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.
1 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 [1-7]). 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:
where
Over the past decades, a variety of local optimization algorithms have been developed for effectively solving the special forms or general form of the quadratically constrained quadratic programs problem (QCQP). For instances, convexification approach [10], interval newton method [11], simplicial branch-and-bound algorithm [12], semidefinite relaxation approach [13], matrix cone decomposition and polyhedral approximation algorithm [14], duality bound algorithm [15], robust optimization algorithm [16], rectangle branch-and-bound algorithms [17-28]. However, to our knowledge, although some algorithms can be also used to solve the QCQP, due to the complication of the investigated problem, it is rather challenging to globally solve the QCQP problem.
The goal of this article is to present a branch-delete-bound algorithm for globally solving the QCQP problem by developing new linearizing technique. To this aim, by utilizing the characteristics of quadratic function, we first introduce a new linearizing method, then by using the linearizing method, we show that the initial problem (QCQP) and its sub-problems can be systematically converted into the corresponding linear relaxed programs problems, and the optimal solutions of these converted linear relaxed programs problems can infinitely approximate the global optimum of the associated quadratically constrained quadratic programs problem. Moreover, the established linear relaxed programs problems are embedded within a branch-and-bound framework without introducing any new variables and constrained functions, which can be easily solved by any effective linear programming algorithms, and which are easier to be solved than any convex relaxation programs problem. We then combine the established linear relaxed programs problem, the branching operation, deleting operation and bounding operation together, so an effective branch-delete-bound algorithm is presented for globally solving the QCQP. Finally, compared with some known algorithms, numerical experimental results show that our method has higher computational efficiency.
The rest of this paper is organized as follows. Based on the characteristics of quadratic function, Section 2 formulates a new linearizing method, and the linear relaxed programs problems of the initial problem (QCQP) and its sub-problems are established. Based on the linear relaxed programs problem derived in Section 2, Section 3 presents a branch-delete-bound algorithm, and its global convergence is discussed and proved. In Section 4, compared with some known algorithms, numerical results demonstrate the computational efficiency of the proposed algorithm. Finally some concluding remarks are drawn.
2 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 = {z ∈ Rn|l ≤ z ≤ u} ⊆ Z0. For ∀ z ∈ Z, for any j, k ∈ {1, 2, …, n}, define
Theorem 2.1
For any z ∈ Z = [l, u] ⊆ Z0, for any j, k ∈{1, 2, …, n}, we have the following conclusions:
Δj(z)→ 0, ∇j(z)→ 0, Δ(zj − zk) → 0, ∇(zj − zk) → 0, Δjk(z) → 0 and ∇jk(z) → 0 as ∥u − l∥ → 0.
Proof
By the convex characters of the quadratic function fj(z) =
i.e.,
Since (zj − zk)2 is a convex function about (zj − zk) over the interval [(lj − uk), (uj − lk)], similarly by the conclusion (i), we have
and
and
i.e.,
Since
is a convex function about zj over the interval [lj, uj]. Thus, Δj(z) can obtain the maximum value at the point lj or uj, i.e.
Similarly, since
is a concave function about zj over the interval [lj, uj], therefore ∇j(z) can obtain maximum value at the point
Since
is a convex function about (zj − zk) over the interval [(lj − uk), (uj − lk)]. Thus, Δ(zj − zk) can obtain the maximum value at the point (lj − uk) or (uj − lk), i.e.,
Similarly, since
is a concave function about (zj − zk) over the interval [(lj − uk), (uj − lk)], therefore ∇(zj − zk) can obtain maximum value at the point
By (7) and (8), we have: as ∥ u − l ∥→ 0,
Since
By (6) and (9), we have Δjk(z) → 0 as ∥u − l∥ → 0.
Similarly, we can prove that ∇jk(z) → 0 as ∥u − l∥→ 0. Therefore, the proof is complete. □
For convenience, without loss of generality, for any j and k ∈ {1, …, n}, we let
Obviously, we have
And for each i = 1, …, p, m = 1, …, M, for any z ∈ Z, define
Theorem 2.2
For ∀ z ∈ Z = [l, u] ⊆ Z0, for each i = 1, …, p, m = 1, …, M, we have the following conclusions:
Proof
(i) Obviously, from (10) we can easily get that
(ii) Considering the error
By the conclusions of Theorem 2.1, we have Δj(z), ∇j(z), Δjk(z) and ∇jk(z) → 0, as ∥u − l ∥ → 0.
