑋(𝑑_𝑖𝑥_𝑖−1𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗−𝑐_𝑖𝑑_𝑖)−𝑑_𝑖𝑥_𝑖,𝑡>𝑡₀,𝑥_𝑖(𝑡₀)=𝑥_𝑖0,𝑞_𝑖𝑗≤𝑞_𝑖𝑗≤𝑞_𝑖𝑗,𝑖=1,…,𝑛,𝑗=1,…,𝑛. (2.12) Figure 1 shows the architecture of the neural network (2.11), where 𝑀=(𝑚_𝑖𝑗)_𝑛×𝑛=𝐷−𝐷⁻¹𝑄, 𝐶=𝐷⁻¹𝑐, and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛.

Figure 1 

Architecture of the proposed neural network in (2.11).

Definition 2.2. The point 𝑥^∗ is said to be an equilibrium point of interval projection neural network (2.11), if 𝑥^∗ satisfies 0=𝑃_𝑋(𝐷𝑥^∗−𝐷⁻¹𝑄𝑥^∗−𝐷⁻¹𝑐)−𝐷𝑥^∗. (2.13)

By Proposition 2.1 and Definition 2.2, we have the following theorem.

Theorem 2.3. The point 𝑥^∗ is an equilibrium point of the interval projection neural network (2.11) if and only if it is an optimal solution of the interval quadratic program (2.1).

Definition 2.4. The equilibrium point 𝑥^∗ of the neural network (2.11) is said to be globally exponentially stable, if the trajectory 𝑥(𝑡) of the neural network (2.11) with the initial value 𝑥₀ satisfies ‖𝑥(𝑡)−𝑥^∗‖≤𝑐₀exp(−𝛽(𝑡−𝑡₀)),∀𝑡≥𝑡₀, (2.14) where 𝛽>0 is a constant independent of the initial value 𝑥₀ and 𝑐₀>0 is a constant dependent on the initial value 𝑥₀. ‖⋅‖ denotes the 1-norm of 𝑅^𝑛, that is, ‖𝑥‖=∑^𝑛_𝑖=1|𝑥_𝑖|.

Lemma 2.5 (see [17]). Let Ω⊂𝑅^𝑛 be a closed convex set. Then, (𝑣−𝑃_Ω(𝑣))^𝑇(𝑃_Ω(𝑣)−𝑢)≥0,∀𝑢∈Ω,𝑣∈𝑅^𝑛,‖𝑃_Ω(𝑢)−𝑃_Ω(𝑣)‖≤‖𝑢−𝑣‖,∀𝑢,𝑣∈𝑅^𝑛, (2.15) where 𝑃_Ω(𝑢) is a project function on Ω, given by 𝑃_Ω(𝑢)=argmin_𝑦∈Ω‖𝑢−𝑦‖.

3. Stability Analysis

In order to obtain the results in this paper, we make the following assumption for the neural network (2.11):𝐻₁:𝑞_𝑖𝑖≤𝑑²_𝑖,𝑑²_𝑖−𝑑_𝑖𝑑^∗<𝑞_𝑖𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖<𝑑²_𝑖,𝑖=1,2,…,𝑛, (3.1) where 𝑑^∗=max_{1≤𝑖≤𝑛}1/𝑑_𝑖, 𝑞^∗_𝑗𝑖=max{|𝑞_𝑗𝑖|,|𝑞_𝑗𝑖|}.

Theorem 3.1. If the assumption 𝐻₁ is satisfied, then there exists a unique equilibrium point for the neural network (2.11).

