An Effective Global Optimization Algorithm for Quadratic Programs with Quadratic Constraints

Abstract

This paper will present an effective algorithm for globally solving quadratic programs with quadratic constraints. In this algorithm, we propose a new linearization method for establishing the linear programming relaxation problem of quadratic programs with quadratic constraints. The proposed algorithm converges with the global optimal solution of the initial problem, and numerical experiments show the computational efficiency of the proposed algorithm.

1. Introduction

Quadratic programs with quadratic constraints (QPWQC) have attracted the attention of many researchers for several decades. On the one hand, it is since these classes of problems have a broad applications in multistage shipping, path planning, finance, and portfolio optimization, among others. [1,2,3,4,5,6,7,8,9,10,11]. On the other hand, it is because these classes of problems exist as important theoretical complexities and computational difficulties, that is to say, they are known to generally possess multiple local optimal solutions, which are not optimal solutions.

In the last several decades, many algorithms have been developed for globally solving the (QPWQC) and its special cases, such as branch-and-bound method [12,13], approximation approach [14], robust approach [15], branch-reduce-bound algorithm [16,17,18,19], geometric programming approach [20,21,22,23], and others. Except for the above approaches, some global optimization algorithms [24,25,26,27,28,29,30,31,32,33,34,35,36,37,38] for linear multiplicative programming problems and generalized linear fractional programming problems can be used to solve the quadratic programs with quadratic constraints (QPWQC) considered in this paper. Although these algorithms can be employed to solve the QPWQC and its special cases, less work has been done for globally solving the QPWQC considered in this paper.

In this paper, first of all, by making use of the characteristics of simple variable quadratic function, we construct a new linearization method for establishing the linear programming relaxation problem of the QPWQC. Next, we present a global optimization algorithm based on the branch-and-bound scheme for solving the QPWQC. Finally, the global convergence of the proposed algorithm is proved, and numerical experimental results demonstrate the higher computational efficiency of the proposed algorithm.

The main features of the proposed algorithm are given as follows. (1) A new linearization method is proposed for systematically converting the QPWQC into a sequence of linear programming relaxation problems, and the solutions of these linear programming relaxation problems can infinitely approximate the global optimal solution of the original QPWQC by subdividing the linear relaxation of the feasible region of the QPWQC and solving a series of linear programming relaxation problems. (2) The constructed linear programming relaxation problems are embedded within a branch-and-bound framework, which can be effectively solved by any efficient linear programming method. (3) Combining the proposed linear programming relaxation problem with the branch-and-bound framework, an effective algorithm is proposed for solving the problem of QPWQC. (4) Compared with the exist algorithms [37,39,40,41,42,43,44,45,46,47], numerical results show that the proposed algorithm in this paper can be used to globally solve the QPWQC with higher computational efficiency.

The remaining sections of this paper are organized as follows. Firstly, the aim of Section 2 is to propose a new linearization method for establishing the linear programming relaxation problem of the initial QPWQC. Secondly, based on the branch-and-bound scheme, Section 3 proposes a global optimization algorithm, and its global convergence is proved. Thirdly, compared with the existing methods, Section 4 describes some numerical examples to show the computational efficiency of the proposed algorithm. Finally, some conclusions are given.

2. New Linearization Method for Deriving Linear Programming Relaxation Problem

In this paper, the mathematical modeling of quadratic programs with quadratic constraints is given as follows:

$$(\mathrm{QPWQC})\{\begin{array}{l}\min {\psi }_0(x)={\displaystyle \sum _{k=1}^n{c}_k^o{x}_k+}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^n{d}_{ij}^0{x}_j{x}_k}}\\ s.t. {\psi }_i(x)={\displaystyle \sum _{k=1}^n{c}_k^i{x}_k+}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^n{d}_{ij}^i{x}_j{x}_k}\le b_i, i=1,2,\dots ,m,}\\ \hspace{1em}\hspace{1em}x\in X^0=\{ x\in R^n: l^0\le x\le u^0\},\end{array}$$

(1)

where $d_{jk}^i,c_k^i$, and $b_i$ are all arbitrary real numbers; $l^0={(l_1^0,\dots ,l_n^0)}^T>-\infty , u^0={(u_1^0,\dots ,u_n^0)}^T<+\infty$.

