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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

## Volume 1 (2003)

Hongwei Jiao
• Corresponding author
• Postdoctoral Research Base, Henan Institute of Science and Technology, Xinxiang 453003, China
• Postdoctoral Research Station of Control Science and Engineering, Henan University of Science and Technology, Luoyang 471023, China
• Email
• Other articles by this author:
/ Rongjiang Chen
Published Online: 2018-04-20 | DOI: https://doi.org/10.1515/math-2018-0037

## Abstract

In this paper we propose a new parametric linearizing approach for globally solving quadratically inequality constrained quadratic programs. By utilizing this approach, we can derive the parametric linear programs relaxation problem of the investigated problem. To accelerate the computational speed of the proposed algorithm, an interval deleting rule is used to reduce the investigated box. The proposed algorithm is convergent to the global optima of the initial problem by subsequently partitioning the initial box and solving a sequence of parametric linear programs relaxation problems. Finally, compared with some existing algorithms, numerical results show higher computational efficiency of the proposed algorithm.

MSC 2010: 90C20; 90C26; 65K05

## 1 Introduction

In this paper we consider the following quadratically inequality constrained quadratic programs: $(QICQP):minH0(z)=∑k=1ndk0zk+∑j=1n∑k=1npjk0zjzks.t.Hi(z)=∑k=1ndkizk+∑j=1n∑k=1npjkizjzk≤bi,i=1,…,m,z∈Z0={z∈Rn:l0≤z≤u0},$

where $\begin{array}{}{p}_{jk}^{i},{d}_{k}^{i}\end{array}$ and bi are all arbitrary real numbers; $\begin{array}{}{l}^{0}=\left({l}_{1}^{0},\dots ,{l}_{n}^{0}{\right)}^{T},{u}^{0}=\left({u}_{1}^{0},\dots ,{u}_{n}^{0}{\right)}^{T}.\end{array}$ The investigated problem (QICQP) has a broad applications in investment portfolio, management decision, route optimization, engineering optimization, production planning and so on. In addition, the investigated problem (QICQP) usually owns multiple local optima which are not global optima, that is to say, in this kind of problems there are important theoretical and computational complexities. Therefore, it is very necessary to present an effective global optimization algorithm for solving the (QICQP).

In last decades, for the problem (QICQP) and its special cases many methods have been developed and described in the existent literature. For example, semi-definite relaxation method [1], reformulation-convexification approach [2], branch-and-reduce approaches [3, 4, 5, 6, 7], approximation algorithms [8, 9, 10], simplicial branch-and-bound method [11], branch-and-cut method [12], rectangle branch-and-bound algorithms [13,14], robust solution approach [15], and so on. In addition, some algorithms for geometric programming [16, 17, 18, 19, 20] and multiplicative (or fractional) programming [21, 22, 23, 24, 25] also can be used to solve the (QICQP). Although these methods can be used to solve the investigated problem (QICQP) or its special forms, less work has been still done for globally solving the investigated quadratically inequality constrained quadratic programs.

In this paper, we will present a new branch-and-bound algorithm for globally solving the (QICQP). Firstly, we present a new parametric linearizing technique. By utilizing this method, we can convert the (QICQP) into a parametric linear programs relaxation problem, which can be used to compute the lower bounds of the optimal values of the initial problem (QICQP) and its subproblems. Secondly, based on the branch-and-bound framework, by successive partitioning of the initial box and by solving those derived parametric linear programs relaxation problems, a new branch-and-bound algorithm is designed for globally solving the (QICQP). Thirdly, to accelerate the computational efficiency of the proposed branch-and-bound algorithm, an interval deleting rule is used to reduce the investigated box. Fourthly, the proposed algorithm is convergent to the global optima of the initial problem (QICQP) by successively partitioning of the initial box and by solving those derived parametric linear programs relaxation problems. Finally, compared with some existent algorithms, numerical results demonstrate the computational efficiency of the proposed algorithm.

The remaining sections of this article are organized as follows. First of all, we present a new parametric linearizing technique for deriving the parametric linear programs relaxation problem of the (QICQP) in Section 2. Secondly, based on the branch-and-bound framework, by combing the derived parametric linear programs relaxation problem with the interval deleting rule, a branch-and-bound algorithm is established for globally solving the (QICQP) in Section 3. Thirdly, compared with the existent methods, some numerical examples in existent literatures are used to verify the computational efficiency of the proposed algorithm in Section 4. Finally, some concluding remarks are presented.