Proof. Let 𝒯(𝑥)=𝐷⁻¹𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐),𝑥∈𝑅^𝑛. By Definition 2.2, it is obvious that the neural network (2.11) has a unique equilibrium point if and only if 𝒯 has a unique fixed point in 𝑅^𝑛. In the following, by using fixed point theorem, we prove that 𝒯 has a unique fixed point in 𝑅^𝑛. For any 𝑥,𝑦∈𝑅^𝑛, by Lemma 2.5 and the assumption 𝐻₁, we can obtain that ‖𝒯(𝑥)−𝒯(𝑦)‖=‖𝐷⁻¹𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝐷⁻¹𝑃_𝑋(𝐷𝑦−𝐷⁻¹𝑄𝑦−𝐷⁻¹𝑐)‖≤‖𝐷⁻¹‖‖𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝑃_𝑋(𝐷𝑦−𝐷⁻¹𝑄𝑦−𝐷⁻¹𝑐)‖≤‖𝐷⁻¹‖‖(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−(𝐷𝑦−𝐷⁻¹𝑄𝑦−𝐷⁻¹𝑐)‖=‖𝐷⁻¹‖‖(𝐷−𝐷⁻¹𝑄)(𝑥−𝑦)‖≤‖𝐷⁻¹‖‖𝐷−𝐷⁻¹𝑄‖‖𝑥−𝑦‖=max_{1≤𝑖≤𝑛}1𝑑_𝑖⋅max_{1≤𝑖≤𝑛}(𝑑_𝑖−1𝑑_𝑖𝑞_𝑖𝑖+1𝑑_𝑖^𝑛_{𝑗=1,𝑗≠𝑖}|𝑞_𝑗𝑖|)⋅‖𝑥−𝑦‖≤𝑑^∗max_{1≤𝑖≤𝑛}(𝑑_𝑖−1𝑑_𝑖𝑞_𝑖𝑖+1𝑑_𝑖^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖)‖𝑥−𝑦‖=max_{1≤𝑖≤𝑛}ℓ_𝑖‖𝑥−𝑦‖, (3.2) where ℓ_𝑖=𝑑^∗(𝑑_𝑖−(1/𝑑_𝑖)𝑞_𝑖𝑖+(1/𝑑_𝑖)∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖). By the assumption 𝐻₁, 0<𝑑^∗(𝑑_𝑖−(1/𝑑_𝑖)𝑞_𝑖𝑖+(1/𝑑_𝑖)∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖)<1, 𝑖=1,2,…,𝑛. This implies that 0<max_{1≤𝑖≤𝑛}ℓ_𝑖<1. Equation (3.2) shows that 𝒯 is a contractive mapping, and hence 𝒯 has a unique fixed point. This completes the proof.

Proposition 3.2. If the assumption 𝐻₁ holds, then for any 𝑥₀∈𝑅^𝑛, there exists a solution with the initial value 𝑥(0)=𝑥₀ for the neural network (2.11).

Proof. Let ℱ=𝐷(𝒯−ℐ), where ℐ is an identity mapping, then ℱ(𝑥)=𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝐷𝑥. By (3.2), we have ‖ℱ(𝑥)−ℱ(𝑦)‖=‖𝐷(𝒯−ℐ)(𝑥)−𝐷(𝒯−ℐ)(𝑦)‖≤max_{1≤𝑖≤𝑛}𝑑_𝑖(1+max_{1≤𝑖≤𝑛}ℓ_𝑖)‖𝑥−𝑦‖,∀𝑥,𝑦∈𝑅^𝑛. (3.3) Equation (3.3) means that the mapping ℱ is globally Lipschitz. Hence, for any 𝑥₀∈𝑅^𝑛, there exists a solution with the initial value 𝑥(0)=𝑥₀ for the neural network (2.11). This completes the proof.

Proposition 3.2 shows the existence of the solution for the neural network (2.11).

Theorem 3.3. If the assumption 𝐻₁ is satisfied, then the equilibrium point of the neural network (2.11) is globally exponentially stable.

