# A Global Optimization Algorithm for Generalized Quadratic Programming

Authors: Hongwei Jiao; Yongqiang Chen
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We present a global optimization algorithm for solving generalized quadratic programming (GQP), that is, nonconvex quadratic programming with nonconvex quadratic constraints. By utilizing a new linearizing technique, the initial nonconvex programming problem (GQP) is reduced to a sequence of relaxation linear programming problems. To improve the computational efficiency of the algorithm, a range reduction technique is employed in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (GQP) by means of the subsequent solutions of a series of relaxation linear programming problems. Finally, numerical results show the robustness and effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

In this paper, we consider the following generalized quadratic programming problem:(1)(GQP):ming0(x)=xTQ0x+d0Txs.t.gi(x)=xTQix+diTx≤bi,i=1,…,m,x∈X0={x∈Rn:l0≤x≤u0},
where Qi∈Rn×n (i=0,1,…,m) are all symmetric matrices, d0,di∈Rn, bi∈R,   i=1,…,m; l0=(l10,…,ln0)T,   u0=(u10,…,un0)T.Generalized quadratic programming problems are worthy of study, and it is not only because they have broad applications in heat exchanger network design engineering, financial management, statistics, optimal controllers, and so on [1, 2], but also because many other nonlinear problems can be transformed into this form [3, 4], and we often approximate nonlinear programming via solution of a series of quadratic programming problems. In addition, from a research point of view, these problems exist as significant theoretical and computational challenges. This is mainly because these problems are global optimization problem; that is, they are known to generally possess multiple local optimal solutions that are not globally optimal. So it is necessary to establish good solution algorithm for generalized quadratic programming problem (GQP).In the last decades, many solution methods have been developed for globally solving special form and general form of the (GQP). For examples, when constraint condition is a box constraint, a branch-and-cut algorithm [5] has been proposed by Vandenbussche and Nemhauser; when each function gi(x) (i=1,…,m) is a linear function, that is, GQP is an indefinite quadratic programming with linear constraints, then two efficient algorithms [6, 7] have been available; when each function gi(x) (i=1,…,m) is a convex function, that is, GQP is one nonconvex quadratic programming with convex constraints, then two feasible algorithms [8, 9] have been presented for globally solving the GQP; by utilizing lagrangian underestimates and the interval newton method, Voorhis [10] has proposed a global optimization algorithm for generalized quadratic programming; by applying a reformulation linearization technique, Sherali and Tuncbilek [11] have presented a feasible method for polynomial programming problems. In addition, Qu et al. [12] Shen and Liu [13] have proposed two different branch and bound algorithms for generalized quadratic programming problem using linearizing method, respectively.In this paper, we will present a global optimization algorithm for solving the (GQP) by combining branch and bound operation with a range reduction technique. The main features of our algorithm are as follows.(1) A new linearizing technique is presented; utilize the technique to systematically convert the GQP problem into a series of relaxation linear programming problems, and the solutions of these converted problems can infinitely approximate the global optimum of the original GQP problem by a successive refinement process. (2) The proposed linearization techniques for generating relaxation linear programming problems are embedded within a branch and bound scheme without increasing new variables and constraints. (3) The proposed relaxation linear programming for the GQP can be easily solved, which is more convenient in computation than the relaxation convex programming; thus, any effective linear programming algorithm can be used to solve this problem. (4) To accelerate the convergent speed of the proposed algorithm, a range reduction technique is applied in the proposed branch and bound procedure, which provides a theoretical possibility to delete a large part of the currently investigated region where there exists no global optimal solution of the (GQP). (5) Combing the new linearizing technique, branch and bound scheme, and range reduction technique, a branch-reduction-bound algorithm is presented. Finally, numerical results show that the proposed method can globally solve all test problems in obtaining global optimal solutions within a given tolerance error.The remainder of this paper is organized as follows. The next section describes the linearizing method and the relaxation linear programming of the original problem is established. Section3 states the branch-reduction-bound algorithm and its convergence. In Section 4 numerical experimental results are given. Finally, the conclusions are presented.