## 2 New parametric linearizing approach

In this section, we propose a new parametric linearizing approach for deriving the parametric linear programs relaxation problem of the (QICQP). The detailed parametric linearizing approach is presented as follows:

Assume that Z = {(z1, z2, …, zn)TRn : ljzjuj, j = 1, …, n} ⊆ Z0, λ = (λjk)n × nRn × n is a symmetric matrix, and λjk ∈ {0, 1}. For convenience, for any zZ, for any k ∈ {1, 2, …, n}, some expressions are introduced as follows: $zk(λkk)=lk+λkk(uk−lk),zk(1−λkk)=lk+(1−λkk)(uk−lk),hkk(z)=zk2,h_kk(z,Z,λkk)=[zk(λkk)]2+2zk(λkk)[zk−zk(λkk)],h¯kk(z,Z,λkk)=[zk(λkk)]2+2zk(1−λkk)[zk−zk(λkk)].$

Obviously, we have zk(0) = lk, zk(1) = uk.

#### Theorem 2.1

For any k ∈ {1, 2, …, n}, for any zZ, consider the functions hkk(z), hkk(z, Z, λkk) and hkk(z, Z, λkk), then, the following conclusions hold: $h_kk(z,Z,λkk)≤hkk(z)≤h¯kk(z,Z,λkk);$(1) $lim∥u−l∥→0[hkk(z)−h_kk(z,Z,λkk)]=0$(2)

and $lim∥u−l∥→0[h¯kk(z,Z,λkk)−hkk(z)]=0.$(3)

#### Proof

1. By the mean value theorem, for any zZ, there exists a point ξk = α zk+(1 − α)zkkk), where α ∈ [0, 1], such that $zk2=[zk(λkk)]2+2ξk[zk−zk(λkk)].$

If λkk = 0, then we have $ξk≥lk=zk(λkk)andzk−zk(λkk)=zk−lk≥0.$

If λkk = 1, then it follows that $ξk≤uk=zk(λkk)andzk−zk(λkk)=zk−uk≤0.$

Thus, we can get that $hkk(z)=zk2=[zk(λkk)]2+2ξk[zk−zk(λkk)]≥[zk(λkk)]2+2zk(λkk)[zk−zk(λkk)]=h_kk(z,Z,λkk).$

Similarly, if λkk = 0, then we have $ξk≤uk=zk(1−λkk)andzk−zk(λkk)=zk−lk≥0.$

If λkk = 1, then it follows that $ξk≥lk=zk(1−λkk)andzk−zk(λkk)=zk−uk≤0.$

Thus, we can get that $hkk(z)=zk2=[zk(λkk)]2+2ξk[zk−zk(λkk)]≤[zk(λkk)]2+2zk(1−λkk)[zk−zk(λkk)]=h¯kk(z,Z,λkk).$

Therefore, for any zZ, we have that $h_kk(z,Z,λkk)≤hkk(z)≤h¯kk(z,Z,λkk).$

2. Since $hkk(z)−h_kk(z,Z,λkk)=zk2−{[zk(λkk)]2+2zk(λkk)[zk−zk(λkk)]}=(zk−zk(λkk))2≤(uk−lk)2,$(4)

we have $lim∥u−l∥→0[hkk(z)−h_kk(z,Z,λkk)]=0.$

Also since $h¯kk(z,Z,λkk)−hkk(z)=[zk(λkk)]2+2zk(1−λkk)[zk−zk(λkk)]−zk2=(zk(λkk)+zk)(zk(λkk)−zk)+2zk(1−λkk)(zk−zk(λkk))=[zk−zk(λkk)][2zk(1−λkk)−zk(λkk)−zk]=[zk−zk(λkk)][zk(1−λkk)−zk(λkk)]+[zk−zk(λkk)][zk(1−λkk)−zk]≤2(uk−lk)2.$(5)

