A Branch and Bound Reduced Algorithm for Quadratic Programming Problems with Quadratic Constraints

Abstract

We propose a branch and bound reduced algorithm for quadratic programming problems with quadratic constraints. In this algorithm, we determine the lower bound of the optimal value of original problem by constructing a linear relaxation programming problem. At the same time, in order to improve the degree of approximation and the convergence rate of acceleration, a rectangular reduction strategy is used in the algorithm. Numerical experiments show that the proposed algorithm is feasible and effective and can solve small- and medium-sized problems.

1. Introduction

Quadratic programming problems with quadratic constraints play a very important role in global optimization because quadratic functions are relatively simple functions among all nonlinear functions, and quadratic functions can approach many other functions. Therefore, it is necessary for us to research quadratic problems for researching nonlinear problems better, and quadratic programming problems with quadratic constraints have an important applications in Science and technology. Then, in spite of researching local optimization problems or global optimization problems, quadratic programming problems have got extensive attention; it is obvious that researching this kind of problems is very necessary. In this paper, we consider the following quadratic programming problems with quadratic constraints: min𝑓⁰(𝑥)=𝑥^𝑇𝑄⁰𝑥+𝑑^0𝑇𝑥+𝑐₀,s.t.𝑓^𝑖(𝑥)=𝑥^𝑇𝑄^𝑖𝑥+𝑑^𝑖𝑇𝑥+𝑐_𝑖≤0,𝑖=1,2,…,𝑝,𝑥∈𝑆=𝑥∈𝑅^𝑛,()∶𝑙≤𝑥≤𝑢𝑄𝑃 where 𝑄^𝑖=(𝑞^𝑖_𝑗1𝑗2)_𝑛×𝑛 are 𝑛-dimension symmetric matrices, 𝑑^𝑖=(𝑑^𝑖₁,𝑑^𝑖₂,…,𝑑^𝑖_𝑛)^𝑇∈𝑅^𝑛, 𝑙∈𝑅^𝑛,  𝑢∈𝑅^𝑛,  𝑐_𝑖∈𝑅, and  𝑖=0,1,…,𝑝.

In recent years, many researchers have researched this kind of problems and made certain progress. In [1], an effective lower bound of the optimal value of original problem is provided using Lagrange lower estimate, and the local optimal solutions are obtained by Newton methods; then to accelerate the convergence of the global optimal solutions, the local Newton methods are used. A decompose-approach method is put forward in [2]. Literature [3] organically combines the outer approximation method with the branch and bound technique and presents a new branch-reduce algorithm. Literature [4] combines the cutting plane algorithm with the branch and bound algorithm, and puts forwards a new algorithm. Literature [5] presents a branch and bound algorithm by the linear lower function of the bilinear function. Based on [5], literature [6] puts forward a branch-reduce method aiming at objective function and constraint conditions of the linear relaxation programming. A simplex branch and bound algorithm is raised in [7]. There are many different methods for solving quadratic programming problems with quadratic constraints in [8–15].

The rest of this paper is organized as follows. In Section 2, we give the linear relaxation programming problem (𝐿𝑃) of the problem (𝑄𝑃). In Section 3, we give the rectangle subdivision and reduce strategy. We explain the branch and bound algorithm in detail in Section 4, and the convergence of the algorithm is proved. Finally, some numerical results turn out the effectiveness of the present algorithm.

2. Linear Relaxation Programming

In this section, we construct a linear relaxation programming problem of the original problem.

Assume that 𝜆^𝑖_min is the minimum eigenvalue of the matrices 𝑄^𝑖, for 𝑖=1,2,…,𝑝. If 𝜆^𝑖_min≥0, let 𝜃_𝑖=0; otherwise, let 𝜃_𝑖=|𝜆^𝑖_min|+𝜏^𝑖, where 𝜏^𝑖≥0; then 𝑄^𝑖+𝜃_𝑖𝐼 is semipositive definite.

On the rectangle 𝑆^𝑘={𝑥∈𝑅^𝑛∶𝑙^𝑘≤𝑥≤𝑢^𝑘}, for each 𝑖, we construct a linear lower function𝑓^𝑖(𝑥) on 𝑆^𝑘.