In this section, we construct a new linearization method for deriving the linear programming relaxation problem of the QPWQC, and the detailed construction process of the linearization method is described as follows.

For convenience, we assume without loss of generality that $X=\{{(x_1,x_2,\dots ,x_n)}^T\in R^n:$ $l_j\le x_j\le u_j,j=1,2,\dots ,n\}\subseteq X^0$.

Theorem 1.

For any$x\in X,$$k\in \{1,2,\dots ,n\},$we consider the functions$x_k^2,$$u_k^2+2u_k(x_k-u_k)$and$u_k^2+2l_k(x_k-u_k)$, we have the following conclusions:

$$u_k^2+2u_k(x_k-u_k)\le x_k^2\le u_k^2+2l_k(x_k-u_k);$$

(2)

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{x_k^2-[u_k^2+2u_k(x_k-u_k)]\}=0$$

(3)

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{u_k^2+2l_k(x_k-u_k)-x_k^2\}=0$$

(4)

Proof. 

(i) From the mean value theorem, there exists a point ${\xi }_k=\alpha l_k+(1-\alpha )u_k\in [l_k,u_k]$, where $\alpha \in [0,1]$, which satisfies that

$$x_k^2=u_k^2+2{\xi }_k(x_k-u_k).$$

(5)

From $l_k\le {\xi }_k\le u_k$, it follows that

$$u_k^2+2l_k(x_k-u_k)\ge u_k^2+2{\xi }_k(x_k-u_k)=x_k^2\ge u_k^2+2u_k(x_k-u_k).$$

(6)

(ii) From

$$x_k^2-[u_k^2+2u_k(x_k-u_k)]=u_k^2-x_k^2\le {(u_k-l_k)}^2,$$

(7)

it follows that

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{x_k^2-[u_k^2+2u_k(x_k-u_k)]\}=0.$$

(8)

Also from

$$u_k^2+2l_k(x_k-u_k)-x_k^2=(x_k-u_k)[2l_k-u_k-x_k]\le 2{(u_k-l_k)}^2.$$

(9)

Therefore, we have

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{u_k^2+2l_k(x_k-u_k)-x_k^2\}=0.$$

(10)

The proof is completed. □

From the conclusion (2), it follows that

$$u_j^2+2u_j(x_j-u_j)\le x_j^2\le u_j^2+2l_j(x_j-u_j),$$

(11)

$${(x_j-x_k)}^2\ge {(u_j-l_k)}^2+2(u_j-l_k)[x_j-x_k-(u_k-l_k)],$$

(12)

$${(x_j-x_k)}^2\le {(u_j-l_k)}^2+2(l_j-u_k)[x_j-x_k-(u_k-l_k)].$$

(13)

From the conclusions (3) and (4), it follows that

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{x_j^2-[u_j^2+2u_j(x_j-u_j)]\}=0,$$

(14)

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{u_j^2+2l_j(x_j-u_j)-x_j^2\}=0,$$

(15)

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{{(u_j-l_k)}^2+2(l_j-u_k)[x_j-x_k-(u_k-l_k)]-{(x_j-x_k)}^2\}=0.$$

(16)

and

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{{(x_j-x_k)}^2-\{ {(u_j-l_k)}^2+2(u_j-l_k)[x_j-x_k-(u_j-l_k)]\}\}=0.$$

(17)

For any $x\in X,$ $j,k\in \{1,2,\dots ,n\},j\ne k,$ without loss of generality, we define

$$\begin{array}{ll}{\underset{\_}{\psi }}_{jk}(x)=\frac{1}{2}& \{u_j^2+2u_j(x_j-u_j)+u_k^2+2u_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(l_j-u_k)[x_j-x_k-(u_k-l_k)]\}]\end{array}$$

(18)

and

$$\begin{array}{ll}{\overline{\psi }}_{jk}(x)=\frac{1}{2}& \{u_j^2+2l_j(x_j-u_j)+u_k^2+2l_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(u_j-l_k)[x_j-x_k-(u_k-l_k)]\}],\end{array}$$

(19)

Theorem 2.