## 2. Relaxation Linear Programming

The principal structure in the development of a solution procedure for solving the (GQP) is the construction of lower bounds for this problem, as well as for its partitioned subproblems. A lower bound on the solution of the (GQP) and its partitioned subproblems can be obtained by solving a relaxation linear programming of the (GQP). In order to generate the relaxation linear programming, the proposed strategy is that underestimates each quadratic functiongi(x) (i=0,1,…,m) with a linear function.In the following, we assume thatXk={x∈Rn∣lk≤x≤uk}⊆X0 with lk=(l1k,…,lnk)T,  uk=(u1k,…,unk)T. Let λi be the minimum eigenvalues of the matrix Qi. Select θi such that
(2)θi={0,ifλi≥0,|λi|+ρ,ifλi<0,where ρ is a small positive number.For convenience, for anyx∈Xk, for each i=0,1,…,m, define the function
(3)giL(x)=[di+2(Qi+θiI)Tuk-2θiuk]Tx+2θi(uk)Tlk-θi(lk)T(lk)-(uk)T(Qi+θiI)uk.Theorem 1.
For eachi=0,1,…,m, for all x∈Xk={x∈Rn∣lk≤x≤uk}⊆X0, by the definition of gi(x) and giL(x), we have the following conclusions:(i)
giL(x)≤gi(x);(ii)
∥gi(x)-giL(x)∥→0 as ∥uk-lk∥→0.Proof.
(
1
) For each j=1,…,n, by the mean value theorem, for the function xj2 over the interval [ljk,ujk], we have
(4)xj2=2ξjk(xj-ljk)+(ljk)2≤2ujk(xj-ljk)+(ljk)2=2ujkxj-2ujkljk+(ljk)2,
where ξjk=αljk+(1-α)ujk,α∈[0,1].
From the definition of the functiongiL(x) and the equivalent deformation of the function gi(x), we can get that
(5)gi(x)=xTQix+diTx=(uk-x)T(Qi+θiI)(uk-x)+diTx-θi∑j=1n‍xj2+2(uk)T×(Qi+θiI)x-(uk)T(Qi+θiI)uk≥diTx-θi∑j=1n‍xj2+2(uk)T×(Qi+θiI)x-(uk)T(Qi+θiI)uk≥diTx+θi∑j=1n‍[-2ujkxj+2ujkljk-(ljk)2]+2(uk)T(Qi+θiI)x-(uk)T(Qi+θiI)uk=[di+2(Qi+θiI)Tuk-2θiuk]Tx+2θi(uk)Tlk-θi(lk)T(lk)-(uk)T(Qi+θiI)uk=giL(x).
That is,
(6)giL(x)≤gi(x),i=0,1,…,m.
(
2
) By the definition of the function giL(x) and the proof of conclusion (i), we have
(7)∥gi(x)-giL(x)∥=∥(uk)T(Qi+θiI)ukxTQix+diTx-{-(uk)T(Qi+θiI)uk[di+2(Qi+θiI)Tuk-2θiuk]Tx+2θi(uk)Tlk-θi(lk)T(lk)-(uk)T(Qi+θiI)uk[di+2(Qi+θiI)Tuk-2θiuk]T}∥=∥[∑j=1n‍(uk-x)T(Qi+θiI)(uk-x)+diTx-θi∑j=1n‍xj2+2(uk)T×(Qi+θiI)x-(uk)T(Qi+θiI)uk∑j=1n‍]-{(uk)T(Qi+θiI)uk[di+2(Qi+θiI)Tuk-2θiuk]Tx+2θi(uk)Tlk-θi(lk)T(lk)-(uk)T(Qi+θiI)uk}∑j=1n‍∥=∥∑j=1n‍(uk-x)T(Qi+θiI)(uk-x)+θi∑j=1n‍[2ujkxj-2ujkljk+(ljk)2-xj2]∥=∥∑j=1n‍(uk-x)T(Qi+θiI)(uk-x)+θi∑j=1n‍(xj-ljk)(ujk-xj+ujk-ljk)∥≤∥(uk-x)T(Qi+θiI)(uk-x)∥+θi∥∑j=1n‍(xj-ljk)(ujk-xj+ujk-ljk)∥≤∥(uk-x)T(Qi+θiI)(uk-x)∥+θi∥(x-lk)T(uk-x)∥+θi∥(x-lk)T(uk-lk)∥≤∥(uk-lk)T(Qi+θiI)(uk-lk)∥+2θi∥(uk-lk)T(uk-lk)∥=∥Qi+θiI∥∥uk-lk∥2+2θi∥uk-lk∥2.
Since
(8)∥Qi+θiI∥∥uk-lk∥2+2θi∥uk-lk∥2⟶0as∥uk-lk∥⟶0,
we have
(9)∥gi(x)-giL(x)∥⟶0as∥uk-lk∥⟶0.
The proof is complete.By Theorem1, we can construct the corresponding approximation relaxation linear programming (RLP) of the (GQP) in Xk as follows:(10)RLP(Xk):ming0L(x)s.t.giL(x)≤bi,i=1,…,m,x∈Xk={x:lk≤x≤uk},
where
(11)giL(x)=[di+2(Qi+θiI)Tuk-2θiuk]Tx+2θi(uk)Tlk-θi(lk)T(lk)-(uk)T(Qi+θiI)uk.Based on the relaxation linear programming, every feasible point of the (GQP) in subdomainXk is feasible to the (RLP); and the objective function value of the (RLP) is less than or equal to that of (GQP) over the partitioned set Xk. Thus, the RLP(Xk) provides a valid lower bound for the solution of the GQP(Xk).