We have 𝑓^𝑖(𝑥)=𝑥^𝑇𝑄^𝑖𝑥+𝑑^𝑖𝑇𝑥+𝑐_𝑖=𝑥^𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥+𝑑^𝑖𝑇𝑥+𝑐_𝑖−𝜃_𝑖‖𝑥‖²=𝑥−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥−𝑙^𝑘+𝑑^𝑖𝑇𝑥+𝑐_𝑖−𝜃_𝑖^𝑛_𝑗=1𝑥²_𝑗+2𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑙^𝑘.(1)

Suppose that 𝑙^𝑘_𝑗 and 𝑢^𝑘_𝑗 are the 𝑗th indicators of 𝑙^𝑘 and 𝑢^𝑘, respectively. We know that, for each 𝑗∈{1,2,…,𝑛}, a linear lower function of −𝑥²_𝑗 is −(𝑢^𝑘_𝑗+𝑙^𝑘_𝑗)𝑥_𝑗+𝑢^𝑘_𝑗𝑙^𝑘_𝑗 on the interval [𝑙^𝑘_𝑗,𝑢^𝑘_𝑗]. Therefore, 𝜑_𝑆𝑘(𝑥)≜^𝑛_𝑗=1−𝑢^𝑘_𝑗+𝑙^𝑘_𝑗𝑥_𝑗+𝑢^𝑘_𝑗𝑙^𝑘_𝑗=−𝑙^𝑘+𝑢^𝑘𝑇𝑥+𝑙^𝑘𝑇𝑢^𝑘(2) is a linear lower function of −∑^𝑛_𝑗=1𝑥²_𝑗 on the rectangle [𝑙^𝑘,𝑢^𝑘]; we construct the following linear function: 𝑙^𝑖_𝑆𝑘(𝑥)=𝑎^𝑖_𝑆𝑘^𝑇𝑥+𝑏^𝑖_𝑆𝑘,(3) where 𝑎^𝑖_𝑆𝑘=𝑑^𝑖+2𝑄^𝑖+𝜃_𝑖𝐼𝑙^𝑘−𝜃_𝑖𝑙^𝑘+𝑢^𝑘,𝑏^𝑖_𝑆𝑘=𝑐_𝑖−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑙^𝑘+𝜃_𝑖𝑙^𝑘𝑇𝑢^𝑘.(4)

We can obtain the following two theorems.

Theorem 1. For each 𝑖∈{0,1,…,𝑝}, let 𝑄^𝑖+𝜃_𝑖𝐼 be semipositive definite. For each 𝑖∈{0,1,…,𝑝}, the linear function 𝑙^𝑖_𝑆𝑘(𝑥) is a lower function of 𝑓^𝑖(𝑥) on the rectangle 𝑆^𝑘; that is 𝑓^𝑖(𝑥)≥𝑙^𝑖_𝑆𝑘(𝑥), for all 𝑥∈𝑆^𝑘.

Proof. From the formula (1) and the definitions of the functions 𝜑_𝑆𝑘(𝑥) and 𝑙^𝑖_𝑆𝑘(𝑥), for each 𝑖∈{0,1,…,𝑝}, we have 𝑓^𝑖(𝑥)≥𝑥−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥−𝑙^𝑘+𝑙^𝑖_𝑆𝑘(𝑥),∀𝑥∈𝑆^𝑘.(5) Moreover, the matrix 𝑄^𝑖+𝜃_𝑖𝐼 is semipositive definite; then, 𝑥−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥−𝑙^𝑘≥0,∀𝑥∈𝑆^𝑘.(6) Consequently, 𝑓^𝑖(𝑥)≥𝑙^𝑖_𝑆𝑘(𝑥), for all 𝑥∈𝑆^𝑘,  𝑖∈{0,1,…,𝑝}.

Theorem 2. Assume that 𝜌(𝑄^𝑖+𝜃_𝑖𝐼) is the spectral radius of the rectangle 𝑄^𝑖+𝜃_𝑖𝐼; then max𝑓^𝑖(𝑥)−𝑙^𝑖_𝑆𝑘(𝑥)∶𝑥∈𝑆^𝑘≤𝜌𝑄^𝑖+𝜃_𝑖𝐼+𝜃_𝑖𝑢^𝑘−𝑙^𝑘2,𝑖∈0,1,…,𝑝.(7)

Proof. From the formula (1) and the definitions of the functions 𝜑_𝑆𝑘(𝑥) and 𝑙^𝑖_𝑆𝑘(𝑥), we have 𝑓^𝑖(𝑥)−𝑙^𝑖_𝑆𝑘(𝑥)=𝑥−𝑙^𝑘𝑇𝑄^𝑖+𝜃_𝑖𝐼𝑥−𝑙^𝑘+𝜃_𝑖−‖𝑥‖²−𝜑_𝑆𝑘(𝑥)≤𝜌𝑄^𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑙^𝑘2+𝜃_𝑖𝑥−𝑙^𝑘𝑇𝑢^𝑘𝑄−𝑥≤𝜌^𝑖+𝜃_𝑖𝐼𝑢^𝑘−𝑙^𝑘2+𝜃_𝑖𝑢^𝑘−𝑙^𝑘2=𝜌𝑄^𝑖+𝜃_𝑖𝐼+𝜃_𝑖𝑢^𝑘−𝑙^𝑘2.(8) Hence, the conclusion is established.