For any$x\in X,$$j,k\in \{1,2,\dots ,n\},j\ne k,$consider the functions${\underset{\_}{\psi }}_{jk}(x),$$x_j{x}_k$and${\overline{\psi }}_{jk}(x) ,$the following conclusions hold:

$${\underset{\_}{\psi }}_{jk}(x)\le x_j{x}_k=\frac{1}{2}[x_j^2+x_k^2-{(x_j-x_k)}^2]\le {\overline{\psi }}_{jk}(x),$$

(20)

$$\underset{\Vert u-l\Vert \to 0}{\lim }[x_j{x}_k-{\underset{\_}{\psi }}_{jk}(x)]=0,$$

(21)

and

$$\underset{\Vert u-l\Vert \to 0}{\lim }[{\overline{\psi }}_{jk}(x)-x_j{x}_k]=0.$$

(22)

Proof. 

(i) By the conclusions of Theorem 1, it follows that

$$\begin{array}{ll}{\overline{\psi }}_{jk}(x)=& \frac{1}{2}\{u_j^2+2l_j(x_j-u_j)+u_k^2+2l_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(u_j-l_k)[x_j-x_k-(u_k-l_k)]\}\}\\ & \ge \frac{1}{2}[x_j^2+x_k^2-(x_j-x_k{)}^2]=x_j{x}_k\\ & \ge \frac{1}{2}\{u_j^2+2u_j(x_j-u_j)+u_k^2+2u_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(l_j-u_k)[x_j-x_k-(u_j-l_k)]\}\}\\ & ={\underset{\_}{\psi }}_{jk}(x).\end{array}$$

(23)

(ii) From the inequalities (7) and (9), we have

$$\begin{array}{ll}{x}_j{x}_k-{\underset{\_}{\psi }}_{jk}(x)=\frac{1}{2}& [x_j^2+x_k^2-(x_j-x_k{)}^2]- \frac{1}{2}\{u_j^2+2u_j(x_j-u_j)+u_k^2+2u_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(l_j-u_k)[x_j-x_k-(u_j-l_k)]\}\}\\ & \le \frac{1}{2}(u_j-l_j{)}^2+\frac{1}{2}(u_k-l_k{)}^2+(u_k+u_j-l_j-l_k{)}^2.\end{array}$$

(24)

Thus, we can get that $\underset{\Vert u-l\Vert \to 0}{\lim }[x_j{x}_k-{\underset{\_}{\psi }}_{jk}(x)]=0.$ □

Also from the proof of Theorem 2 and the inequalities (7) and (9), we get that

$$\begin{array}{ll}{\overline{\psi }}_{jk}(x)-x_j{x}_k=\frac{1}{2}& \{u_j^2+2l_j(x_j-u_j)+u_k^2+2l_k(x_k-u_k)\\ & -\{(u_j-l_k{)}^2+2(u_j-l_k)[x_j-x_k-(u_j-l_k)]\}\}\\ & -\frac{1}{2}[x_j^2+x_k^2-(x_j-x_k{)}^2]\\ & \le (u_j-l_j{)}^2+(u_k-l_k{)}^2+\frac{1}{2}(u_k+u_j-l_k-l_j{)}^2.\end{array}$$

(25)

Thus, we can get that $\underset{\Vert u-l\Vert \to 0}{\lim }[{\overline{\psi }}_{jk}(x)-x_j{x}_k]=0.$

Without loss of generality, for any $X=[l,u]\subseteq X^0$ for any $x\in X,$ and $i\in \{0,1,2,\dots ,m\}$ we let

$${\underset{\_}{f}}_{kk}^i=\{\begin{array}{cc}{d}_{kk}^i\{u_k^i+2u_k(x_k-u_k)\},& if d_{kk}^i>0 ,\\ d_{kk}^i\{u_k^i+2l_k(x_k-u_k)\},& if d_{kk}^i<0, \end{array}$$