## 3. Algorithm and Its Convergence

In this section, a branch-reduction-bound algorithm is proposed for globally solving the (GQP). To find a global optimal solution, the main computation is to solve a sequence of relaxation linear programming problems over partitioned subsets ofX0. Furthermore, in order to improve the computational efficiency of the proposed algorithm, a range reduction technique is employed to enhance the solution procedure.The proposed algorithm is based on partitioning the setX0 into subhyperrectangles, and each subhyperrectangle is associated with a node of the branch and bound tree, and each node is associated with a relaxation linear programming problem in corresponding subhyperrectangle. Therefore, at any stage k of the algorithm, assume that we obtain a collection of active nodes denoted by Δk, say, each is associated with a hyperrectangle X⊆X0, for all X∈Δk. For each such node X, compute a corresponding lower bound LB(X) of the GQP by solving the RLP, so that the lower bound of optimal value for the (GQP) over the X0 at iteration k is updated by LBk=min{LB(X),∀X∈Δk}. Until the solution of the (RLP) turns out to be feasible to the (GQP), we renew the upper bound g¯ if necessary. Then, the set of the active nodes Δk will satisfy LB(X)<g¯, for all X∈Δk, for each iteration k. We now select an active node to partition its associated hyperrectangle into two subhyperrectangles according to the following branching rule, and calculate the lower bounds for each new node as before. Upon fathoming any nonimproving nodes, we get a set of active nodes for the next iteration, and this process is repeated until termination condition is satisfied.The critical factor in guaranteeing convergence to a global optimum is the choice of a suitable branching rule. In this paper, we will choose a simple and standard bisection rule. This rule is sufficient to ensure convergence since it drives all the intervals to zero for all variables. This branching rule is stated as follows. Consider any node subproblem identified by the rectangleX′=[x_′,x¯′]⊆X0. Let q=argmax{x¯i′-x_i′:i=1,2,…,n}, and partition X′ into two subrectangles X1′ and X2′ by bisection the interval [x_q′,x¯q′] into the subintervals [x_q′,(x_q′+x¯q′)/2] and [(x_q′+x¯q′)/2,x¯q′].


### 3.1. Range Reduction Technique

To improve computational efficiency of the proposed algorithm, a range reduction technique proposed in [13, 14] is applied in the algorithm, which is reformulated as the following Theorems 2 and 3. In iteration k, we will check whether or not there exists global optimal solution of the GQP(X0) over X. The proposed technique aims at reducing the rectangle X without deleting any global optimal solution of the GQP(X0) over X. For convenience, for all x∈X={x∈Rn∣l≤x≤u}⊆X0 with l=(l1,l2,…,ln)T and u=(u1,u2,…,un)T. Without loss of generality, we rewrite the RLP(X) as the following form:
(12)RLP(X):ming0L(x),s.t.giL(x)≤bi,i=1,…,m,x∈X={x:l≤x≤u},
where
(13)giL(x)=∑j=1n‍cijxj+δi,i=0,1,…,m.Letg¯ be a known upper bound of the optimal value of the GQP(X0), and set
(14)RLBi=∑j=1,cij>0n‍cijlj+∑j=1,cij<0n‍cijuj+δi,i=0,1,…,m,RgipL=∑j=1,j≠p,cij>0n‍cijlj+∑j=1,j≠p,cij<0n‍cijuj+δi,i=0,1,…,m,p=1,…,n,γp=(g¯-Rg0pL)c0p,p=1,…,n,μip=(bi-RgipL)cip,i=1,…,m,p=1,…,n.Theorem 2.
For any subrectangleX⊆X0, the following conclusions hold.(i)
IfRLB0>g¯, then there exists no global optimal solution of the GQP(X0) over X.(ii)
IfRLB0≤g¯, then, for each p∈{1,2,…,n}, if c0p>0, then there does not exist global optimal solution of the GQP(X0) over X¯¯=(X¯¯j)n×1; if c0p<0, there is no global optimal solution of the GQP(X0) over X_=(X_j)n×1, where
(15)X¯¯j={Xj,j≠p,j=1,…,n,(γp,up]⋂Xp,j=p;X_j={Xj,j≠p,j=1,…,n,[lp,γp)⋂Xp,j=p.Proof.
(i) IfRLB0>g¯, then for all x∈X, g0L(x)≥RLB0>g¯; that is,
(16)g0(x)≥g0L(x)≥RLB0>g¯.
Therefore, there exists no global optimal solution of the GQP(X0) over X.
(ii) IfRLB0≤g¯, then, for each p∈{1,…,n}, if c0p>0, for all x∈X¯¯, we have xp>γp; that is,
(17)xp>g¯-Rg0pLc0porc0pxp>g¯-Rg0pL.
Therefore, we get that
(18)g0L(x)=∑j=1,j≠pn‍c0jxj+c0pxp+δ0≥∑j=1,j≠p,c0j>0n‍c0jlj+∑j=1,j≠p,c0j<0n‍cijuj+δ0+c0pxp=Rg0pL+c0pxp>Rg0pL+g¯-Rg0pL=g¯.
Thus, we have
(19)g0(x)≥g0L(x)>g¯.
Therefore, there does not exist global optimal solution of the GQP(X0) over X¯¯.
Similarly, ifc0p<0, there is no global optimal solution of problem GQP(X0) over X_.Theorem 3.
For any subrectangleX⊆X0, for all i=1,…,m, the following conclusions hold.(i)
IfRLBi>bi for some i∈{1,…,m}, then there exists no global optimal solution of the GQP(X0) over X.(ii)
IfRLBi≤bi for each i∈{1,…,m}, then, for each p∈{1,2,…,n}, if cip>0, then there is no global optimal solution of the GQP(X0) over X~=(X~j)n×1; if cip<0, there does not exist global optimal solution of the GQP(X0) over X^=(X^j)n×1, where
(20)X~j={Xj,j≠p,j=1,…,n,(μip,up]⋂Xp,j=p;X^j={Xj,j≠p,j=1,…,n,[lp,μip)⋂‍Xp,j=p.Proof.
(
i
) If RLBi>bi for some i∈{1,…,m}, then for all x∈X, we have
(21)gi(x)≥giL(x)≥RLBi>bi.
Therefore, there exists no global optimal solution of the GQP(X0) over X.
(
ii
) If RLBi≤bi for each i∈{1,…,m}, then, for each p∈{1,2,…,n}, if cip>0, for all x∈X~, we have xp>μip; that is,
(22)xp>bi-RgipLciporcipxp>bi-RgipL.
Therefore, we get that
(23)giL(x)=∑j=1,j≠pn‍cijxj+cipxp+δi≥∑j=1,j≠p,cij>0n‍cijlj+∑j=1,j≠p,cij<0n‍cijuj+δi+cipxp=RgipL+cipxp>RgipL+bi-RgipL=bi.
Thus, we have
(24)gi(x)≥giL(x)>bi.
Hence, there does not exist global optimal solution of the GQP(X0) over X~.
Similarly, ifcip<0, there is no global optimal solution of the GQP(X0) over X^.By Theorems2 and 3, we can give a range reduction technique to reject some regions where there does not exist the global optimal solution of the GQP(X0). Let X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n) be any subrectangle of X0.Range Reduction TechniqueReduction Rule (i). Calculate RLB0.IfRLB0>g¯, then let X=∅; otherwise, calculate Rg0pL and γp (p=1,…,n).Ifc0p>0 and γp<up for some p∈{1,…,n}, then let up=γp and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Ifc0p<0 and γp>lp for some p∈{1,…,n}, then let lp=γp and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Reduction Rule (ii). For each i=1,…,M, calculate RLBi.IfRLBi>bi for some i∈{1,…,M}, let X=∅; otherwise, calculate RgipL and μip (i=1,…,M,p=1,…,n).Ifcip>0 and μip<up for some i∈{1,…,M} and p∈{1,…,n}, then let up=μip and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Ifc0p<0 and μip>lp for some i∈{1,…,M} and p∈{1,…,n}, then let lp=μip and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).


