Branch-delete-bound algorithm for globally solving quadratically constrained quadratic programs

Abstract This paper presents a branch-delete-bound algorithm for effectively solving the global minimum of quadratically constrained quadratic programs problem, which may be nonconvex. By utilizing the characteristics of quadratic function, we construct a new linearizing method, so that the quadratically constrained quadratic programs problem can be converted into a linear relaxed programs problem. Moreover, the established linear relaxed programs problem is embedded within a branch-and-bound framework without introducing any new variables and constrained functions, which can be easily solved by any effective linear programs algorithms. By subsequently solving a series of linear relaxed programs problems, the proposed algorithm can converge the global minimum of the initial quadratically constrained quadratic programs problem. Compared with the known methods, numerical results demonstrate that the proposed method has higher computational efficiency.


Introduction
The quadratically constrained quadratic programs problem (QCQP) has attracted a huge attention of practitioners and researchers for many years. In part, this is because the quadratically constrained quadratic programs problem finds a wide range of applications in management science and engineering, product subassembly, production programs, portfolio decision optimization, chance problem, production design, finance and economy, etc. (see [1][2][3][4][5][6][7]). In particular, many practical problems (such as stochastic programs problem, packing problem, 0-1 programs problem, etc. [8,9]) can be transformed into the quadratically constrained quadratic programs problem. In addition, the problem (QCQP) possess multiple local optimum points which are not globally optimum, i.e., from a research point of view, this problem (QCQP) poses significant theoretical and computational complication.
In this paper, the mathematical modelling of the investigated quadratically constrained quadratic programs problem is given as follows:

New linearizing method
The main operation in the proposed branch-delete-bound algorithm is computation of the lower bounds of the initial problem and its partitioned subproblems. The lower bounds for the initial problem and its partitioned subproblems can be computed by solving their corresponding linear relaxed programs problems, which are derived by the following new linearizing method.
Let Z D fz 2 R n jl Ä z Ä ug Â Z 0 . For 8z 2 Z; for any j; k 2 f1; 2; : : : ; ng; define f j .z/ D z 2 j ; f l j .z/ D .l j C u j /z j .l j C u j / 2 4 ; f u j .z/ D .l j C u j /z j l j u j ; 4 j .z/ D f j .z/ f l j .z/; r j .z/ D f u j .z/ f j .z/; f j k .z/ D z j z k ; f l j k .z/ D 1 2 OE.l j C u j /z k C .l k C u k /z j .l j C u j / 2 4 .l k C u k / 2 4 C .l j u k /.u j l k /; f u j k .z/ D 1 2 OE.l j C u j /z k C .l k C u k /z j l j u j l k u k C .l j u k C u j l k / 2 4 ; 4 j k .z/ D f j k .z/ f l j k .z/; r j k .z/ D f u j k .z/ f j k .z/; 4.z j z k / D .z j z k / 2 OE.l j u k C u j l k /.z j z k / .l j u k C u j l k / 2 4 ; r.z j z k / D OE.l j u k C u j l k /.z j z k / .l j u k /.u j l k / .z j z k / 2 : Theorem 2.1. For any z 2 Z D OEl; u Â Z 0 , for any j; k 2 f1; 2; : : : ; ng; we have the following conclusions: Proof. (i) By the convex characters of the quadratic function f j .z/ D z 2 j ; we have (ii) Since .z j z k / 2 is a convex function about .z j z k / over the interval OE.l j u k /; .u j l k /, similarly by the conclusion (i), we have and .l j u k C u j l k /.z j z k / .l j u k /.u j l k / .z j z k / 2 : .3/ By (1), (2) and (3), we have is a convex function about z j over the interval OEl j ; u j : Thus, 4 j .z/ can obtain the maximum value at the point l j or u j , i.e. max z j 2OEl j ;u j is a concave function about z j over the interval OEl j ; u j ; therefore r j .z/ can obtain maximum value at the point l j Cu j 2 , i.e., max z j 2OEl j ;u j r j .z/ D .u j l j / 2 4 : .5/ By (4) and (5), we have max z j 2OEl j ;u j 4 j .z/ D max z j 2OEl j ;u j r j .z/ ! 0; as ku lk ! 0: is a convex function about .z j z k / over the interval OE.l j u k /; .u j l k /: Thus, 4.z j z k / can obtain the maximum value at the point .l j u k / or .u j l k /, i.e., max .z j z k /2OE.l j u k /;.u j l k / is a concave function about .z j z k / over the interval OE.l j u k /; .u j l k /; therefore r.z j z k / can obtain maximum value at the point l j u k Cu j l k 2 , i.e., max .z j z k /2OE.l j u k /;.u j l k / r.z j z k / D .u j l k l j C u k / 2 4 : .8/ By (7) and (8), we have: as ku lk ! 0, max .z j z k /2OE.l j u k /;.u j l k / 4.z j z k / D max .z j z k /2OE.l j u k /;.u j l k / r.z j z k / ! 0: .l j u k /;.u j l k / r.z j z k /: By (6) and (9), we have 4 j k .z/ ! 0 as ku lk ! 0: Similarly, we can prove that r j k .z/ ! 0 as ku lk ! 0: Therefore, the proof is complete.
For convenience, without loss of generality, for any j and k 2 f1; : : : ; ng; we let Obviously, we have .10/ And for each i D 1; : : : ; p; m D 1; : : : ; M; for any z 2 Z, define Proof. (i) Obviously, from (10) we can easily get that By the conclusions of Theorem 2.1, we have 4 j .z/; r j .z/; 4 j k .z/ and r j k .z/ ! 0; as ku lk ! 0: Therefore, G i .z/ G L i .z/ ! 0 as ku lk ! 0: Using the similar method as in the above proof, we can conclude that The proof of the conclusion (ii) is complete.
By Theorem 2.2, we can construct the corresponding approximation linear relaxed programs problem (LRPP) of the QCQP in Z as follows.
From the constructing method of the above linear relaxed programs, for any Z Â Z 0 ; every feasible point of the QCQP in sub-rectangle Z is also feasible to the LRPP in sub-rectangle Z; and the optimum value of the LRPP in sub-rectangle Z is less than or equal to that of the QCQP in sub-rectangle Z. Thus, the LRPP in sub-rectangle Z provides a valid lower bound for the global minimum of the QCQP in sub-rectangle Z.