(26)

$${\underset{\_}{f}}_{jk}^i=\{\begin{array}{cc}{d}_{jk}^i{\underset{\_}{\psi }}_{jk}(x),& if d_{jk}^i>0 ,j\ne k,\\ d_{jk}^i{\overline{\psi }}_{jk}(x),& if d_{jk}^i<0 ,j\ne k,\end{array}$$

(27)

$${\psi }_i^L(x)={\displaystyle \sum _{k=1}^n(c_k^i{x}_k+{\underset{\_}{f}}_{kk}^i(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{f}}_{jk}^i(x).}}$$

(28)

Theorem 3.

For any$x\in X=[l,u]\subseteq X^0,$for each$i=0,1,2,\dots ,m,$we get that${\psi }_i^L(x)\le {\psi }_i(x)$and$\underset{\Vert u-l\Vert \to 0}{\lim }[{\psi }_i(x)-{\psi }_i^L(x)]=0.$

Proof. 

(i) From (2) and (12), we have

$${\underset{\_}{f}}_{kk}^i\le d_{kk}^i{x}_k^2\le {\overline{f}}_{kk}^i(x) and {\underset{\_}{f}}_{jk}^i\le d_{jk}^i{x}_j{x}_k\le {\overline{f}}_{jk}^i(x).$$

(29)

By (29), it follows that ${\psi }_i^L(x)\le {\psi }_i(x).$

(ii)

$$\begin{array}{ll}{\psi }_i(x)-{\psi }_i^L(x)=& {\displaystyle \sum _{k=1}^n{c}_k^i{x}_k+}{\displaystyle \sum _{k=1}^n{d}_{kk}^i{x}_k^2+}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{d}_{jk}^i{x}_j}{x}_k}\\ & \hspace{1em}-\left[{\displaystyle \sum _{k=1}^n{c}_k^i{x}_k+{\displaystyle \sum _{k=1}^n{\underset{\_}{f}}_{kk}^i(x)}}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{f}}_{jk}^i(x)}}\right]\\ & ={\displaystyle \sum _{k=1}^n(d_{kk}^i{x}_k^2-{\underset{\_}{f}}_{kk}^i(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n\left[d_{jk}^i{x}_j{x}_k-{\underset{\_}{f}}_{jk}^i(x)\right]}}\\ & ={\displaystyle \sum _{k=1,d_{kk}^i>0}^n{d}_{kk}^i\{x_k^2-[u_k^2+2u_k(x_k-u_k)]\}}\\ & \hspace{1em}+{\displaystyle \sum _{k=1,d_{kk}^i<0}^n{d}_{kk}^i\{x_k^2-[u_k^2+2l_k(x_k-u_k)]\}}\\ & \hspace{1em}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j,d_{kk}^i>0}^n{d}_{jk}^i\left[x_j{x}_k-{\underset{\_}{\psi }}_{jk}(x)\right]}}\\ & \hspace{1em}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j,d_{kk}^i<0}^n{d}_{jk}^i\left[x_j{x}_k-{\overline{\psi }}_{jk}(x)\right]}}\end{array}$$

(30)

From (3), (4), (14), and (15), we get

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{x_k^2-[u_k^2+2u_k(x_k-u_k)]\}=0,$$

(31)

$$\underset{\Vert u-l\Vert \to 0}{\lim }\{[u_k^2+2l_k(x_k-u_k)]-x_k^2\}=0,$$

(32)

$$\underset{\Vert u-l\Vert \to 0}{\lim }[x_j{x}_k-{\underset{\_}{\psi }}_{jk}(x)]=0$$

(33)

and

$$\underset{\Vert u-l\Vert \to 0}{\lim }[{\overline{\psi }}_{jk}(x)-x_j{x}_k]=0.$$

(34)