### 3.2. Branch-Reduction-Bound Algorithm

LetLB(Xk) refer to the optimal value of the (RLP) over the subhyperrectangles Xk and let xk=x(Xk) refer to an element of corresponding argmin. Combining the former relaxation linear programming problem, branching operation, and range reduction technique together, the basic steps of the proposed algorithm for globally solving the (GQP) may be stated as follows.Algorithm 4.
Step0 (initialization). Let the iteration counter k=0, the set of all active nodes Δ0={X0}, and the convergence tolerance ϵ>0. Let the initial upper bound  g¯=+∞, and the set of feasible points F=∅.
Solve the RLP(X0), obtain optimal solution x0=x(X0) and optimal value LB(X0) for the RLP(X0), and let LB0=LB(X0). If x0 is feasible to the GQP(X0), let g¯=g0(x0) and F=F∪{x0}. If g¯-LB0≤ϵ, then stop with x0 as the global ϵ-optimal solution for the GQP(X0). Otherwise, proceed to Step 1.
Step1 (branching). Using the proposed branching rule to partition Xk into two new subhyperrectangles, and denote the set of new partition rectangles by X¯k.
Step2 (reduction). For each subhyperrectangle X∈X¯k, utilize the proposed range reduction technique to reject a part of the region or the whole region X, and still denote the remaining region by X, and still denote the remaining partition set by X¯k.
Step3 (updating the upper bound). If X¯k≠∅, solve the RLP(X) to obtain LB(X) and x(X) for each X∈X¯k. For each X∈X¯k, if LB(X)>g¯, set X¯k:=X¯k∖X; otherwise, select the midpoint xmid of X; if xmid is feasible to the GQP(X0), then let F:=F∪{xmid}, and if x(X) is feasible to the GQP(X0), then let F:=F∪{x(X)}.
IfF≠∅, update the upper bound g¯:=minx∈Fg0(x), and denote the best known feasible point by x*:=argminx∈Fg0(x).
Step4 (updating the lower bound). The partition set remaining is now Δk:=(Δk∖Xk)∪X¯k, and update the lower bound LBk:=infX∈ΔkLB(X).
Step5 (convergence checking). Set Δk+1=Δk∖{X:g¯-LB(X)≤ϵ,X∈Δk}. If Δk+1=∅, then stop with g¯ as the global optimal value of the (GQP) and x* as the global optimal solution. Otherwise, select an active node Xk+1 such that Xk+1=argminX∈Δk+1LB(X), xk+1:=x(Xk+1). Set k:=k+1, and return to Step 1.


