A parametric linearizing approach for quadratically inequality constrained quadratic programs

: 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


Introduction
In this paper we consider the following quadratically inequality constrained quadratic programs: where p i jk , d i k and b i are all arbitrary real numbers; l = (l , . . ., l n ) T , u = (u , . . ., u n ) T .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 e ective global optimization algorithm for solving the (QICQP).
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 branchand-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 e ciency 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 e ciency 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 e ciency of the proposed algorithm in Section 4. Finally, some concluding remarks are presented.

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 R n×n is a symmetric matrix, and λ jk ∈ { , }.For convenience, for any z ∈ Z, for any k ∈ { , , . . ., n}, some expressions are introduced as follows: Theorem 2.1.For any k ∈ { , , . . ., n}, for any z ∈ Z, consider the functions h kk (z), h kk (z, Z, λ kk ) and h kk (z, Z, λ kk ), then, the following conclusions hold: Proof.(i) By the mean value theorem, for any z ∈ Z, there exists a point If λ kk = , then we have If λ kk = , then it follows that Thus, we can get that Similarly, if λ kk = , then we have If λ kk = , then it follows that Thus, we can get that Therefore, for any z ∈ Z, we have that we have lim The proof is completed.
Without loss of generality, for any z ∈ Z, for any j ∈ { , , . . ., n}, k ∈ { , , . . ., n}, j ≠ k, we de ne Obviously, we have In a similar way as in Theorem 2.1, we can get the following Theorem 2.2: Theorem 2.2.For each j = , , . . ., n, k = , , . . ., n, for any z ∈ Z, we have: (i) The following inequalities hold: (ii) The following limitations hold: Proof.(i) From the inequality (1), replacing λ kk by λ jk , and replacing z k by z j , we can get that From the inequality (1), replacing λ kk by λ jk , we can get that From (1), replacing λ kk and z k by λ jk and (z j − z k ), respectively, we can get that (ii) From the limitations (2) and (3), replacing λ kk and z k by λ jk and z j , we have Unauthenticated Download Date | 7/8/19 6:37 AM From the limitations (2) and (3), replacing λ kk by λ jk , it follows that By the limitations ( 2) and ( 3), replacing λ kk and z k by λ jk and (z j − z k ), respectively, we can get that lim 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: where 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) o ers 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 su ciently approximate the optimal solution of the (QICQP) as u − l → , and this guarantees the global convergence of the proposed algorithm.

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 re ned box partition.Here we choose a typical boxbisection method, which is su cient to ensure the global convergence of the proposed branch-and-bound method.For any selected box Bounding the lower bound: For each sub-box Z ⊆ Z , 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 LB s = min{LB(Z) Z ∈ Ω s }, where Ω s 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 subbox Z, where Z ∈ Ω s .In addition, we need to calculate the objective function values of each known feasible solutions for the (QICQP), and denote by UB s = min{H (z) ∶ z ∈ Θ} 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 z ∈ Z, i ∈ { , , . . ., m}, q ∈ { , . . ., n}, and denote by HUB the current upper bound of the (QICQP), we let Theorem 3.1.For any investigated sub-box Z = (Z j ) ×n ⊆ Z , we have the following conclusions: (i) If RLB (λ) > HUB, then the whole investigated sub-box Z should be deleted.
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.
Therefore, z s is an -global optimal solution of the initial problem (QICQP).
If the proposed algorithm does not terminate after nite iterations, for this case, the detailed convergent conclusions are given as follows.
Theorem 3.2.If the proposed algorithm does not terminate after nite iterations, then it will generate an in nite partitioning sequence {Z s } of the initial box Z , and any accumulation point of the sequence {Z s } 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 , i.e., u − l → .
Secondly, from Theorem 2.4, the optimal solution of the (PLPRP) will su ciently approximate the optimal solution of the (QICQP) as u − l → , and this ensures that the limitation lim s→∞ (UB s − LB s ) = 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 su cient 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).

Numerical experiments
Given the convergent error = − and the parameter matrix λ = (λ jk ) n×n ∈ R n×n , where λ jk ∈ { , }, 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.Example 4.1 ([16]).([5,14,17,18]).
Example 4.5 ([4,6,14]).14,20]).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 e ciency.

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 e ciency of the proposed algorithm.

Table 2 .
Numerical results for Example 4.8