Proof. By Theorem 3.1, the neural network (2.11) has a unique equilibrium point. We denote the equilibrium point of the neural network (2.11) by 𝑥^∗.
Consider Lyapunov function 𝒱(𝑡)=‖𝑥(𝑡)−𝑥^∗‖=∑^𝑛_𝑖=1|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|. Calculate the derivative of 𝒱(𝑡) along the solution 𝑥(𝑡) of the neural network (2.11). When 𝑡>𝑡₀, we have 𝑑𝒱(𝑡)=𝑑𝑡^𝑛_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|𝑑(𝑥_𝑖(𝑡)−𝑥^∗_𝑖)=𝑑𝑡_𝑛𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|(𝑃_𝑋(𝑑_𝑖𝑥_𝑖−1𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗−𝑐_𝑖𝑑_𝑖)−𝑑_𝑖𝑥_𝑖)=^𝑛_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|(𝑃_𝑋(𝑑_𝑖𝑥_𝑖−1𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗−𝑐_𝑖𝑑_𝑖)−𝑑_𝑖𝑥^∗_𝑖+𝑑_𝑖𝑥^∗_𝑖−𝑑_𝑖𝑥_𝑖)=−^𝑛_𝑖=1𝑑_𝑖|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|+^𝑛_𝑖=1𝑥_𝑖(𝑡)−𝑥^∗_𝑖|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|(𝑃_𝑋(𝑑_𝑖𝑥_𝑖−1𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗−𝑐_𝑖𝑑_𝑖)−𝑑_𝑖𝑥^∗_𝑖)≤(−min_{1≤𝑖≤𝑛}𝑑_𝑖)^𝑛_𝑖=1|𝑥_𝑖(𝑡)−𝑥^∗_𝑖|+^𝑛_𝑖=1|𝑃_𝑋(𝑑_𝑖𝑥_𝑖−1𝑑_𝑖^𝑛_𝑗=1𝑞_𝑖𝑗𝑥_𝑗−𝑐_𝑖𝑑_𝑖)−𝑑_𝑖𝑥^∗_𝑖|=(−min_{1≤𝑖≤𝑛}𝑑_𝑖)‖𝑥−𝑥^∗‖+‖𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝐷𝑥^∗‖. (3.4) Noting 𝐷𝑥^∗=𝑃_𝑋(𝐷𝑥^∗−𝐷⁻¹𝑄𝑥^∗−𝐷⁻¹𝑐), by Lemma 2.5, we have ‖𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝐷𝑥^∗‖=‖𝑃_𝑋(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−𝑃_𝑋(𝐷𝑥^∗−𝐷⁻¹𝑄𝑥^∗−𝐷⁻¹𝑐)‖≤‖(𝐷𝑥−𝐷⁻¹𝑄𝑥−𝐷⁻¹𝑐)−(𝐷𝑥^∗−𝐷⁻¹𝑄𝑥^∗−𝐷⁻¹𝑐)‖≤‖(𝐷−𝐷⁻¹𝑄)(𝑥−𝑥^∗)‖≤‖𝐷−𝐷⁻¹𝑄‖‖𝑥−𝑥^∗‖. (3.5) Hence, 𝑑𝒱(𝑡)𝑑𝑡≤(−min_{1≤𝑖≤𝑛}𝑑_𝑖+max_{1≤𝑖≤𝑛}(𝑑_𝑖−1𝑑_𝑖|𝑞_𝑖𝑖1|+𝑑_𝑖^𝑛_{𝑗=1,𝑗≠𝑖}|𝑞_𝑗𝑖|))⋅‖𝑥−𝑥^∗‖1≤(−𝑑^∗+max_{1≤𝑖≤𝑛}(𝑑_𝑖−1𝑑_𝑖𝑞_𝑖𝑖+1𝑑_𝑖^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖))⋅‖𝑥−𝑥^∗‖=max_{1≤𝑖≤𝑛}ℓ^′_𝑖‖𝑥−𝑥^∗‖, (3.6) where ℓ^′_𝑖=𝑑_𝑖−(1/𝑑_𝑖)𝑞_𝑖𝑖+(1/𝑑_𝑖)∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑞^∗_𝑗𝑖−(1/𝑑^∗). By the assumption 𝐻₁, ℓ^′_𝑖<0. Hence, max_{1≤𝑖≤𝑛}ℓ^′_𝑖<0. Let ℓ^∗=min_{1≤𝑖≤𝑛}|ℓ^′_𝑖|, then ℓ^∗>0. Equation (3.6) can be rewritten as 𝑑𝒱(𝑡)/𝑑𝑡≤−ℓ^∗‖𝑥−𝑥^∗‖. It follows easily that ‖𝑥(𝑡)−𝑥^∗‖≤‖𝑥₀−𝑥^∗‖exp(−ℓ^∗(𝑡−𝑡₀)), for all 𝑡>𝑡₀. This shows that the equilibrium point 𝑥^∗ of the neural network (2.11) is globally exponentially stable. This completes the proof.

4. Illustrative Examples

Example 4.1. Consider the interval quadratic program defined by 𝐷=diag(2,1), 𝑔=(2,2)^𝑇, ℎ=(3,2)^𝑇, 𝑐^𝑇=(−1,−1), 𝑄=(^30.1_0.10.7), 𝑄=(^3.10.2_0.20.8), and 𝑄^∗=(𝑞^∗_𝑗𝑖)=(^3.10.2_0.20.8).