### 3.3. Convergence Analysis

In this subsection, the global convergence properties of the proposed algorithm are presented. If the algorithm does not terminate finitely, then the branching rule guarantees all the intervals shrinking to a singleton for all the variables. On the other hand, Theorem1 guarantees that as ∥uk-lk∥→0 the relaxation linear programming problem RLP(Xk) approaches GQP(Xk). So it is not surprising that the algorithm is shown to be the convergent to the global optimal solution.Theorem 5.
The proposed algorithm either terminates finitely with the globalϵ-optimal solution x* of problem GQP(X0) or generates an infinite sequence of iteration such that along any infinite branch of the branch and bound tree, any accumulation point of the sequence {xk} will be the global solution of problem GQP(X0).Proof.
If the algorithm is finite, suppose that it terminates in iterationk, k≥0. Upon termination, we have
(25)g¯-LBk≤ϵ.
By Steps 0 and  3 in Algorithm 4, we can find a feasible solution x* for the problem GQP(X0) such that g0(x*)=g¯, which implies
(26)g0(x*)-LBk≤ϵ.
Set v denotes the optimal value of problem GQP(X0); then, by computational method of lower bound, we have
(27)LBk≤v.
Since x* is a feasible solution of problem GQP(X0), we have
(28)v≤g0(x*).
By inequalities (26)–(28), we can get that
(29)v≤g0(x*)≤LBk+ϵ≤v+ϵ.
That is,
(30)v≤g0(x*)≤v+ϵ.
Therefore, x* is a global ϵ-optimal solution of the problem GQP(X0).
If the algorithm is infinite, by the proposed algorithm, we know that the lower bound sequence{LBk} is nondecreasing and bounded by the upper bound minx∈Fg0(x); therefore, there exists limitation such that
(31)LB=limk→∞LBk≤minx∈Fg0(x).
Since {xk}∈X0, where X0 is a bounded close set, there exists a convergent subsequence {xβ}⊆{xk} and assume limβ→∞xβ=x*. Then by the proposed algorithm, there exists a decreasing subsequence {Xσ}⊆Xβ where Xβ∈Δβ with xσ∈Xσ, LBσ=LB(Xσ)=g0L(xσ) and limσ→∞Xσ={x*}. By Theorem 1 and the continuity of the function g0(x), we can follow that
(32)limσ→∞LBσ=limσ→∞g0L(xσ)=limσ→∞g0(xσ)=g0(x*).
Then all remaining is to demonstrate thatx* is feasible to the GQP(X0).
Firstly, sinceX0 is bounded and closed set, obviously we have x*∈X0.
Secondly, we will prove thatgi(x*)≤bi, i=1,…,m.
By contradiction, assumegτ(x*)>bτ for some τ∈{1,…,m}. Since gτL(x) is a continuous function and by Theorem 1, the sequence {gτL(xσ)} is convergent to gτ(x*); that is, there exists a σ¯ such that
(33)|gτL(xσ)-gτ(x*)|<gτ(x*)-bτforanyσ>σ¯.
Hence, for any σ>σ¯, we have gτL(xσ)>bτ, which shows that the LRP(Xσ) is infeasible and violates the assumption that xσ=x(Xσ). This will bring about a contradiction; thus, the proof is competed.