Therefore,
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.
where
From the constructing method of the above linear relaxed programs, for any Z ⊆ Z0, 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.
3 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.
3.1 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 Zk into two sub-rectangles Zk,1 and Zk,2, it generates a more refined partition that cannot yet be excluded from further consideration in finding the global minimum of the QCQP in Z0. 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 Zk = [lk, uk] ⊆ Z0. This branching operation is formulated as follows.
Let ξ = arg max{uj − lj : i = 1, …, n}.
Let
So that the identified sub-rectangle Zk is divided into two sub-rectangles Zk,1 and Zk,2.
3.2 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 Z0. For convenience, for any z ∈ Z = (Zj)n×1 with Zj = [lj, uj] (j = 1, …, n), without loss of generality, we rewrite the LRPP into the following linear programming problem in sub-rectangle Z:
Let UBk be a currently known upper bound of the global optimal value for the QCQP in Z0, which is obtained after k iterations, and set
Similarly as in Theorem 3 in [23], for any sub-rectangle Z ⊆ Z0, we can easily prove that the following conclusions hold:
If LB0 > UBk, then the sub-rectangle Z can be deleted; else if LB0 ≤ UBk, then: for each p ∈ {1, 2, …, n}, if γ0p > 0, then the sub-rectangle
If LBi > bi for some i ∈ {1, …, m}, then the sub-rectangle Z can be deleted; else if LBj ≤ bj for some i ∈ {1, …, m}, then: for each p ∈ {1, 2, …, n}, if γip > 0, then the sub-rectangle
By utilizing the deleting operation to delete a part of the investigated rectangle where the global optimal solution of the QCQP in Z0 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.
3.3 Bounding operation
The bounding operations are used to update the lower bounds and upper bounds of the global optimal value of the QCQP in Z0. 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 Zk, respectively.
3.4 Branch-delete-bound algorithm
Let LB(Zk) and zk = z(Zk) be the optimum value and optimum solution for the LRPP in the sub-rectangle Zk, 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.
Branch-Delete-Bound Algorithm
Initializing Step
Initializing the counter of iteration k := 0, the active node set Λ0 = {Z0}, the feasible point set F = ∅, the convergence judgement error ϵ > 0, the initial upper bound UB0 = +∞. Compute the LRPP(Z0), obtain LB0 := LB(Z0) and z0 := z(Z0). If Gi(z0) ≤ bi holds for all i = 1, …, m, then we update the feasible point set F = {z0} and the upper bound UB0 = G0(z0). If UB0 − LB0≤ϵ holds, then the algorithm stops with z0 as the global ϵ-optimal solution of the QCQP in Z0; else go on the following Branching Step.
Branching Step
Select a rectangle Zk ∈ Λk to determine a branching variable zq, and employ the selected branching operation to divide the selected rectangle Zk into two new sub-rectangles, and represent the new subdivided sub-rectangles set as Žk.
Deleting Step
For any sub-rectangle Z ∈ Žk, compute LBj(i = 0, 1, …, m), βp(p = 1, …, n), λjp(i = 1, …, m, p = 1, …, n).
For each i ∈ {1, …, m}, if LBj > bj, then delete the investigated sub-rectangle Z;
else
if γip > 0 and λip < up for some p ∈ {1, …, n}, then let up = λip;
else if γip < 0 and λip > lp for some p ∈ {1, …, n}, then let lp = λip.
If LB0 > UBk, then delete the investigated sub-rectangle Z;
else
if γ0p > 0 and βp < up for some p ∈ {1, …, n}, then let up = βp;
else if γ0p < 0 and βp > lp for some p ∈ {1, …, n}, then let lp = βp.
Finally, still denote the remaining sub-rectangle by Z, and denote the remaining partition sub-rectangle set by Žk.
Bounding Step
Solve the LRPP (Z) for each sub-rectangle Z ∈ Žk to get LB(Z) and z(Z), if LB(Z) > UBk, then set Žk := Žk \ Z; else if the midpoint zmid of Zk satisfies constrained condition for the QCQP in Z0, then set F := F ∪ {zmid}, and if z(Z) satisfies constrained condition for the QCQP in Z0, then let F := F ∪ {z(Z)}, at the same time, we update the upper bound UBk := minz∈F G0(z). If F ≠ ∅, denote zk := argminz∈F G0(z) as the current best feasible point. Let Λk := (Λk \ Zk) ∪Žk, we then update the lower bound LBk := infZ∈Λk LB(Z).