Therefore, we have

$$\underset{\Vert u-l\Vert \to 0}{\lim }[{\psi }_i(x)-{\psi }_i^L(x)]=0.$$

(35)

The proof is completed. □

By Theorem 3, we can establish the linear programming relaxation problem (LPRP) of the QPWQC over X as follows:

$$(\mathrm{LPRP}):\{\begin{array}{l}\min {\psi }_0^L(x)={\displaystyle \sum _{k=1}^n(c_k^0{x}_k+{\underset{\_}{\phi }}_{kk}^0(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{\phi }}_{jk}^0(x),}}\\ s.t. {\psi }_i^L(x)={\displaystyle \sum _{k=1}^n(c_k^i{x}_k+{\underset{\_}{\phi }}_{kk}^i(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{\phi }}_{jk}^i(x)\le b_i, i=1,2,\dots ,m,}}\\ x\in X=\{x:l\le x\le u\}.\end{array}$$

(36)

From the construction process of the former linearizing method, it is obvious that for any given X, each feasible point of the QPWQC is also feasible to the LPRP, and the optimal value of the LPRP is less than or equal to that of QPWQC. Therefore, the LPRP offers a reliable lower bound for the optimal value of the QPWQC. Except for the above approach, Theorem 3 also ensures the global convergence of the proposed algorithm.

3. New Global Optimization Algorithm

In this section, based on the former LPRP, we shall present a new global optimization algorithm for solving the QPWQC. In this algorithm, there are the following several key operations: branching, bounding, and space reduction.

Firstly, we choose a simple branching operation, which is called an interval bisection method. For any selected box $X^{\prime }=[l^{\prime },u^{\prime }]\subseteq X^0.$ Let $\delta \in \arg \max \{u_i{}^{\prime }-l_i{}^{\prime }: i=1,2,\dots ,n\},$ subdivide $[{\underset{\_}{x}}_{\delta }{}^{\prime }-{\overline{x}}_{\delta }{}^{\prime }]$ into $[{\underset{\_}{x}}_{\delta }{}^{\prime },({\underset{\_}{x}}_{\delta }{}^{\prime }+{\overline{x}}_{\delta }{}^{\prime })/2]$ and $[({\underset{\_}{x}}_{\delta }{}^{\prime }+{\overline{x}}_{\delta }{}^{\prime })/2,{\overline{x}}_{\delta }{}^{\prime }],$ $X^{\prime }$ can be subdivide into ${X^{\prime }}^1$ and ${X^{\prime }}^2$. The selected branching operation is sufficient to ensure the global convergence of this algorithm.

Secondly, for each investigated sub-box $X\subseteq X^0,$ we must solve the LPRP, and set $L{B}_s=\min \{LB(X)|X\in {\mathsf{\Omega }}_s\},$ where ${\mathsf{\Omega }}_s$ is still not fathomed as a sub-box set. In order to update the upper bound, we need to fathom the feasible point, and set $\mathsf{\Theta }$ be the known feasible point set and $U{B}_s=\min \{{\psi }_0(x)|x\in \mathsf{\Theta }\},$ to be the existent best upper bound.

In addition, we can introduce an interval reduction operation from Theorem 3 [6] to improve the convergent speed of the proposed algorithm.

3.1. Steps for Global Optimization Algorithm

For any investigated box $X^s\subseteq X^0,$ let $LB(X^s)$ and $x^s=x(X^s)$ be the optimal value and optimal solution of the LPRP over $X^s$. Based on the branch-and-bound scheme and the former LPRP, a new global optimization algorithm is described as follows.

Algorithm Steps:

Step 1. Set $\epsilon ={10}^{-6}$, solve the (LPRP) over $X^0$ to obtain its optimal solution $x^0$ and the optimal value $LB(X^0)$, respectively.

Let the lower bound $L{B}_0=LB(X^0)$. If $x^0$ is feasible to the QPWQC, let the upper bound be $U{B}_0={\psi }_0(x^0)$, otherwise let the initial upper bound be $U{B}_0=+\infty$.