## 3.1. Range Reduction Technique

To improve computational efficiency of the proposed algorithm, a range reduction technique proposed in [13, 14] is applied in the algorithm, which is reformulated as the following Theorems 2 and 3. In iteration k, we will check whether or not there exists global optimal solution of the GQP(X0) over X. The proposed technique aims at reducing the rectangle X without deleting any global optimal solution of the GQP(X0) over X. For convenience, for all x∈X={x∈Rn∣l≤x≤u}⊆X0 with l=(l1,l2,…,ln)T and u=(u1,u2,…,un)T. Without loss of generality, we rewrite the RLP(X) as the following form:
(12)RLP(X):ming0L(x),s.t.giL(x)≤bi,i=1,…,m,x∈X={x:l≤x≤u},
where
(13)giL(x)=∑j=1n‍cijxj+δi,i=0,1,…,m.Letg¯ be a known upper bound of the optimal value of the GQP(X0), and set
(14)RLBi=∑j=1,cij>0n‍cijlj+∑j=1,cij<0n‍cijuj+δi,i=0,1,…,m,RgipL=∑j=1,j≠p,cij>0n‍cijlj+∑j=1,j≠p,cij<0n‍cijuj+δi,i=0,1,…,m,p=1,…,n,γp=(g¯-Rg0pL)c0p,p=1,…,n,μip=(bi-RgipL)cip,i=1,…,m,p=1,…,n.Theorem 2.
For any subrectangleX⊆X0, the following conclusions hold.(i)
IfRLB0>g¯, then there exists no global optimal solution of the GQP(X0) over X.(ii)
IfRLB0≤g¯, then, for each p∈{1,2,…,n}, if c0p>0, then there does not exist global optimal solution of the GQP(X0) over X¯¯=(X¯¯j)n×1; if c0p<0, there is no global optimal solution of the GQP(X0) over X_=(X_j)n×1, where
(15)X¯¯j={Xj,j≠p,j=1,…,n,(γp,up]⋂Xp,j=p;X_j={Xj,j≠p,j=1,…,n,[lp,γp)⋂Xp,j=p.Proof.
(i) IfRLB0>g¯, then for all x∈X, g0L(x)≥RLB0>g¯; that is,
(16)g0(x)≥g0L(x)≥RLB0>g¯.
Therefore, there exists no global optimal solution of the GQP(X0) over X.
(ii) IfRLB0≤g¯, then, for each p∈{1,…,n}, if c0p>0, for all x∈X¯¯, we have xp>γp; that is,
(17)xp>g¯-Rg0pLc0porc0pxp>g¯-Rg0pL.
Therefore, we get that
(18)g0L(x)=∑j=1,j≠pn‍c0jxj+c0pxp+δ0≥∑j=1,j≠p,c0j>0n‍c0jlj+∑j=1,j≠p,c0j<0n‍cijuj+δ0+c0pxp=Rg0pL+c0pxp>Rg0pL+g¯-Rg0pL=g¯.
Thus, we have
(19)g0(x)≥g0L(x)>g¯.
Therefore, there does not exist global optimal solution of the GQP(X0) over X¯¯.
Similarly, ifc0p<0, there is no global optimal solution of problem GQP(X0) over X_.Theorem 3.
For any subrectangleX⊆X0, for all i=1,…,m, the following conclusions hold.(i)
IfRLBi>bi for some i∈{1,…,m}, then there exists no global optimal solution of the GQP(X0) over X.(ii)
IfRLBi≤bi for each i∈{1,…,m}, then, for each p∈{1,2,…,n}, if cip>0, then there is no global optimal solution of the GQP(X0) over X~=(X~j)n×1; if cip<0, there does not exist global optimal solution of the GQP(X0) over X^=(X^j)n×1, where
(20)X~j={Xj,j≠p,j=1,…,n,(μip,up]⋂Xp,j=p;X^j={Xj,j≠p,j=1,…,n,[lp,μip)⋂‍Xp,j=p.Proof.
(
i
) If RLBi>bi for some i∈{1,…,m}, then for all x∈X, we have
(21)gi(x)≥giL(x)≥RLBi>bi.
Therefore, there exists no global optimal solution of the GQP(X0) over X.
(
ii
) If RLBi≤bi for each i∈{1,…,m}, then, for each p∈{1,2,…,n}, if cip>0, for all x∈X~, we have xp>μip; that is,
(22)xp>bi-RgipLciporcipxp>bi-RgipL.
Therefore, we get that
(23)giL(x)=∑j=1,j≠pn‍cijxj+cipxp+δi≥∑j=1,j≠p,cij>0n‍cijlj+∑j=1,j≠p,cij<0n‍cijuj+δi+cipxp=RgipL+cipxp>RgipL+bi-RgipL=bi.
Thus, we have
(24)gi(x)≥giL(x)>bi.
Hence, there does not exist global optimal solution of the GQP(X0) over X~.
Similarly, ifcip<0, there is no global optimal solution of the GQP(X0) over X^.By Theorems2 and 3, we can give a range reduction technique to reject some regions where there does not exist the global optimal solution of the GQP(X0). Let X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n) be any subrectangle of X0.Range Reduction TechniqueReduction Rule (i). Calculate RLB0.IfRLB0>g¯, then let X=∅; otherwise, calculate Rg0pL and γp (p=1,…,n).Ifc0p>0 and γp<up for some p∈{1,…,n}, then let up=γp and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Ifc0p<0 and γp>lp for some p∈{1,…,n}, then let lp=γp and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Reduction Rule (ii). For each i=1,…,M, calculate RLBi.IfRLBi>bi for some i∈{1,…,M}, let X=∅; otherwise, calculate RgipL and μip (i=1,…,M,p=1,…,n).Ifcip>0 and μip<up for some i∈{1,…,M} and p∈{1,…,n}, then let up=μip and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).Ifc0p<0 and μip>lp for some i∈{1,…,M} and p∈{1,…,n}, then let lp=μip and X=(Xj)n×1 with Xj=[lj,uj] (j=1,…,n).