Therefore, it follows that $lim∥u−l∥→0[h¯kk(z,Z,λkk)−hkk(z)]=0.$

The proof is completed. □

Without loss of generality, for any zZ, for any j ∈ {1, 2, …, n}, k ∈ {1, 2, …, n}, jk, we define $zj(λjk)=lj+λjk(uj−lj),zk(λjk)=lk+λjk(uk−lk),zj(1−λjk)=lj+(1−λjk)(uj−lj),zk(1−λjk)=lk+(1−λjk)(uk−lk),(zj−zk)(λjk)=(lj−uk)+λjk(uj−lk−lj+uk),(zj−zk)(1−λjk)=(lj−uk)+(1−λjk)(uj−lk−lj+uk).$

Obviously, we have (zjzk)(0) = ljuk, (zjzk)(1) = ujlk.

In a similar way as in Theorem 2.1, we can get the following Theorem 2.2:

#### Theorem 2.2

For each j = 1, 2, …, n, k = 1, 2, …, n, for any zZ, we have:

1. The following inequalities hold: $[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]≤zj2≤[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)],zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]≤zk2≤[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)],(zj−zk)2≤[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)],(zj−zk)2≥[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)].$

2. The following limitations hold: $lim∥u−l∥→0[zj2−{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]}]=0,lim∥u−l∥→0[[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]−zj2]=0,lim∥u−l∥→0[zk2−{[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]}]=0,lim∥u−l∥→0[[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−zk2]=0,lim∥u−l∥→0[[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]−(zj−zk)2]=0,lim∥u−l∥→0[(zj−zk)2−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[(zj−zk)−(zj−zk)(λjk)]}]=0.$

#### Proof

1. From the inequality (1), replacing λkk by λjk, and replacing zk by zj, we can get that $[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]≤zj2≤[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)].$

From the inequality (1), replacing λkk by λjk, we can get that $[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]≤zk2≤[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)].$

From (1), replacing λkk and zk by λjk and (zjzk), respectively, we can get that $(zj−zk)2≤[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[(zj−zk)−(zj−zk)(λjk)],(zj−zk)2≥[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[(zj−zk)−(zj−zk)(λjk)].$

2. From the limitations (2) and (3), replacing λkk and zk by λjk and zj, we have $lim∥u−l∥→0[zj2−{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]}]=0$

and $lim∥u−l∥→0[[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]−zj2]=0.$

From the limitations (2) and (3), replacing λkk by λjk, it follows that $lim∥u−l∥→0[zk2−{[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]}]=0$

and $lim∥u−l∥→0[[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−zk2]=0.$

By the limitations (2) and (3), replacing λkk and zk by λjk and (zjzk), respectively, we can get that $lim∥u−l∥→0[[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]−(zj−zk)2]=0$

and $lim∥u−l∥→0[(zj−zk)2−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)]}]=0.$

The proof is completed. □

Without loss of generality, for any zZ, for any j ∈ {1, 2, …, n}, k ∈ {1, 2, …, n}, jk, define $hjk(z)=zjzk=zj2+zk2−(zj−zk)22,h_jk(z,Z,λjk)=12{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]}},h¯jk(z,Z,λjk)=12{[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)]}}.$

#### Theorem 2.3

For each k = 1, 2, …, n, consider the functions hjk(z, Z, λjk), hjk(z) and hjk(z, Z, λjk), then, for any zZ, we have the following conclusions: $h_jk(z,Z,λjk)≤hjk(z)≤h¯jk(z,Z,λjk),$(6) $lim∥u−l∥→0[hjk(z)−h_jk(z,Z,λjk)]=0$(7)

and $lim∥u−l∥→0[h¯jk(z,Z,λjk)−hjk(z)]=0.$(8)

#### Proof

First of all, from the conclusions (i) of Theorem 2.2, it follows that $hjk(z)=zjzk=zj2+zk2−(zj−zk)22≥12{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]}}=h_jk(z,Z,λjk)$

and $hjk(z)=zjzk=zj2+zk2−(zj−zk)22≤12{[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)]}}=h¯jk(z,Z,λjk).$

Secondly, from the inequalities (4) and (5), we have $hjk(z)−h_jk(z,Z,λjk)=zjzk−h_jk(z,Z,λjk)=zj2+zk2−(zj−zk)22−12{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]}},=12[zj2−{[zj(λjk)]2+2zj(λjk)[zj−zj(λjk)]}]+12[zk2−{[zk(λjk)]2+2zk(λjk)[zk−zk(λjk)]}]+12{{[(zj−zk)(λjk)]2+2(zj−zk)(1−λjk)[zj−zk−(zj−zk)(λjk)]}−(zj−zk)2}≤12(uj−lj)2+12(uk−lk)2+(uk+uj−lj−lk)2.$

Thus, we can get that limul∥ → 0[hjk(z) − hjk(z, Z, λjk)] = 0.