If $U{B}_0-L{B}_0\le \epsilon$, let the global $\epsilon –$ optimal solution of the QPWQC be $x^0$, otherwise let ${\mathsf{\Omega }}_0=\{X^0\}, \mathsf{\Lambda }=\varphi , s=1.$

Step 2. Let the upper bound be $U{B}_s=U{B}_{s-1}$, partition $X^{s-1}$ into $X^{s,1}$ and $X^{s,2}$, and let $\mathsf{\Lambda }=\mathsf{\Lambda }\cup \{X^{s-1}\}$ be the deleted sub-boxes set.

For each $X^{s,t},t=1,2,$ utilize the interval reduction method to compress the investigated box, and let $X^{s,t}$ be the remaining box.

For each remaining box $X^{s,t},t=1,2,$ solve the LPRP to obtain its optimal solution $x^{s,t}$ and optimal value $LB(X^{s,t})$, respectively.

Set ${\mathsf{\Omega }}_s=\{X|X\in {\mathsf{\Omega }}_{s-1}\cup \{X^{s,1},X^{s,2}\},X\notin \mathsf{\Lambda }\}$ and $L{B}_s=\min \{LB(X)|X\in {\mathsf{\Omega }}_s\}.$

Step 3. For each $X^{s,t},t=1,2,$ if $x^{mid}$ is the feasible point of the QPWQC, let $\mathsf{\Theta }:=\mathsf{\Theta }\cup \{x^{mid}\}$, and let the new upper bound $U{B}_s=\underset{x\in \mathsf{\Theta }}{\min }\{{\psi }_0(x)\}$; if $x^{s,t}$ is feasible to the QPWQC, let the new upper bound $U{B}_s=\min \{U{B}_s,{\psi }_0(x^{s,t})\}$, and let the best known feasible point be $x^s$, which satisfies $U{B}_s={\psi }_0(x^s).$

Step 4. If $U{B}_s-L{B}_s\le \epsilon$, then we let the $\epsilon –$global optimal solution of the QPWQC be $x^s$, otherwise let $s=s+1$, and return to Step 2.

3.2. Global Convergence of the Proposed Algorithm

If the proposed algorithm terminates after finite iterations, then, when it terminates, we can obtain the global optimal solution of the QPWQC. Otherwise, the proposed algorithm will generate an infinite sequence, whose limitation is the global optimal solution of the QPWQC; the detailed proof is given as follows.

Theorem 4.

If the proposed algorithm does not terminate after finite iterations, then the proposed algorithm will generate an infinite sequence $\{X^s\}$, whose accumulation point will be the global optimal solution of the QPWQC.

Proof. 

First of all, in the proposed algorithm, the selected branching method is the rectangle bisection, which is exhaustive, and which guarantees that the intervals of all variables converge to 0.

Secondly, as $\Vert u-l\Vert \to 0$, from the conclusions of Theorem 3, it follows that the LPRP will sufficiently approximate the QPWQC, which is to say, $\underset{s\to \infty }{\lim }(U{B}_s-L{B}_s)=0$, i.e., the proposed algorithm satisfies that the bounding operation is consistent. Thirdly, in the proposed algorithm, the subdivided box which achieved the actual lower bound is immediately selected for the later partition, and the proposed algorithm satisfies that the selected operational bound is improving. By Theorem 4.3 of Reference [39], the proposed branch-and-bound algorithm satisfies the global convergent sufficient condition. Hence, the proposed algorithm converges to the global optimal solution of the QPWQC. □

4. Numerical Experiments

Let $\epsilon ={10}^{-6}$ be the convergence error. Some numerical examples in recent literature are solved in C++ program on microcomputer, and the simplex approach is employed to solve the LPRP. Compared with the existent algorithms, these numerical examples are given as follows, and their computational results are listed in

Table 1. Numerical comparisons for Examples 1–7.

