An Effective Global Optimization Algorithm for Quadratic Programs with Quadratic Constraints

This paper will present an effective algorithm for globally solving quadratic programs with quadratic constraints. In this algorithm, we propose a new linearization method for establishing the linear programming relaxation problem of quadratic programs with quadratic constraints. The proposed algorithm converges with the global optimal solution of the initial problem, and numerical experiments show the computational efficiency of the proposed algorithm.


Introduction
Quadratic programs with quadratic constraints (QPWQC) have attracted the attention of many researchers for several decades.On the one hand, it is since these classes of problems have a broad applications in multistage shipping, path planning, finance, and portfolio optimization, among others.[1][2][3][4][5][6][7][8][9][10][11].On the other hand, it is because these classes of problems exist as important theoretical complexities and computational difficulties, that is to say, they are known to generally possess multiple local optimal solutions, which are not optimal solutions.
In this paper, first of all, by making use of the characteristics of simple variable quadratic function, we construct a new linearization method for establishing the linear programming relaxation problem of the QPWQC.Next, we present a global optimization algorithm based on the branch-and-bound scheme for solving the QPWQC.Finally, the global convergence of the proposed algorithm is proved, and numerical experimental results demonstrate the higher computational efficiency of the proposed algorithm.
The main features of the proposed algorithm are given as follows.
(1) A new linearization method is proposed for systematically converting the QPWQC into a sequence of linear programming relaxation problems, and the solutions of these linear programming relaxation problems can infinitely approximate the global optimal solution of the original QPWQC by subdividing the linear relaxation of the feasible region of the QPWQC and solving a series of linear programming relaxation problems.(2) The constructed linear programming relaxation problems are embedded within a branch-and-bound framework, which can be effectively solved by any efficient linear programming method.(3) Combining the proposed linear programming relaxation problem with the branch-and-bound framework, an effective algorithm is proposed for solving the problem of QPWQC.(4) Compared with the exist algorithms [37,[39][40][41][42][43][44][45][46][47], numerical results show that the proposed algorithm in this paper can be used to globally solve the QPWQC with higher computational efficiency.
The remaining sections of this paper are organized as follows.Firstly, the aim of Section 2 is to propose a new linearization method for establishing the linear programming relaxation problem of the initial QPWQC.Secondly, based on the branch-and-bound scheme, Section 3 proposes a global optimization algorithm, and its global convergence is proved.Thirdly, compared with the existing methods, Section 4 describes some numerical examples to show the computational efficiency of the proposed algorithm.Finally, some conclusions are given.

New Linearization Method for Deriving Linear Programming Relaxation Problem
In this paper, the mathematical modeling of quadratic programs with quadratic constraints is given as follows: (QPWQC) where d i jk , c i k , and b i are all arbitrary real numbers;

+∞.
In this section, we construct a new linearization method for deriving the linear programming relaxation problem of the QPWQC, and the detailed construction process of the linearization method is described as follows.
For convenience, we assume without loss of generality that X = {(x 1 , x 2 , . . ., x n ) T ∈ R n : l j ≤ x j ≤ u j , j = 1, 2, . . ., n} ⊆ X 0 .Theorem 1.For any x ∈ X, k ∈ {1, 2, . . ., n}, we consider the functions x 2 k , u 2 k + 2u k (x k − u k ) and u 2 k + 2l k (x k − u k ), we have the following conclusions: Proof.(i) From the mean value theorem, there exists a point From it follows that lim Also from Therefore, we have lim The proof is completed.
By Theorem 3, we can establish the linear programming relaxation problem (LPRP) of the QPWQC over X as follows: (LPRP) : From the construction process of the former linearizing method, it is obvious that for any given X, each feasible point of the QPWQC is also feasible to the LPRP, and the optimal value of the LPRP is less than or equal to that of QPWQC.Therefore, the LPRP offers a reliable lower bound for the optimal value of the QPWQC.Except for the above approach, Theorem 3 also ensures the global convergence of the proposed algorithm.