Also from the inequalities (4) and (5), we get that $h¯jk(z,Z,λjk)−hjk(z)=h¯jk(z,Z,λjk)−zjzk=12{[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]+[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)]}}−zj2+zk2−(zj−zk)22=12[[zj(λjk)]2+2zj(1−λjk)[zj−zj(λjk)]−zj2]+12[[zk(λjk)]2+2zk(1−λjk)[zk−zk(λjk)]−zk2]+12{(zj−zk)2−{[(zj−zk)(λjk)]2+2(zj−zk)(λjk)[zj−zk−(zj−zk)(λjk)]}}=(zj−zj(λjk))(zk−zk(λjk))≤(uj−lj)2+(uk−lk)2+(uk+uj−lk−lj)2.$

Thus, it follows that limul∥ → 0[hjk(z, Z, λjk) − hjk(z)] = 0. □

Without loss of generality, for any Z = [l, u] ⊆ Z0, for any parameter matrix λ = (λjk)n × n, for any zZ and i ∈ {0, 1, …, m}, we let $f_kki(z,Z,λkk)=pkkih_kk(z,Z,λkk),ifpkki>0,pkkih¯kk(z,Z,λkk),ifpkki<0,f¯kki(z,Z,λkk)=pkkih¯kk(z,Z,λkk),ifpkki>0,pkkih_kk(z,Z,λkk),ifpkki<0,f_jki(z,Z,λjk)=pjkih_jk(z,Z,λjk),ifpjki>0,j≠k,pjkih¯jk(z,Z,λjk),ifpjki<0,j≠k,f¯jki(z,Z,λjk)=pjkih¯jk(z,Z,λjk),ifpjki>0,j≠k,pjkih_jk(z,Z,λjk),ifpjki<0,j≠k.HiL(z,Z,λ)=∑k=1n(dkizk+f_kki(z,Z,λkk))+∑j=1n∑k=1,k≠jnf_jki(z,Z,λjk).HiU(z,Z,λ)=∑k=1n(dkizk+f¯kki(z,Z,λkk))+∑j=1n∑k=1,k≠jnf¯jki(z,Z,λjk).$

#### Theorem 2.4

ForzZ = [l, u] ⊆ Z0, for any given parameter matrix λ = (λjk)n × n, for each i = 0, 1, …, m, we have the following conclusions: $HiL(z,Z,λ)≤Hi(z)≤HiU(z,Z,λ),lim∥u−l∥→0[Hi(z)−HiL(z,Z,λ)]=0$

and $lim∥u−l∥→0[HiU(z,Z,λ)−Hi(z)]=0.$

#### Proof

First of all, from the inequalities (1) and (6), for any j, k ∈ {1, …, n}, we have $f_kki(z,Z,λkk)≤pkkizk2≤f¯kki(z,Z,λkk),$(9) $f_jki(z,Z,λjk)≤pjkizjzk≤f¯jki(z,Z,λjk).$(10)

By the above inequalities (9) and (10), for any zZZ0, we can get that $HiL(z,Z,λ)=∑k=1n(dkizk+f_kki(z,Z,λkk))+∑j=1n∑k=1,k≠jnf_jki(z,Z,λjk)≤∑k=1ndkizk+∑k=1npkkizk2+∑j=1n∑k=1,k≠jnpjkizjzk=Hi(z)≤∑k=1n(dkizk+f¯kki(z,Z,λkk))+∑j=1n∑k=1,k≠jnf¯jki(z,Z,λjk)=HiU(z,Z,λ).$

Therefore, we have $\begin{array}{}{H}_{i}^{L}\left(z,Z,\lambda \right)\le {H}_{i}\left(z\right)\le {H}_{i}^{U}\left(z,Z,\lambda \right).\end{array}$