Therefore, from Theorem 1, we obtain the linear relaxation programming problem of (𝑄𝑃) on the rectangle 𝑆^𝑘: min𝑙⁰_𝑆𝑘(𝑥)s.t.𝑙^𝑖_𝑆𝑘(𝑥)≤0,𝑖=1,2,…,𝑝,𝑥∈𝑆^𝑘.𝐿𝑃𝑆^𝑘 Solving the problem 𝐿𝑃𝑆^𝑘, its optimal value is obtained, which is a lower bound of the global optimum of the problem (𝑄𝑃) on the rectangle 𝑆^𝑘.

3. The Subdivision and Reduction of the Rectangle

In this section, we give the bisection and reduction methods of the rectangle. Let 𝑆^𝑘={𝑙^𝑘≤𝑥≤𝑢^𝑘} be a rectangle on 𝑅^𝑛, and 𝑥^𝑘∈𝑆^𝑘.

3.1. The Subdivision of the Rectangle

The method of the subdivision of the rectangle is described as follows.(i)Select the longest edge of the rectangle 𝑆^𝑘; that is 𝑈^𝑘_𝑠−𝐿^𝑘_𝑠=max{𝑈^𝑘_𝑗−𝐿^𝑘_𝑗∶𝑗=1,2,…,𝑛}.(ii)Let 𝑉^𝑘_𝑠=(𝑈^𝑘_𝑠+𝐿^𝑘_𝑠)/2. Then 𝑆^𝑘1=^𝑠−1_𝑗=1𝐿^𝑘_𝑗,𝑈^𝑘_𝑗×𝐿^𝑘_𝑠,𝑉^𝑘_𝑠×^𝑛_𝑗=𝑠+1𝐿^𝑘_𝑗,𝑈^𝑘_𝑗,𝑆^𝑘2=^𝑠−1_𝑗=1𝐿^𝑘_𝑗,𝑈^𝑘_𝑗×𝑉^𝑘_𝑠,𝑈^𝑘_𝑠×^𝑛_𝑗=𝑠+1𝐿^𝑘_𝑗,𝑈^𝑘_𝑗.(9)

3.2. The Reduction of the Rectangle

Based on [8], in order to improve the convergence of the algorithm, we give two pruning methods of problem (𝐿𝑃). For all 𝑆^𝑘={𝑥∈𝑅^𝑛∶𝑙^𝑘≤𝑥≤𝑢^𝑘}⊆𝑆, 𝑆^𝑘_𝑗=[𝑙^𝑘_𝑗,𝑢^𝑘_𝑗], suppose that the objective function of 𝐿𝑃𝑆^𝑘 is 𝜑^𝑘₀(𝑥)=∑^𝑛_𝑗=1𝑐^𝑘_𝑗𝑥_𝑗+𝑐^𝑘₀, the constraint functions are ∑^𝑛_𝑗=1𝑎^𝑘_𝑖𝑗𝑥_𝑗≤𝑏^𝑘_𝑖, and the upper bound of (𝑄𝑃) is denoted by 𝑈𝐵; let 𝑟𝐶^𝑘=^𝑛_𝑗=1min𝑐^𝑘_𝑗𝑙^𝑘_𝑗,𝑐^𝑘_𝑗𝑢^𝑘_𝑗,𝑟𝐿^𝑘_𝑖=^𝑛_𝑗=1min𝑎^𝑘_𝑖𝑗𝑙^𝑘_𝑗,𝑎^𝑘_𝑖𝑗𝑢^𝑘_𝑗,𝑖=1,…,𝑝.(10)