The optimal solution of this quadratic program is (1,2) under 𝑄=𝑄 or 𝑄=𝑄. It is easy to check that3.0<𝑞₁₁<3.1,𝑞₁₁<𝑑²₁=4,0.7<𝑞₂₂<0.8,𝑞₂₂<𝑑²₂𝑑=1,₂₁−𝑑₁𝑑^∗=2<𝑞₁₁−𝑞^∗₂₁=3.0−0.2<𝑑²₁𝑑=4,₂₂−𝑑₂𝑑^∗=0<𝑞₂₂−𝑞^∗₁₂=0.7−0.2<𝑑²₂=1. (4.1) The assumption 𝐻₁ holds. By Theorems 3.1 and 3.3, the neural network (2.11) has a unique equilibrium point which is globally exponentially stable, and the unique equilibrium point (1,2) is the optimal solution of this quadratic programming problem.

In the case of 𝑄=𝑄, Figure 2 reveals that the projection neural network (2.11) with random initial value (2.5,−0.5) has a unique equilibrium point (1,2) which is globally exponentially stable. In the case of 𝑄=𝑄, Figure 3 reveals that the projection neural network (2.11) with random initial value (−2.5,3) has the same unique equilibrium point (1,2) which is globally exponentially stable. These are in accordance with the conclusion of Theorems 3.1 and 3.3.

Figure 2 

Convergence of the state trajectory of the neural network with random initial value (2.5,$-0.5$); $Q=\underline{Q}$ in this example.

Figure 3 

Convergence of the state trajectory of the neural network with random initial value ($-2.5$,3); $Q=\overline{Q}$ in this example.

Example 4.2. Consider the interval quadratic program defined by 𝐷=diag(1,2,2), 𝑔=(1,2,3)^𝑇, ℎ=(2,3,4)^𝑇, 𝑐^𝑇=(1,1,1), 𝑄=(^{0.80.20.30.230.1}_0.30.13.5), 𝑄=(^{0.90.30.40.33.10.2}_0.40.23.6), and 𝑄^∗=(𝑞^∗_𝑗𝑖)=(^{0.90.30.40.33.10.2}_0.40.23.6).

The optimal solution of this quadratic program is (1,1,1.5) under 𝑄=𝑄 or 𝑄=𝑄. It is easy to check that0.8<𝑞₁₁<0.9,𝑞₁₁<𝑑²₁=1,3.0<𝑞₂₂<3.1,𝑞₂₂<𝑑²₂=4,3.5<𝑞₃₃<3.6,𝑞₃₃<𝑑²₃𝑑=4,₂₁−𝑑₁𝑑^∗=0<𝑞₁₁−(𝑞^∗₂₁+𝑞^∗₃₁)=0.8−(0.3+0.4)<𝑑²₁𝑑=1,₂₂−𝑑₂𝑑^∗=2<𝑞₂₂−(𝑞^∗₁₂+𝑞^∗₃₂)=3.0−(0.3+0.2)<𝑑²₂𝑑=4,₂₃−𝑑₃𝑑^∗=0<𝑞₃₃−(𝑞^∗₁₃+𝑞^∗₂₃)=3.5−(0.4+0.2)<𝑑²₃=4. (4.2) The assumption 𝐻₁ holds. By Theorems 3.1 and 3.3, the neural network (2.11) has a unique equilibrium point which is globally exponentially stable, and the unique equilibrium point (1,1,1.5) is the optimal solution of this quadratic programming problem.

In the case of 𝑄=𝑄, Figure 4 reveals that the projection neural network (2.11) with random initial value (−0.5,0.6,−0.8) has a unique equilibrium point (1,1,1.5) which is globally exponentially stable. In the case of 𝑄=𝑄, Figure 5 reveals that the projection neural network (2.11) with random initial value (0.8,−0.6,0.3) has the same unique equilibrium point (1,1,1.5) which is globally exponentially stable. These are in accordance with the conclusion of Theorems 3.1 and 3.3.

Figure 4 

Convergence of the state trajectory of the neural network with random initial value ($-0.5$,$0.6$,$-0.8$); $Q=\underline{Q}$ in this example.

Figure 5 

Convergence of the state trajectory of the neural network with random initial value ($0.8$,$-0.6$,$0.3$); $Q=\overline{Q}$ in this example.

5. Conclusion

In this paper, we have developed a new projection neural network for solving interval quadratic programs, the equilibrium point of the proposed neural network is equivalent to the solution of interval quadratic programs. A condition is derived which ensures the existence, uniqueness, and global exponential stability of the equilibrium point. The results obtained are highly valuable in both theory and practice for solving interval quadratic programs in engineering.

Acknowledgment

This paper was supported by the Hebei Province Education Foundation of China (2009157).