## 3.2. Branch-Reduction-Bound Algorithm

LetLB(Xk) refer to the optimal value of the (RLP) over the subhyperrectangles Xk and let xk=x(Xk) refer to an element of corresponding argmin. Combining the former relaxation linear programming problem, branching operation, and range reduction technique together, the basic steps of the proposed algorithm for globally solving the (GQP) may be stated as follows.Algorithm 4.
Step0 (initialization). Let the iteration counter k=0, the set of all active nodes Δ0={X0}, and the convergence tolerance ϵ>0. Let the initial upper bound  g¯=+∞, and the set of feasible points F=∅.
Solve the RLP(X0), obtain optimal solution x0=x(X0) and optimal value LB(X0) for the RLP(X0), and let LB0=LB(X0). If x0 is feasible to the GQP(X0), let g¯=g0(x0) and F=F∪{x0}. If g¯-LB0≤ϵ, then stop with x0 as the global ϵ-optimal solution for the GQP(X0). Otherwise, proceed to Step 1.
Step1 (branching). Using the proposed branching rule to partition Xk into two new subhyperrectangles, and denote the set of new partition rectangles by X¯k.
Step2 (reduction). For each subhyperrectangle X∈X¯k, utilize the proposed range reduction technique to reject a part of the region or the whole region X, and still denote the remaining region by X, and still denote the remaining partition set by X¯k.
Step3 (updating the upper bound). If X¯k≠∅, solve the RLP(X) to obtain LB(X) and x(X) for each X∈X¯k. For each X∈X¯k, if LB(X)>g¯, set X¯k:=X¯k∖X; otherwise, select the midpoint xmid of X; if xmid is feasible to the GQP(X0), then let F:=F∪{xmid}, and if x(X) is feasible to the GQP(X0), then let F:=F∪{x(X)}.
IfF≠∅, update the upper bound g¯:=minx∈Fg0(x), and denote the best known feasible point by x*:=argminx∈Fg0(x).
Step4 (updating the lower bound). The partition set remaining is now Δk:=(Δk∖Xk)∪X¯k, and update the lower bound LBk:=infX∈ΔkLB(X).
Step5 (convergence checking). Set Δk+1=Δk∖{X:g¯-LB(X)≤ϵ,X∈Δk}. If Δk+1=∅, then stop with g¯ as the global optimal value of the (GQP) and x* as the global optimal solution. Otherwise, select an active node Xk+1 such that Xk+1=argminX∈Δk+1LB(X), xk+1:=x(Xk+1). Set k:=k+1, and return to Step 1.


## 3.3. Convergence Analysis

In this subsection, the global convergence properties of the proposed algorithm are presented. If the algorithm does not terminate finitely, then the branching rule guarantees all the intervals shrinking to a singleton for all the variables. On the other hand, Theorem1 guarantees that as ∥uk-lk∥→0 the relaxation linear programming problem RLP(Xk) approaches GQP(Xk). So it is not surprising that the algorithm is shown to be the convergent to the global optimal solution.Theorem 5.
The proposed algorithm either terminates finitely with the globalϵ-optimal solution x* of problem GQP(X0) or generates an infinite sequence of iteration such that along any infinite branch of the branch and bound tree, any accumulation point of the sequence {xk} will be the global solution of problem GQP(X0).Proof.
If the algorithm is finite, suppose that it terminates in iterationk, k≥0. Upon termination, we have
(25)g¯-LBk≤ϵ.
By Steps 0 and  3 in Algorithm 4, we can find a feasible solution x* for the problem GQP(X0) such that g0(x*)=g¯, which implies
(26)g0(x*)-LBk≤ϵ.
Set v denotes the optimal value of problem GQP(X0); then, by computational method of lower bound, we have
(27)LBk≤v.
Since x* is a feasible solution of problem GQP(X0), we have
(28)v≤g0(x*).
By inequalities (26)–(28), we can get that
(29)v≤g0(x*)≤LBk+ϵ≤v+ϵ.
That is,
(30)v≤g0(x*)≤v+ϵ.
Therefore, x* is a global ϵ-optimal solution of the problem GQP(X0).
If the algorithm is infinite, by the proposed algorithm, we know that the lower bound sequence{LBk} is nondecreasing and bounded by the upper bound minx∈Fg0(x); therefore, there exists limitation such that
(31)LB=limk→∞LBk≤minx∈Fg0(x).
Since {xk}∈X0, where X0 is a bounded close set, there exists a convergent subsequence {xβ}⊆{xk} and assume limβ→∞xβ=x*. Then by the proposed algorithm, there exists a decreasing subsequence {Xσ}⊆Xβ where Xβ∈Δβ with xσ∈Xσ, LBσ=LB(Xσ)=g0L(xσ) and limσ→∞Xσ={x*}. By Theorem 1 and the continuity of the function g0(x), we can follow that
(32)limσ→∞LBσ=limσ→∞g0L(xσ)=limσ→∞g0(xσ)=g0(x*).
Then all remaining is to demonstrate thatx* is feasible to the GQP(X0).
Firstly, sinceX0 is bounded and closed set, obviously we have x*∈X0.
Secondly, we will prove thatgi(x*)≤bi, i=1,…,m.
By contradiction, assumegτ(x*)>bτ for some τ∈{1,…,m}. Since gτL(x) is a continuous function and by Theorem 1, the sequence {gτL(xσ)} is convergent to gτ(x*); that is, there exists a σ¯ such that
(33)|gτL(xσ)-gτ(x*)|<gτ(x*)-bτforanyσ>σ¯.
Hence, for any σ>σ¯, we have gτL(xσ)>bτ, which shows that the LRP(Xσ) is infeasible and violates the assumption that xσ=x(Xσ). This will bring about a contradiction; thus, the proof is competed.


