An effective algorithm for globally solving quadratic programs using parametric linearization technique

1 Introduction

This paper considers the following quadratic programs with quadratic constraints:

where l0=(l10,…,ln0)T,u0=(u10,…,un0)T;pjki,dkiand β_i are all arbitrary real numbers. QP has wide application in route optimization, engineering design, investment portfolio, management decision, production programs, etc. In addition, QP usually owns multiple local optimal solutions which are not global optimal solutions, i.e., in these classes of problems there exist important theoretical difficulties and computational complexities. Thus, it is necessary to present an efficient algorithm for globally solving QP.

In last several decades, many algorithms have been developed for solving QP and its special cases, such as duality-bounds algorithm [1], branch-and-reduce methods [2, 3, 4, 5, 6], approximation approach [7], branch-and-bound approaches [8, 9, 10], and so on. Except for the above ones, some algorithms for polynomial programming [11, 12, 13, 14, 15, ] and quadratic fractional programming [16, 17, ] also can be used to solve QP. Although these algorithms can be used to solve QP and its special cases, less work has been still done for globally solving the investigated quadratic programs with quadratic constraints.

This paper will present a new global optimization branch-and-bound algorithm for solving QP. First of all, we derive a new parametric linearization technique. By utilizing this linearization technique, the initial QP can be converted into a parametric linear programming relaxation problem, which can be used to determine the lower bounds of the global optimal values of the initial QP and its subproblems. Based on the branch-and-bound framework, a new global optimization branch-and-bound algorithm is designed for solving QP, the proposed algorithm is convergent to the global optimal solution of the initial QP by successively subdividing the initial box and by solving the converted parametric linear programming relaxation problems. To improve the computational speed of the proposed branch-and-bound algorithm, some interval reduction operations are used to compress the investigated interval. Finally, compared with some known algorithms, numerical experimental results show higher computational efficiency of the proposed branch-and-bound algorithm.

The remaining sections of this paper are listed as follows. Firstly, in order to derive the parametric linear programming relaxation problem of QP, Section 2 presents a new parametric linearization technique. Secondly, based on the branch-and-bound framework in Section 3, by combing the derived parametric linear programming relaxation problem with the interval reduction operations, an effective branch-and-bound algorithm is constructed for globally solving QP. Thirdly, compared with some known methods, some existent test problems are used to verify the computational feasibility of the proposed algorithm in Section 4. Finally, some conclusions are obtained.

2 New parametric linearization approach

In this section, we will present a new parametric linearization approach for constructing the parametric linear programming relaxation problem of QP. The detailed deriving process of the parametric linearization approach is given as follows. Without loss of generality, we assume that Y = {(y₁, y₂, . . ., y_n)^T ϵ Rⁿ : l_j ≤ y_j ≤ u_j, j = 1, . . ., n}⊆ Y⁰, = (γ_jk)_n×n ϵ R^n×n is a symmetric matrix, and _γjk ϵ {0, 1}.

For convenience in expression, for any y ϵ Y, for any j ϵ {1, 2, . . ., n}, k ϵ {1, 2, . . ., n}, j ≠ k, we define

It is obvious that

Theorem 2.1. For any k ϵ {1, 2, . . ., n}, for any y ϵ Y, then we have:

• (i) The following inequalities hold:

and

1. (ii) The following limitations hold:

and

Proof. (i) By the mean value theorem, for any y ϵ Y, there exists a point ξ_k = αy_k + (1 − α)y_k(γ_kk), where α ϵ [0, 1], such that

If γ_kk = 0, then we have

If γ_kk = 1, then it follows that

Thus, we can get that

Similarly, if γ_kk = 0, then we have

If γ_kk = 1, then it follows that

Thus, we can get that

Therefore, for any y ϵ Y, we have that

From the inequality (1), replacing γ_kk by γ_jk, and replacing y_k by y_j, we can get that

From the inequality (1), replacing γ_kk by γ_jk, we can get that

From (1), replacing γ_kk and y_k by γ_jk and (y_j + y_k), respectively, we can get that

From the former several inequalities, it is easy to follow that

and

Therefore, we have

(ii) Since

we have

Also since

Therefore, it follows that

From the limitations (3) and (4), replacing γ_kk and y_k by γ_jk and y_j, respectively, we have

and

From the limitations (3) and (4), replacing γ_kk by γ_jk, it follows that

and

By the limitations (3) and (4), replacing γ_kk and y_k by γ_jk and (y_j + y_k), respectively, we can get that

and

From the inequalities (7) and (8), we have

Thus, we can get that

Also from the inequalities (7) and (8), we get that

Thus, it follows that

Without loss of generality, for any Y = [l, u] b Y⁰, for any parameter matrix = (γ_jk)_n×n, for any y ϵ Y and i ϵ {0, 1, . . ., m}, we let

Theorem 2.2. For any y ϵ Y = [l, u] ⊆ Y⁰, for any parameter matrix = (γ_jk)_n×n, for any i = 0, 1, . . ., m, we can get the following conclusions:

and

Proof. (i) From (1) and (2), for any j, k ϵ {1, . . ., n}, we can get that

and

By (9) and (10), for any y ϵ Y ⊆ Y⁰, we have that

Therefore, we obtain that

(ii)