Secondly, $Hi(z)−HiL(z,Z,λ)=∑k=1ndkizk+∑k=1npkkizk2+∑j=1n∑k=1,k≠jnpjkizjzk−[∑k=1ndkizk+∑k=1nf_kki(z,Z,λkk)+∑j=1n∑k=1,k≠jnf_jki(z,Z,λjk)]=∑k=1n[pkkizk2−f_kki(z,Z,λkk)]+∑j=1n∑k=1,k≠jn[pjkizjzk−f_jki(z,Z,λjk)]=∑k=1,pkki>0npkki[hkk(z)−h_kk(z,Z,λkk)]+∑k=1,pkki<0npkki[hkk(z)−h¯kk(z,Z,λkk)]+∑j=1n∑k=1,k≠j,pjki>0npjki[hjk(z)−h_jk(z,Z,λjk)]+∑j=1n∑k=1,k≠j,pjki<0npjki[hjk(z)−h¯jk(z,Z,λjk)].$

From the limitations (2), (3), (7) and (8), limul∥ → 0[hkk(z) − hkk(z, Z, λkk)] = 0, limul∥ → 0[hkk(z, Z, λkk) − hkk(z)] = 0, limul∥ → 0[hjk(z) − hjk(z, Z, λjk)] = 0 and limul∥ → 0[hjk(z, Z, λjk) − hjk(z)] = 0.

Therefore, it follows that $lim∥u−l∥→0[Hi(z)−HiL(z,Z,λ)]=0.$

Similarly, we can prove that $lim∥u−l∥→0[HiU(z,Z,λ)−Hi(z)]=0.$

The proof is completed. □

By Theorem 2.4, we can construct the parametric linear programs relaxation problem (PLPRP) of the (QICQP) over Z as follows: $(PLPRP):minH0L(z,Z,λ),s.t.HiL(z,Z,λ)≤bi,i=1,…,m,z∈Z={z:l≤z≤u}.$

where $HiL(z,Z,λ)=∑k=1n(dkizk+f_kki(z,Z,λkk))+∑j=1n∑k=1,k≠jnf_jki(z,Z,λjk).$

Based on the former parametric linearizing technique, each feasible solution of the (QICQP) must be also feasible to the (PLPRP) in the sub-region Z; and the minimum value of the (PLPRP) must be less than or equal to that of the (QICQP) in the sub-region Z. Hence, the (PLPRP) offers a reliable lower bound for the minimum value of the (QICQP) in the sub-region Z. In addition, Theorem 2.4 ensures that the optimal solution of the (PLPRP) will sufficiently approximate the optimal solution of the (QICQP) as ∥ul∥ → 0, and this guarantees the global convergence of the proposed algorithm.

## 3 Branch-and-bound global optimization algorithm

In this section, a new branch-and-bound global optimization algorithm is proposed for solving the (QICQP). In this algorithm, there are the following several important techniques: branching, bounding the lower bound, bounding upper bound and interval deleting.

Branching: The branching step will generate a more refined box partition. Here we choose a typical box-bisection method, which is sufficient to ensure the global convergence of the proposed branch-and-bound method. For any selected box Z = [l, u] ⊆ Z0. Set η ∈ arg max $\begin{array}{}\left\{{u}_{i}^{{}^{\prime }}-{l}_{i}^{{}^{\prime }}:i=1,2,\dots ,n\right\},\end{array}$ by partitioning $\begin{array}{}\left[{\underset{_}{z}}_{\eta }^{{}^{\prime }},{\overline{z}}_{\eta }^{{}^{\prime }}\right]\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{into}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left[{\underset{_}{z}}_{\eta }^{{}^{\prime }},\left({\underset{_}{z}}_{\eta }^{{}^{\prime }}+{\overline{z}}_{\eta }^{{}^{\prime }}\right)/2\right]\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left[\left({\underset{_}{z}}_{\eta }^{{}^{\prime }}+{\overline{z}}_{\eta }^{{}^{\prime }}\right)/2,{\overline{z}}_{\eta }^{{}^{\prime }}\right],\end{array}$ we can subdivide Z into two new sub-boxes Z′1 and Z′2.