Branch-delete-bound algorithm
In this section, based on the linear relaxed programs problem derived by new linearizing method in Section 2, we will present an effective branch-delete-bound algorithm for globally solving the QCQP. In this algorithm, there are three fundamental operations: branching operation, deleting operation and bounding operation. We then introduce this three fundamental operations as follows.

Branching operation
Here, we select a standard branching operation, which is called as bisection method of rectangle maximum edge. The selected branching operation iteratively subdivides the investigated rectangle Z k into two sub-rectangles Z k;1 and Z k;2 , it generates a more refined partition that cannot yet be excluded from further consideration in finding the global minimum of the QCQP in Z 0 . This selected branching operation is enough to ensure the global convergence of the proposed algorithm since the interval of each variable is shrank into a singleton through infinite rectangle bisection. For any identified sub-rectangle Z k D OEl k ; u k Â Z 0 . This branching operation is formulated as follows.
(a) Let D arg maxfu i l i W i D 1; : : : ; ng.
So that the identified sub-rectangle Z k is divided into two sub-rectangles Z k;1 and Z k;2 .

Deleting operation
Based upon the linear relaxed programs in section 2 and branch-and-bound structure, we will introduce a deleting operation to improve the convergent speed of the proposed algorithm, which is used to delete a part of the rectangle Z or the whole rectangle Z without rejecting any global optimal solution of the initial problem (QCQP) in Z 0 . For convenience, for any z 2 Z D .Z j / n 1 with Z j D OEl j ; u j .j D 1; : : : ; n/, without loss of generality, we rewrite the LRPP into the following linear programming problem in sub-rectangle Z: LP.Z/ W Let UB k be a currently known upper bound of the global optimal value for the QCQP in Z 0 , which is obtained after k iterations, and set .
( Z j ; j ¤ p; j D 1; : : : ; n; Similarly as in Theorem 3 in [23], for any sub-rectangle Z Â Z 0 , we can easily prove that the following conclusions hold: (i) If LB 0 > UB k , then the sub-rectangle Z can be deleted; else if LB 0 Ä UB k , then: for each p 2 f1; 2; : : : ; ng, if 0p > 0, then the sub-rectangle Z D .Z j / n 1 can be deleted; if 0p < 0, then the sub-rectangle Z D .Z j / n 1 can be deleted.
(ii) If LB i > b i for some i 2 f1; : : : ; mg, then the sub-rectangle Z can be deleted; else if LB i Ä b i for some i 2 f1; : : : ; mg, then: for each p 2 f1; 2; : : : ; ng, if ip > 0, then the sub-rectangle e Z D . e Z j / n 1 can be deleted; if ip < 0, then the sub-rectangle b Z D . b Z j / n 1 can be deleted. By utilizing the deleting operation to delete a part of the investigated rectangle where the global optimal solution of the QCQP in Z 0 does not exist, we can improve the computational speed of the proposed branch-and-bound procedure, and accelerate the global convergence of the proposed branch-and-bound algorithm.

Bounding operation
The bounding operations are used to update the lower bounds and upper bounds of the global optimal value of the QCQP in Z 0 . This main computations for updating lower bounds need to solve a sequence of linear relaxed programs problems, which can be easily solved by using simplex methods. In additions, the upper bounds can be updated by computing the objective function value of the QCQP, which is corresponding to the optimal solution of each linear relaxed programs problem or midpoint of the investigated rectangle Z k , respectively.

Branch-delete-bound algorithm
Let LB.Z k / and z k D z.Z k / be the optimum value and optimum solution for the LRPP in the sub-rectangle Z k , respectively. Combining the former linear relaxed programs, the branching operation, deleting operation and bounding operation together, we can establish an effective branch-delete-bound algorithm for globally solving the problem (QCQP) as follows.