New Global Optimization Algorithm
In this section, based on the former LPRP, we shall present a new global optimization algorithm for solving the QPWQC.In this algorithm, there are the following several key operations: branching, bounding, and space reduction.
Firstly, we choose a simple branching operation, which is called an interval bisection method.For any selected box X can be subdivide into X 1 and X 2 .The selected branching operation is sufficient to ensure the global convergence of this algorithm.
Secondly, for each investigated sub-box X ⊆ X 0 , we must solve the LPRP, and set LB s = min{LB(X)|X ∈ Ω s }, where Ω s is still not fathomed as a sub-box set.In order to update the upper bound, we need to fathom the feasible point, and set Θ be the known feasible point set and UB s = min{ψ 0 (x)|x ∈ Θ}, to be the existent best upper bound.
In addition, we can introduce an interval reduction operation from Theorem 3 [6] to improve the convergent speed of the proposed algorithm.

Steps for Global Optimization Algorithm
For any investigated box X s ⊆ X 0 , let LB(X s ) and x s = x(X s ) be the optimal value and optimal solution of the LPRP over X s .Based on the branch-and-bound scheme and the former LPRP, a new global optimization algorithm is described as follows.
Let the lower bound LB 0 = LB(X 0 ).If x 0 is feasible to the QPWQC, let the upper bound be UB 0 = ψ 0 (x 0 ), otherwise let the initial upper bound be UB 0 = +∞.
If UB 0 − LB 0 ≤ ε, let the global ε-optimal solution of the QPWQC be x 0 , otherwise let Step 2. Let the upper bound be UB s = UB s−1 , partition X s−1 into X s,1 and X s,2 , and let Λ = Λ ∪ {X s−1 } be the deleted sub-boxes set.
For each X s,t , t = 1, 2, utilize the interval reduction method to compress the investigated box, and let X s,t be the remaining box.
For each remaining box X s,t , t = 1, 2, solve the LPRP to obtain its optimal solution x s,t and optimal value LB(X s,t ), respectively. Set Step 3.For each X s,t , t = 1, 2, if x mid is the feasible point of the QPWQC, let Θ := Θ ∪ {x mid }, and let the new upper bound UB s = min x∈Θ {ψ 0 (x)}; if x s,t is feasible to the QPWQC, let the new upper bound UB s = min{UB s , ψ 0 (x s,t )}, and let the best known feasible point be x s , which satisfies UB s = ψ 0 (x s ).
Step 4. If UB s − LB s ≤ ε, then we let the ε-global optimal solution of the QPWQC be x s , otherwise let s = s + 1, and return to Step 2.

Global Convergence of the Proposed Algorithm
If the proposed algorithm terminates after finite iterations, then, when it terminates, we can obtain the global optimal solution of the QPWQC.Otherwise, the proposed algorithm will generate an infinite sequence, whose limitation is the global optimal solution of the QPWQC; the detailed proof is given as follows.
Theorem 4. If the proposed algorithm does not terminate after finite iterations, then the proposed algorithm will generate an infinite sequence {X s }, whose accumulation point will be the global optimal solution of the QPWQC.
Proof.First of all, in the proposed algorithm, the selected branching method is the rectangle bisection, which is exhaustive, and which guarantees that the intervals of all variables converge to 0.
Secondly, as u − l → 0 , from the conclusions of Theorem 3, it follows that the LPRP will sufficiently approximate the QPWQC, which is to say, lim s→∞ (UB s − LB s ) = 0, i.e., the proposed algorithm satisfies that the bounding operation is consistent.Thirdly, in the proposed algorithm, the subdivided box which achieved the actual lower bound is immediately selected for the later partition, and the proposed algorithm satisfies that the selected operational bound is improving.By Theorem 4.3 of Reference [39], the proposed branch-and-bound algorithm satisfies the global convergent sufficient condition.Hence, the proposed algorithm converges to the global optimal solution of the QPWQC.

Numerical Experiments
Let ε = 10 −6 be the convergence error.Some numerical examples in recent literature are solved in C++ program on microcomputer, and the simplex approach is employed to solve the LPRP.Compared with the existent algorithms, these numerical examples are given as follows, and their computational results are listed in Tables 1 and 2.

Example 6 (Reference [46])
Example 7 (References [26,43]) Comparing with the existent algorithms, numerical results show that the proposed algorithm has the higher computational efficiency.
To demonstrate robustness of the proposed algorithm, we give a large-scale random numerical example as follows.2.

Concluding Remarks
This paper presents an effective algorithm for globally solving quadratic programs with quadratic constraints.In this algorithm, a new linearization method is constructed for deriving the linear programming relaxation problem of the QPWQC.The proposed algorithm converges to the global optimal solution of the initial problem of QPWQC, and numerical experimental results show the higher computational efficiency of the proposed algorithm.

Table 2 .
Computational results for Example 8.