Bounding the lower bound: For each sub-box ZZ0, which has not been deleted, the bounding the lower bound step needs to solve the parametric linear programs relaxation problem over each sub-box, and denote by LBs = min{LB(Z)|ZΩs}, where Ωs denotes the set of sub-box which has not been deleted after s iteration.

Bounding the upper bound: The bounding upper bound step needs to judge the feasibility of the midpoint of each investigated sub-box Z and the optimal solution of the (PLPRP) over the investigated sub-box Z, where ZΩs. In addition, we need to calculate the objective function values of each known feasible solutions for the (QICQP), and denote by UBs = min{H0(z) : zΘ} the best upper bound, where Θ is the known feasible point set.

Interval deleting: To improve the convergent speed of the branch-and-bound algorithm, an interval deleting rule is introduced as follows. For convenience, for any zZ, i ∈ {0,1, …, m}, q ∈ {1, …, n}, and denote by HUB the current upper bound of the (QICQP), we let

$HiL(z,Z,λ)=∑j=1nαij(λ)zj+βi(λ),RLBi(λ)=∑j=1nmin{αij(λ)lj,αij(λ)uj}+βi(λ).$

#### Theorem 3.1

For any investigated sub-box Z = (Zj)nZ0, we have the following conclusions:

1. If RLB0(λ) > HUB, then the whole investigated sub-box Z should be deleted.

2. If RLB0(λ) ≤ HUB, then: for any q ∈ {1,2, …, n}, if α0q(λ) > 0, the interval Zq should be replaced by $\begin{array}{}\left[{l}_{q},\frac{HUB-RL{B}_{0}\left(\lambda \right)+min\left\{{\alpha }_{0q}\left(\lambda \right){l}_{q},{\alpha }_{0q}\left(\lambda \right){u}_{q}\right\}}{{\alpha }_{0q}\left(\lambda \right)}\right]\bigcap {Z}_{q};\end{array}$ if α0q(λ) < 0, the interval Zq should be replaced by $\begin{array}{}\left[\frac{HUB-RL{B}_{0}\left(\lambda \right)+min\left\{{\alpha }_{0q}\left(\lambda \right){l}_{q},{\alpha }_{0q}\left(\lambda \right){u}_{q}\right\}}{{\alpha }_{0q}\left(\lambda \right)},{u}_{q}\right]\bigcap {Z}_{q}.\end{array}$

3. If RLBi(λ) > bi for some i ∈ {1, …, m}, then the whole investigated sub-box Z should be deleted.

4. If RLBi(λ) ≤ bi for each i ∈ {1, …, m}, then, for any q ∈ {1,2, …, n}, if αiq(λ) > 0, the interval Zq can be replaced by $\begin{array}{}\left[{l}_{q},\frac{{b}_{i}-RL{B}_{i}\left(\lambda \right)+min\left\{{\alpha }_{iq}\left(\lambda \right){l}_{q},{\alpha }_{iq}\left(\lambda \right){u}_{q}\right\}}{{\alpha }_{iq}\left(\lambda \right)}\right]\bigcap {Z}_{q};\end{array}$ if αiq(λ) < 0, the interval Zq can be replaced by $\begin{array}{}\left[\frac{{b}_{i}-RL{B}_{i}\left(\lambda \right)+min\left\{{\alpha }_{iq}\left(\lambda \right){l}_{q},{\alpha }_{iq}\left(\lambda \right){u}_{q}\right\}}{{\alpha }_{iq}\left(\lambda \right)},{u}_{q}\right]\bigcap {Z}_{q}.\end{array}$

#### Proof

In a similar way as in the proof of Theorem 3 in [14], we may draw the conclusions for Theorem 3.1, so here it is omitted.

From Theorem 3.1, we can construct an interval deleting step to compress the investigated box for improving the convergent speed of the proposed branch-and-bound algorithm.

## 3.1 New branch-and-bound algorithm

For any sub-box ZsZ0, we denote by LB(Zs) the optimal value of the (PLPRP) over the sub-box Zs, and denote by zs = z(Zs) the optimal solution of the (PLPRP) over the sub-box Zs. Based on the branch-and-bound framework, combining the former branching step, bounding the lower bound step, bounding upper bound step and interval deleting step together, a new branch-and-bound algorithm is designed as follows.