Initializing
Step. Initializing the counter of iteration k WD 0, the active node set ƒ 0 D fZ 0 g, the feasible point set F D ;, the convergence judgement error > 0, the initial upper bound UB 0 D C1. Compute the LRPP.Z 0 /, obtain LB 0 WD LB.Z 0 / and z 0 WD z.Z 0 /. If G i .z 0 / Ä b i holds for all i D 1; : : : ; m; then we update the feasible point set F D fz 0 g and the upper bound UB 0 D G 0 .z 0 /. If UB 0 LB 0 Ä holds, then the algorithm stops with z 0 as the global optimal solution of the QCQP in Z 0 ; else go on the following Branching Step.

Branching
Step. Select a rectangle Z k 2 ƒ k to determine a branching variable z q , and employ the selected branching operation to divide the selected rectangle Z k into two new sub-rectangles, and represent the new subdivided sub-rectangles set as L Z k .

Bounding
Step. Solve the LRPP.Z/ for each sub-rectangle Z 2 L Z k to get LB.Z/ and z.Z/, if LB.Z/ > UB k , then set L Z k WD L Z k nZ; else if the midpoint z mid of Z k satisfies constrained condition for the QCQP in Z 0 , then set F WD F [ fz mid g, and if z.Z/ satisfies constrained condition for the QCQP in Z 0 , then let F WD F [ fz.Z/g, at the same time, we update the upper bound UB k WD min z2F G 0 .z/. If F ¤ ;, denote z k WD argmin z2F G 0 .z/ as the current best feasible point. Let ƒ k WD .ƒ k n Z k / [ L Z k , we then update the lower bound LB k WD inf Z2ƒ k LB.Z/.

Optimality Judgement
Step. If UB k LB k Ä ; then the algorithm stops, at the same time, we get that UB k and z k are the global optimal value and the global optimal solution for the initial problem (QCQP), respectively; else let k WD k C 1, and select a new active node Z k satisfying Z k D argmin Z2ƒ k LB.Z/, and return to Branching Step.

Global convergence analysis
The global convergence of the proposed branch-delete-bound algorithm is formulated as follows.
Theorem 3.1. If the proposed branch-delete-bound algorithm terminates after k iterations, then z k is a globaloptimum solution for the (QCQP); else if the branch-delete-bound algorithm does not finitely terminates after k iteration, then it must generate an infinite subsequence fz k q g of iterations, which satisfies that its any accumulation point must be the global optimum solution of the QCQP.
Proof. If the proposed branch-delete-bound algorithm finitely terminates after k iterations, where k 0, then by optimality judgement step, we have UB k LB k Ä : By the bounding operation for the upper bound, this implies that there must exist a feasible point z k satisfying UB k D G 0 .z k /; thus we can follow that G 0 .z k / LB k Ä ; i.e. G 0 .z k / Ä LB k : Denote v as the optimal value of the QCQP, obviously, by the structure of branch-and-bound framework, it follows that LB k Ä v: Since z k is feasible to the QCQP, therefore, it follows that If the proposed algorithm does not finitely terminates after k iterations, a sufficient condition for the branchdelete-bound algorithm that is convergent to the global minimum is that the bounding operation must be consistent and the selection operation must satisfy that bound can be improved.
By the proposed branch-delete-bound algorithm, the employed branching operation is bisection, which satisfies the exhaustiveness, that is to say that any unfathomed partition can be further refined by the branching operation. Therefore, by Theorem 2.2 and the relationship between the QCQP and its linear relaxed programs problem (LRPP), it is so easy to conclude that lim k!1 .UB k LB k / D 0 holds, this implies that the employed bounding operation is consistent.
By the proposed branching operation, the selected sub-rectangle Z k , which actually attained lower bound, is immediately selected for further partition in the later iteration. So that the selecting operation of the branch-deletebound algorithm must satisfy that bound can be improved.
In general, it follows that the bounding operation is consistent and selection operation satisfy that bound can be improved. Finally, by Refs. [1,3], we can follow that the proposed branch-delete-bound algorithm converges to the global minimum of the initial problem (QCQP).

Numerical experiments
To compare the proposed branch-delete-bound algorithm with the known algorithms in computational speed and solution quality, some numerical examples in recent literature are solved on microcomputer. The solving procedure is coded in C++ software, and each linear relaxed programs problem in the solving procedure is solved by using simplex method. These test examples are listed as follows, and compared with the known methods. Numerical results are given in Tables 1-3. In Table 1 the number of algorithm iteration is denoted by "Iter.".
The numerical comparisons of computational results for Example 4.10 are listed in the following Table 3, where n denotes the number of variables, m denotes the number of constraints. The numerical results show that our algorithm has higher computational efficiency than that of [27].

Concluding remarks
In this article, an effective branch-delete-bound algorithm is presented for globally solving the quadratically constrained quadratic programs problem. Based on the characteristics of quadratic function, we first introduce a new linearizing technique, by utilizing this technique the initial quadratically constrained quadratic programs problem can be converted into a linear relaxed programs problem. By utilizing the currently known upper bound and the characters of the linear relaxed programs problem of the QCQP, a deleting operation is introduced, which can be used to accelerate the convergent speed of the proposed algorithm. Next, combining the established linear relaxed