Example	Refs.	Optimal Value	Optimal Solution	Iteration	Time (s)
1	ours	1.177124990	(1.177124344, 2.177124344)	22	0.0091
	[40]	1.177124327	(1.177124327, 2.177124353)	434	1.0000
2	ours	−0.999999202	(2.000000, 1.000000)	22	0.0085
	[40]	−1.0	(2.000000, 1.000000)	24	0.0129
3	ours	6.777809491	(2.000000000, 1.666676181)	13	0.0038
	[37]	6.777778340	(2.000000000, 1.666666667)	30	0.0068
	[41]	6.777782016	(2.000000000, 1.666666667)	40	0.0320
	[42]	6.7780	(2.00003, 1.66665)	44	0.1800
4	ours	0.500000600	(0.500000000, 0.500000000)	26	0.0061
	[41]	0.500004627	(0.5 0.5)	34	0.0560
	[42]	0.5	(0.5, 0.5)	91	0.8500
	[43]	0.500000442	(0.500000000, 0.500000000)	37	0.0193
	[44]	0.5	(0.5, 0.5)	96	1.0000
5	ours	118.381493268	(2.564162744, 3.119857633)	70	0.0435
	[45]	118.383756475	(2.5557793695, 3.1301646393)	210	0.7800
6	ours	−1.162882315	(1.499977112, 1.5)	37	0.0412
	[46]	−1.16288	(1.5, 1.5)	84	0.1257
7	ours	−11.363635682	(1.0,0.181818133, 0.983332175)	229	0.3919
	[43]	−11.363636364	(1.0,0.181818470, 0.983332113)	420	0.2845
	[26]	−10.35	(0.998712, 0.196213, 0.979216)	1648	0.3438

and

Table 2. Computational results for Example 8.

(n,m)	Algorithm of [47]	This Paper
	Computational Time (s)	Computational Time (s)
(4, 6)	2.37678	1.9894
(5, 11)	6.39897	4.9867
(14, 6)	9.22732	6.4567
(18, 7)	15.8410	11.6856
(20, 5)	11.9538	8.9802
(35, 10)	74.8853	56.7866
(37, 9)	77.1476	45.6324
(45, 8)	86.7174	65.6845
(46, 5)	44.2502	32.2150
(60, 11)	315.659	216.534

.

Example 1 (Reference [40])

$$(\mathrm{LPRP}):\{\begin{array}{l}\min {\psi }_0^L(x)={\displaystyle \sum _{k=1}^n(c_k^0{x}_k+{\underset{\_}{\phi }}_{kk}^0(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{\phi }}_{jk}^0(x),}}\\ s.t. {\psi }_i^L(x)={\displaystyle \sum _{k=1}^n(c_k^i{x}_k+{\underset{\_}{\phi }}_{kk}^i(x))}+{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1,k\ne j}^n{\underset{\_}{\phi }}_{jk}^i(x)\le b_i, i=1,2,\dots ,m,}}\\ x\in X=\{x:l\le x\le u\}.\end{array}$$

$$\{\begin{array}{l}\min {\psi }_0(x)=x_1\\ \hspace{1em}s.t. {\psi }_1(x)=\frac{1}{4}{x}_1+\frac{1}{2}{x}_2-\frac{1}{16}{x}_1^2-\frac{1}{16}{x}_2^2\le 1,\\ \hspace{1em}\hspace{1em}{\psi }_2(x)=-\frac{3}{7}{x}_1-\frac{3}{7}{x}_2+\frac{1}{14}{x}_1^2+\frac{1}{14}{x}_2^2\le -1,\\ \hspace{1em}\hspace{1em}1\le x_1\le 5.5, 1\le x_2\le 5.5,\end{array}$$

Example 2 (Reference [40])

$$\{\begin{array}{l}\min {\psi }_0(x)=x_1{x}_2-2x_1+x_2+1\\ s.t. {\psi }_1(x)=-6x_1-16x_2+8x_2^2\le -11,\\ \hspace{1em}{\psi }_2(x)=3x_1+2x_2-x_2^2\le 7,\\ \hspace{1em}1\le x_1\le 2.5, 1\le x_2\le 2.225.\end{array}$$