Theorem 3 (see [8]). For any 𝑆^𝑘⊆𝑆, if 𝑟𝐶^𝑘+𝑐^𝑘₀>𝑈𝐵, then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘; otherwise, if 𝑐^𝑘_𝑟>0  (𝑟∈{1,…,𝑛}), then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘=(𝑆^𝑘_𝑗)_𝑛×1; if 𝑐^𝑘_𝑟<0  (𝑟∈{1,…,𝑛}), then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘=(𝑆^𝑘_𝑗)_𝑛×1, where 𝑆^𝑘_𝑗=𝑆^𝑘_𝑗,𝑗≠𝑟,𝑗=1,…,𝑛,𝑈𝐵−𝑐^𝑘₀−𝑟𝐶^𝑘+𝑐^𝑘_𝑗𝑙^𝑘_𝑗𝑐^𝑘_𝑗,𝑢^𝑘_𝑗∩𝑆^𝑘_𝑗𝑆,𝑗=𝑟,_𝑘𝑗=𝑆^𝑘_𝑗𝑙,𝑗≠𝑟,𝑗=1,…,𝑛,_𝑘𝑗,𝑈𝐵−𝑐^𝑘₀−𝑟𝐶^𝑘+𝑐^𝑘_𝑗𝑢^𝑘_𝑗𝑐^𝑘_𝑗∩𝑆^𝑘_𝑗,𝑗=𝑟.(11)

Theorem 4 (see [8]). For any 𝑖=1,…,𝑝, if 𝑟𝐿^𝑘_𝑖>𝑏^𝑘_𝑖, then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘; otherwise, if 𝑎^𝑘_𝑖𝑟>0  (𝑟∈{1,…,𝑛}), then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘=(𝑆^𝑘_𝑗)_𝑛×1; if 𝑎^𝑘_𝑖𝑟<0  (𝑟∈{1,…,𝑛}), then there is no optimal solution of (𝑄𝑃) on 𝑆^𝑘=(𝑆^𝑘_𝑗)_𝑛×1, where 𝑆^𝑘_𝑗=𝑆^𝑘_𝑗𝑏,𝑗≠𝑟,𝑗=1,…,𝑛,_𝑘𝑖−𝑟𝐿^𝑘_𝑖+𝑎^𝑘_𝑖𝑗𝑙^𝑘_𝑗𝑎^𝑘_𝑖𝑗,𝑢^𝑘_𝑗∩𝑆^𝑘_𝑗𝑆,𝑗=𝑟,_𝑘𝑗=𝑆^𝑘_𝑗𝑙,𝑗≠𝑟,𝑗=1,…,𝑛,_𝑘𝑗,𝑏^𝑘_𝑖−𝑟𝐿^𝑘_𝑖+𝑎^𝑘_𝑖𝑗𝑢^𝑘_𝑗𝑎^𝑘_𝑖𝑗∩𝑆^𝑘_𝑗,𝑗=𝑟.(12)

From Theorems 3 and 4, we can construct the following pruning rules to delete or reduce the rectangle 𝑆^𝑘.

Rule 1. Compute 𝑟𝐶^𝑘, if 𝑟𝐶^𝑘+𝑐^𝑘₀>𝑈𝐵, then 𝑆^𝑘 is deleted; otherwise, for any 𝑗=1,…,𝑛.
If 𝑐^𝑘_𝑗>0, let 𝑢^𝑘_𝑗=min{𝑢^𝑘_𝑗,(𝑈𝐵−𝑐^𝑘₀−𝑟𝐶^𝑘+𝑐^𝑘_𝑗𝑙^𝑘_𝑗)/𝑐^𝑘_𝑗}.
If 𝑐^𝑘_𝑗<0, let 𝑙^𝑘_𝑗=max{𝑙^𝑘_𝑗,(𝑈𝐵−𝑐^𝑘₀−𝑟𝐶^𝑘+𝑐^𝑘_𝑗𝑢^𝑘_𝑗)/𝑐^𝑘_𝑗}.

Rule 2. Compute 𝑟𝐿^𝑘_𝑖, if 𝑟𝐿^𝑘_𝑖>𝑏^𝑘_𝑖, then 𝑆^𝑘 is deleted; otherwise, for any 𝑗=1,…,𝑛.
If 𝑎^𝑘_𝑖𝑗>0, let 𝑢^𝑘_𝑗=min{𝑢^𝑘_𝑗,(𝑏^𝑘_𝑖−𝑟𝐿^𝑘_𝑖+𝑎^𝑘_𝑖𝑗𝑙^𝑘_𝑗)/𝑎^𝑘_𝑖𝑗}.
If 𝑎^𝑘_𝑖𝑗<0, let 𝑙^𝑘_𝑗=max{𝑙^𝑘_𝑗,(𝑏^𝑘_𝑖−𝑟𝐿^𝑘_𝑖+𝑎^𝑘_𝑖𝑗𝑢^𝑘_𝑗)/𝑎^𝑘_𝑖𝑗}, where 𝑖=1,…,𝑝.