## 4. Numerical Experiments

To compare the proposed algorithm with the known methods in computational efficiency and solution quality, some test examples in recent literature are implemented on microcomputer, the algorithm procedure is coded in C++, and each relaxation linear programming is solved by using simplex method, and the convergence toleranceϵ is set to 10-6 in Examples 6–13 and 10-5 in Example 14, respectively. These test examples and numerical results compared with the known algorithms are given  in Tables 1 and 2.Table 1
Numerical results for the test Examples6–13.
Example
References
Optimal value
Optimal solution
6
[ours]
−16.000000000
(5.000000000, 1.000000000)
[15]
−16.0
(5.0, 1.0)
7
[ours]
6.777777779
(2.000000000, 1.666666667)
[16]
6.7780
(2.00003, 1.66665)
[13]
6.777782016
(2.000000000, 1.666666667)
[13]
6.777781963
(2.000000000, 1.666666667)
8
[ours]
0.500000442
(0.500000000, 0.500000000)
[16]
0.5
(0.5, 0.5)
[17]
0.5
(0.5, 0.5)
[13]
0.500004627
(0.5  0.5)
[13]
0.5
(0.5, 0.5)
9
[ours]
118.383671904
(2.555745855, 3.130201688)
[18]
118.383756475281
(2.5557793695324, 3.13016463929768)
10
[ours]
−1.162882693
(1.500000000, 1.500000000)
[19]
−1.16288
(1.5, 1.5)
11
[ours]
1.177125051
(1.177124344, 2.177124344)
[14]
1.177124327
(1.177124327, 2.177124353)
12
[ours]
−0.999999410
(2.000000000, 1.000000000)
[14]
−1.0
(2.000000, 1.000000)
13
[ours]
−11.363636364
(1.000000000, 0.181818470, 0.983332113)
[20]
−10.35
(0.998712, 0.196213, 0.979216)Table 2
Numerical results for the test Examples6–13.
Example
References
Number of algorithm iteration
Computational times
6
[ours]
5
0.00184588
[15]
10
7
[ours]
10
0.00266479
[16]
44
0.18
[13]
40
0.032
[13]
32
0.015
8
[ours]
37
0.0192625
[16]
91
0.85
[17]
96
1
[13]
34
0.056
[13]
29
0.041
9
[ours]
59
0.0385038
[18]
210
0.78
10
[ours]
24
0.0105913
[19]
84
11
[ours]
22
0.00919099
[14]
434
1
12
[ours]
21
0.00849446
[14]
24
<1
13
[ours]
420
0.284541
[20]
1648
0.3438Example 6 (see [15]).
Consider(34)min-x12+x1x2+x22+x1-2x2s.t.x1+x2≤6,-2x12+x22+2x1+x2≤-4,1≤x1≤6,1≤x2≤6.Example 7 (see [13, 16]).
Consider(35)minx12+x22s.t.0.3x1x2≥1,2≤x1≤5,1≤x2≤3.Example 8 (see [13, 16, 17]).
Consider(36)minx1s.t.4x2-4x12≤1,-x1-x2≤-1,0.01≤x1≤15,0.01≤x2≤15.Example 9 (see [18]).
Consider(37)min6x12+4x22+5x1x2s.t.-6x1x2≤-48,0≤x1≤10,0≤x2≤10.Example 10 (see [19]).
Consider(38)min-x1+x1x20.5-x2s.t.-6x1+8x2≤3,3x1-x2≤3,1≤x1≤1.5,1≤x2≤1.5.Example 11 (see [14]).
Consider(39)minx1s.t.14x1+12x2-116x12-116x22≤1,114x12+114x22-37x1-37x2≤-1,1≤x1≤5.5,1≤x2≤5.5.Example 12 (see [14]).
Consider(40)minx1x2-2x1+x2+1s.t.8x22-6x1-16x2≤-11,-x22+3x1+2x2≤7,1≤x1≤2.5,1≤x2≤2.225.Example 13 (see [20]).
Consider(41)min-4x2+(x1-1)2+x22-10x32s.t.x12+x22+x32≤2,(x1-2)2+x22+x32≤2,2-2≤x1≤2,0≤x2,x3≤2.Example 14 (see [21]).
Consider(42)min-∑i=1n‍xi2s.t.∑i=1j‍xi≤j,j∈{1,2,…,n},xi≥0,i∈{1,2,…,n}.The numerical results for Example14 compared with [21] are given in Table 3.Table 3
Numerical results for Example14.
References
Dimension of variable
n
=
5
n
=
10
n
=
20
n
=
30
This paper
CPU time(s)
0.0181791
0.302157
6.01095
44.4965
Number of iteration
12
32
88
206
Ref. [21]
CPU time(s)
10.11
21.86
47.00
106.33
Number of iteration
141
283
651
965The numerical results for test Examples6–14 show that the proposed algorithm is robust and can be used to globally solve generalized quadratic programming (GQP).


## 5. Concluding Remarks

In this paper, a global optimization algorithm based on a new linearizing method and a range reduction technique is presented for solving the (GQP). The relaxation linear programming of the original problem (GQP) is established by underestimating the objective and constraint functions with linear functions. By making full use of the currently known upper bound and relaxation linear programming of the (GQP), a range reduction technique is presented. The algorithm is convergent to the global minimum through the successive refinement of a linear relaxation of the feasible region and/or of the objective function and the subsequent solution of a series of relaxation linear programming problems. Numerical results show effectiveness and robustness of the proposed algorithm.



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*Source: 215312-2013-10-22.xml*