Example 3 (References [37,41,42])

$$\{\begin{array}{l}\min {\psi }_0(x)=x_1^2+x_2^2\\ s.t. {\psi }_1(x)=0.3x_1{x}_2\ge 1,\\ \hspace{1em}2\le x_1\le 5, 1\le x_2\le 3.\end{array}$$

Example 4 (References [41,42,43,44])

$$\{\begin{array}{l}\min {\psi }_0(x)=x_1\\ s.t. {\psi }_1(x)=4x_2-4x_1^2\le 1,\\ \hspace{1em}{\psi }_2(x)=-x_1-x_2\le -1,\\ \hspace{1em}0.01\le x_1,x_2\le 15.\end{array}$$

Example 5 (Reference [45])

$$\{\begin{array}{l}\min {\psi }_0(x)=6x_1^2+4x_2^2+5x_1{x}_2\\ s.t. {\psi }_1(x)=-6x_1{x}_2\le -48,\\ \hspace{1em}0\le x_1,x_2\le 10.\end{array}$$

Example 6 (Reference [46])

$$\{\begin{array}{l}\min {\psi }_0(x)=-x_1+x_1{x}_2^{0.5}-x_2\\ s.t. {\psi }_1(x)=-6x_1+8x_2\le 3,\\ \hspace{1em}{\psi }_2(x)=3x_1-x_2\le 3,\\ \hspace{1em}1\le x_1,x_2\le 1.5.\end{array}$$

Example 7 (References [26,43])

$$\{\begin{array}{l}\min {\psi }_0(x)=4x_2+(x_1-1{)}^2+x_2-10x_3^2\\ s.t. {\psi }_1(x)=x_1^2+x_2^2+x_3^2\le 2,\\ \hspace{1em}{\psi }_2(x)=(x_1-2{)}^2+x_2^2+x_3^2\le 2,\\ \hspace{1em}2-\sqrt{2}\le x_1\le \sqrt{2}, 0\le x_2,x_3\le \sqrt{2}.\end{array}$$

Comparing with the existent algorithms, numerical results show that the proposed algorithm has the higher computational efficiency.

To demonstrate robustness of the proposed algorithm, we give a large-scale random numerical example as follows.

Example 8. (Reference [47])

$$\{\begin{array}{l}\min {\psi }_0(x)={\displaystyle \sum _{k=1}^n{c}_k^o{x}_k+}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^n{d}_{ij}^0{x}_j{x}_k}}\\ s.t. {\psi }_i(x)={\displaystyle \sum _{k=1}^n{c}_k^i{x}_k+}{\displaystyle \sum _{j=1}^n{\displaystyle \sum _{k=1}^n{d}_{ij}^i{x}_j{x}_k}\le b_i, i=1,2,\dots ,m,}\\ \hspace{1em}x\in X^0=\{ x\in R^n: l^0\le x\le u^0\},\end{array}$$

where $c_k^0,k=1,2,\dots ,n,$ is randomly generated in [0, 1], $d_{kj}^0,k=1,2,\dots ,n,j=1,2,\dots ,n,$ is randomly generated in [0, 1]; $c_k^i,i=1,\dots ,m,k=1,2,\dots ,n,$ is randomly generated in [−1, 0], $d_{kj}^i,k=1,2,\dots ,n,j=1,2,\dots ,n,$ is randomly generated in [−1, 0], $b_i,i=1,2,\dots ,m,$ is randomly generated in [−300, −90]. In the Example 8, ‘n’ denotes the number of variables while ‘m’ denotes the number of constraints. Numerical results about the Example 8 are given in the Table 2.

5. Concluding Remarks

This paper presents an effective algorithm for globally solving quadratic programs with quadratic constraints. In this algorithm, a new linearization method is constructed for deriving the linear programming relaxation problem of the QPWQC. The proposed algorithm converges to the global optimal solution of the initial problem of QPWQC, and numerical experimental results show the higher computational efficiency of the proposed algorithm.