Branch-and-Bound Algorithm Steps

Initializing step. Given the initial convergent error ϵ, the initial randomly generated parameter matrix λ.

Solve the (PLPRP) over the initial box Z0 to obtain its optimal solution z0 and optimal value LB(Z0), denote by the initial lower bound LB0 = LB(Z0). If z0 is a feasible solution of the (QICQP), we denote by the initial upper bound UB0 = H0(z0). Otherwise, we denote by the initial upper bound UB0 = + ∞.

If UB0LB0ϵ, the proposed algorithm terminates, z0 is a global ϵ-optimal solution of the initial problem (QICQP). Otherwise, set Ω0 = {Z0}, Λ = ∅, s = 1.

Branching step. Let UBs = UBs−1. Partition the investigated sub-box Zs−1 into two sub-boxes Zs,1, Zs,2 by the selected branching rule, and denote by Λ = Λ ∪ {Zs−1} the set of the deleting sub-boxes.

Interval deleting step. For each investigated sub-box Zs,t, t = 1, 2, use the former interval deleting rule to compress the investigated sub-box, still denote by Zs,t the remaining sub-box.

Bounding the lower bound step. For each remaining sub-box Zs,t, where t = 1, 2, solve the (PLPRP) over Zs,t to obtain its optimal solution zs,t and optimal value LB(Zs,t), and let Ωs = {Z|ZΩs−1 ∪ {Zs,1, Zs,2}, ZΛ} and LBs = min{LB(Z)|ZΩs}.

Bounding the upper bound step. For each sub-box Zs,t, if its midpoint zmid is the feasible point of the initial problem (QICQP), let Θ := Θ ∪ {zmid}, denote by the new upper bound UBs = minzΘ H0(z); if the optimal solution zs,t of the (PLPRP) is the feasible point of the initial problem (QICQP), denote by the new upper bound UBs = min{UBs, H0(zs,t)}, and denote by zs the best existent feasible point such that UBs = H0(zs).

Terminating judgement step. If UBsLBsϵ, the proposed algorithm terminates, zs is a global ϵ-optimal solution of the initial problem (QICQP). Otherwise, denote by s = s + 1, and go to the Branching step.

## 3.2 Global convergence of the proposed algorithm

Without loss of generality, we assume that v is the global optimal value of the initial problem (QICQP). If the proposed algorithm terminates after s finite iterations, where s is a finite number such that s ≥ 0, then it follows that

$UBs≤LBs+ϵ.$

From the bounding the upper bound step of the proposed algorithm, we know that there must exist a feasible point zs of the initial problem (QICQP) such that

$v≤UBs=H0(zs).$

By the branch-and-bound structure of the proposed algorithm, we have

$LBk≤v.$

Combining the above several inequalities together, it follows that

$v≤UBs=H0(zs)≤LBs+ϵ≤v+ϵ.$

Therefore, zs is an ϵ-global optimal solution of the initial problem (QICQP).

If the proposed algorithm does not terminate after finite iterations, for this case, the detailed convergent conclusions are given as follows.

#### Theorem 3.2

If the proposed algorithm does not terminate after finite iterations, then it will generate an infinite partitioning sequence {Zs} of the initial box Z0, and any accumulation point of the sequence {Zs} will be a global optimum solution of the initial problem (QICQP).

#### Proof

First of all, in the proposed algorithm the selected branching method is the bisection of box, so that the branching process is exhaustive, that is to say, the branching step will ensure that the intervals of all variables tend to 0, i.e., ‖ul‖ → 0.

Secondly, from Theorem 2.4, the optimal solution of the (PLPRP) will sufficiently approximate the optimal solution of the (QICQP) as ‖ul‖ → 0, and this ensures that the limitation lims→∞(UBsLBs) = 0 holds. So that the bounding operation is consistent.

Thirdly, in the proposed algorithm the subdivided box which attains the actual lower bound is selected for further partition at the later immediate iteration, so that the used selecting operation is bound improving.