Optimality Judgement Step
If UBk − LBk ≤ ϵ, then the algorithm stops, at the same time, we get that UBk and zk are the global ϵ-optimal value and the global ϵ-optimal solution for the initial problem (QCQP), respectively; else let k := k + 1, and select a new active node Zk satisfying Zk = argminZ∈Λk LB(Z), and return to Branching Step.
3.5 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 zk is a global ϵ-optimum 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 {zkq} 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 UBk − LBk ≤ ϵ. By the bounding operation for the upper bound, this implies that there must exist a feasible point zk satisfying UBk = G0(zk), thus we can follow that G0(zk) − LBk ≤ ϵ, i.e. G0(zk) − ϵ≤LBk. Denote v* as the optimal value of the QCQP, obviously, by the structure of branch-and-bound framework, it follows that LBk ≤ v. Since zk is feasible to the QCQP, therefore, it follows that G0(zk) ≥ v, i.e. G0(zk) − ϵ ≥ v − ϵ. Combing the above inequalities, it follows that v ≥ LBk ≥ G0(zk) − ϵ ≥ v− ϵ, i.e. v ≤ G0(zk) ≤ v + ϵ. Therefore, zk is a global ϵ-optimum solution for the QCQP.
If the proposed algorithm does not finitely terminates after k iterations, a sufficient condition for the branch-delete-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 limk→∞(UBk − LBk) = 0 holds, this implies that the employed bounding operation is consistent.
By the proposed branching operation, the selected sub-rectangle Zk, which actually attained lower bound, is immediately selected for further partition in the later iteration. So that the selecting operation of the branch-delete-bound 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). □
4 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.”.
Numerical results for Examples 4.1-4.8
| Example | Refs. | Optimal solution | Optimal value | Iter. |
|---|---|---|---|---|
| 1 | This paper | (5.000000000, 1.000000000) | −16.000000000 | 1 |
| [15] | (5.0, 1.0) | -16.0 | 10 | |
| [23] | (5.000000000, 1.000000000) | −16.000000000 | 5 | |
| 2 | This paper | (2.000000000, 1.666666667) | 6.777778336 | 3 |
| [17] | (2.00003, 1.66665) | 6.7780 | 44 | |
| [19] | (2.000000000, 1.666666667) | 6.777782016 | 40 | |
| [19] | (2.000000000, 1.666666667) | 6.777781963 | 32 | |
| [23] | (2.000000000, 1.666666667) | 6.777777779 | 10 | |
| 3 | This paper | (0.500000000, 0.500000000) | 0.500000361 | 21 |
| [17] | (0.5, 0.5) | 0.5 | 91 | |
| [19] | (0.5 0.5) | 0.500004627 | 24 | |
| [19] | (0.5, 0.5) | 0.5 | 29 | |
| [23] | (0.500000000, 0.500000000) | 0.500000442 | 37 | |
| [24] | (0.5, 0.5) | 0.5 | 96 | |
| 4 | This paper | (2.555730431, 3.130220581) | 118.383672506 | 44 |
| [22] | (2.555779370, 3.130164639) | 118.383756475 | 210 | |
| [23] | (2.555745855, 3.130201688) | 118.383671904 | 59 | |
| 5 | This paper | (1.500000000, 1.500000000) | −1.162882693 | 11 |
| [23] | (1.500000000, 1.500000000) | −1.162882693 | 24 | |
| [25] | (15, 1.5) | −1.16288 | 84 | |
| 6 | This paper | (1.177124344, 2.177124344) | 1.177125181 | 19 |
| [21] | (1.177124327, 2.177124353) | 1.177124327 | 432 | |
| [23] | (1.177124344, 2.177124344) | 1.177125051 | 22 | |
| 7 | This paper | (2.000000000, 1.000000000) | −1.000000000 | 2 |
| [21] | (2.000000, 1.000000) | −1.0 | 24 | |
| [23] | (2.000000000, 1.000000000) | −0.999999410 | 21 | |
| 8 | This paper | (1.0, 0.181815435, 0.983332674) | −11.363551588 | 148 |
| [23] | (1.0, 0.181818470, 0.983332113) | −11.363636364 | 420 | |
| [26] | (0.998712, 0.196213,0.979216) | −10.35 | 1648 |
Numerical results for Example 4.9