From [26, Theorem IV.3], the sufficient condition of global convergence of the branch-and-bound algorithm is that the branching method is exhaustive, the bounding method is consistent and the selecting method is improvement, therefore, the proposed algorithm is convergent to the global optimal solution of the initial (QICQP). □

## 4 Numerical experiments

Given the convergent error ϵ = 10−6 and the parameter matrix λ = (λjk)n×nRn×n, where λjk ∈ {0, 1}, compared with the existing methods, several numerical examples in existing literature are tested on microcomputer, the procedure is coded in C++ software, the parametric linear programs relaxation problems are solved by the simplex method. These examples and their numerical results are listed as follows. In the following Tables 1 and 2, the number of iteration and running time in seconds for the algorithm are represented by “Iteration” and “Time(s)”, respectively.

Table 1

Numerical comparisons for Examples 4.1-4.7

Table 2

Numerical results for Example 4.8

#### Example 4.1

([16]).

${minH0(z)=z1s.t.H1(z)=14z1+12z2−116z12−116z22≤1,H2(z)=114z12+114z22−37z1−37z2≤−1,1≤z1≤5.5,1≤z2≤5.5.$

#### Example 4.2

([16]).

${minH0(z)=z1z2−2z1+z2+1s.t.H1(z)=8z22−6z1−16z2≤−11,H2(z)=−z22+3z1+2z2≤7,1≤z1≤2.5,1≤z2≤2.225.$

#### Example 4.3

([4,5,17]).

${minH0(z)=z12+z22s.t.H1(z)=0.3z1z2≥1,2≤z1≤5,1≤z2≤3.$

#### Example 4.4

([5,14,17,18]).

${minH0(z)=z1s.t.H1(z)=4z2−4z12≤1,H2(z)=−z1−z2≤−1,0.01≤z1,z2≤15.$

#### Example 4.5

([4,6,14]).

${minH0(z)=6z12+4z22+5z1z2s.t.H1(z)=−6z1z2≤−48,0≤z1,z2≤10.$

#### Example 4.6

([19]).

${minH0(z)=−z1+z1z20.5−z2s.t.H1(z)=−6z1+8z2≤3,H2(z)=3z1−z2≤3,1≤z1,z2≤1.5.$

#### Example 4.7

([14,20]).

${minH0(z)=−4z2+(z1−1)2+z22−10z32s.t.H1(z)=z12+z22+z32≤2,H2(z)=(z1−2)2+z22+z32≤2,2−2≤z1≤2,0≤z2,z3≤2.$

#### Example 4.8

([3,14]).

${maxH0(z)=∑i=1nzi2s.t.Hj(z)=∑i=1jzi≤j,j=1,2,…,n,zi≥0,i=1,2,…,n.$

Compared with the existing algorithms, the numerical results for examples 1-8 show that the proposed algorithm can be used to globally solve the quadratically inequality constrained quadratic programs with higher computational efficiency.

## 5 Concluding remarks

In this paper, we propose a new branch-and-bound algorithm for globally solving the quadratically inequality constrained quadratic programs. In this algorithm, we present a new parametric linearizing technique, which can be used to derive the parametric linear programs relaxation problem of the investigated problem (QICQP). To accelerate the computational speed of the proposed branch-and-bound algorithm, an interval deleting rule is used to reduce the investigated box. By subsequently partitioning the initial box and solving a sequence of parametric linear programs relaxation problems, the proposed algorithm is convergent to the global optima of the initial problem (QICQP). Finally, compared with some existing algorithms, numerical results show higher computational efficiency of the proposed algorithm.

## Acknowledgement

This paper is supported by the National Natural Science Foundation of China (11671122, 11471102), the China Postdoctoral Science Foundation (2017M622340), the Basic and Frontier Technology Research Program of Henan Province (152300410097), the Cultivation Plan of Young Key Teachers in Colleges and Universities of Henan Province (2016GGJS-107), the Higher School Key Scientific Research Projects of Henan Province (18A110019,17A110021), the Major Scientific Research Projects of Henan Institute of Science and Technology (2015ZD07), the High-level Scientific Research Personnel Project for Henan Institute of Science and Technology (2015037), Henan Institute of Science and Technology Postdoctoral Science Foundation.

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Accepted: 2018-02-20

Published Online: 2018-04-20

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 407–419, ISSN (Online) 2391-5455,

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