| Refs. | Dimensionn | Optimal value | Number of iteration | Time(s) |
|---|---|---|---|---|
| This paper | 5 | -25.0 | 1 | 0.00231396 |
| 10 | -100.0 | 1 | 0.0153671 | |
| 20 | -400.0 | 1 | 0.0993256 | |
| 30 | -900.0 | 1 | 0.342342 | |
| 40 | -1600.0 | 1 | 0.919515 | |
| 50 | -2500.0 | 1 | 2.05164 | |
| 60 | -3600.0 | 1 | 3.96127 | |
| 70 | -4900.0 | 1 | 7.05643 | |
| 80 | -6400.0 | 1 | 11.7382 | |
| 90 | -8100.0 | 1 | 18.18 | |
| 100 | -10000.0 | 1 | 27.2642 | |
| 200 | -40000.0 | 1 | 403.016 | |
| [18] | 5 | -25.0 | 141 | 10.11 |
| 10 | -100.0 | 283 | 21.86 | |
| 20 | -400.0 | 651 | 47.00 | |
| 30 | -900.0 | 965 | 106.33 | |
| 50 | -2500.0 | 1891 | 304.30 | |
| [23] | 5 | -25.0 | 12 | 0.0181791 |
| 10 | -100.0 | 32 | 0.302157 | |
| 20 | -400.0 | 88 | 6.01095 | |
| 30 | -900.0 | 206 | 44.4965 | |
| 40 | -1600.0 | 302 | 98.122 | |
| [27] | 5 | -25.0 | 12 | 0.0187141 |
| 10 | -100.0 | 31 | 0.334158 | |
| 20 | -400.0 | 86 | 5.93962 | |
| 30 | -900.0 | 204 | 44.8577 | |
Numerical comparisons with [27] for Example 4.10
| Parameter | Ref.[27] | This paper | ||||
|---|---|---|---|---|---|---|
| m | Iter | Lmax | Time(s) | Iter | Lmax | Time(s) |
| 5 | 1571 | 300 | 2.92998 | 481 | 199 | 0.9604 |
| 10 | 754 | 171 | 2.56231 | 567 | 202 | 1.2962 |
| 20 | 1490 | 462 | 14.4705 | 381 | 153 | 1.4200 |
| 30 | 1829 | 562 | 36.1397 | 394 | 159 | 2.8441 |
| 40 | 1700 | 467 | 60.9473 | 497 | 178 | 6.2495 |
| 50 | 2722 | 429 | 160.367 | 574 | 205 | 11.5478 |
| 60 | 3054 | 518 | 276.695 | 537 | 221 | 16.4134 |
| 70 | 1965 | 521 | 256.028 | 597 | 234 | 25.568 |
| 80 | 2299 | 540 | 434.802 | 506 | 179 | 29.9703 |
| 90 | 1961 | 623 | 485.656 | 526 | 199 | 42.3564 |
Example 4.6
Example 4.8
Example 4.9
Compared with the known algorithms ([18, 23, 27]), the numerical results for Examples 4.1-4.9 show that the proposed algorithm can be used to globally solve the quadratically constrained quadratic programs problem with higher computational efficiency.
Example 4.10
([27]).
All elements of A0 and d0 are all randomly generated in [0, 1], all elements of Ai and di are all randomly generated in [−1, 0]; all bi, i = 1, 2, …, m, are randomly generated in [−300, −90], and n = 5.
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].
5 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 programs problem with branching operation and deleting operation in a branch-and-bound framework, we formulate a branch-delete-bound algorithm for effectively solving the QCQP. By subsequently dividing the initial rectangle and subsequently solving a series of linear relaxed programs problems, the presented algorithm is convergent to the global minimum of the initial problem (QCQP). Compared with the known methods, numerical results demonstrate that the proposed branch-delete-bound method has higher computational efficiency.
-
Competing interests
The authors declare that they have no competing interests.
Acknowledgement
This paper is supported by the National Natural Science Foundation of China under Grant (61304061), the Natural Science Foundation of Henan Province (152300410097), the Higher School Key Scientific Research Projects of Henan Province (18A110019, 17A110021, 16A110014), the Henan Provincial Youth Backbone Teachers Training Program (2016GGJS-107), the Major Scientific Research Projects of Henan Institute of Science and Technology (2015ZD07), the Basic and Advanced Technology Research Project of Henan Province (152300410097), the High-level Scientific Research Personnel Project for Henan Institute of Science and Technology(2015037).
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