4. The Algorithm Description and Convergence Analysis

Next, we can describe a branch and bound reduced algorithm of problem (𝑄𝑃) as follows.

Suppose when the iteration proceeds in step 𝑘, the feasible region of the problem (𝑄𝑃) is denoted by 𝐷, 𝑄 represents the feasible set at present, 𝑆^𝑘 represents the divided rectangle soon, the set of remained rectangle after pruning is denoted by 𝑇, and the current lower bound and upper bound of the global optimal value of the problem (𝑄𝑃) are denoted by 𝛼_𝑘 and 𝛽_𝑘, respectively.

Step 1 (initializing). Set 𝜀>0, and let 𝑇={𝑆},  𝑘=1,  𝑆^𝑘=𝑆, and 𝛽_𝑘=∞. Solving the problem 𝐿𝑃𝑆^𝑘, its optimal solution and optimal value are denoted by 𝑥^𝑘 and 𝛽(𝑆^𝑘), respectively. Let 𝛽_𝑘=𝛽(𝑆^𝑘); then 𝛽_𝑘 is a lower bound of global optimal value of the problem (𝑄𝑃); if 𝑥^𝑘∈𝐷, let 𝑄=𝑄∪{𝑥^𝑘}, the upper bound is 𝑄=𝑄∪{𝑥^𝑘}, and find a current optimal solution 𝑥^∗∈argmin𝛼_𝑘.

Step 2 (termination rule). If there was a condition satisfying between 𝛼_𝑘−𝛽_𝑘≤𝜀  (𝑘=1,2,…) or 𝑇=∅, then stop; the global optimal solution 𝑥^∗ and the global optimal value 𝑓⁰(𝑥^∗) are outputted; otherwise, go to the next step.

Step 3 (selection rule). Select a rectangle which has a minimum lower bound in the rectangle set 𝑇; that is, 𝑆^𝑘=argmin𝛽_𝑘.

Step 4 (subdivision rule). Using the subdivision method in the former section, then the rectangle 𝑆^𝑘 can be divided into subrectangles 𝑆^𝑘1 and 𝑆^𝑘2, and int𝑆^𝑘1∩int𝑆^𝑘2=∅.

Step 5 (reduction technique). Reducing the subrectangles after dividing using the reduction method in the former section, without loss of generality, the new rectangles after reduction are also denoted by 𝑆^𝑘𝑗,  𝑗∈Γ, where Γ is the index set of the rectangles after reduction.

Step 6 (bounding rule). Lower bound is 𝛽^∗_𝑘=min{𝛽_𝑘∶𝑘=1,2,…}; upper bound is 𝛼^∗_𝑘=min{𝑓⁰(𝑥),𝑥∈𝑄}.
The current best feasible solution is 𝑥^∗∈argmin{𝑓⁰(𝑥)∶𝑥∈𝑄}.

Step 7 (pruning rule). Let 𝑇=𝑇⧵{𝑆∶𝛽_𝑘(𝑆)≥𝛼^∗_𝑘,𝑆∈𝑇}.

Step 8. Set 𝑘=𝑘+1; go to Step 2.

Theorem 5. (a) If the algorithm terminates in limited steps, then 𝑥^𝑘 is a 𝜀-global optimal solution of problem (𝑄𝑃).
(b) For each 𝑘≥1, let 𝑥^𝑘 be the solution after step 𝑘. If the algorithm is infinite, then {𝑥^𝑘} is a feasible solution sequence of problem (𝑄𝑃), and any accumulation is a global optimal solution of problem (𝑄𝑃), and lim_𝑘→∞𝛼_𝑘=lim_𝑘→∞𝛽_𝑘=𝜈.

Proof. (a) If the algorithm is finite, suppose that it terminates in step 𝑘  (𝑘≥1). Because 𝑥^𝑘 is obtained by solving 𝐿𝑃𝑆^𝑘, then 𝑥^𝑘∈𝑆^𝑘⊆𝑆, and 𝑥^𝑘 is a feasible solution of problem (𝑄𝑃). When 𝛼_𝑘−𝛽_𝑘≤𝜀, the algorithm terminate. From Steps 1 and 6, we have 𝑓⁰(𝑥^𝑘)−𝛽_𝑘≤𝜀; from the algorithm, 𝛽_𝑘≤𝑣, where 𝑣 is the global optimal value of problem (𝑄𝑃). Because 𝑥^𝑘 is a feasible solution of problem (𝑄𝑃), so 𝑓⁰(𝑥^𝑘)≥𝑣. Thus 𝑣≤𝑓⁰𝑥^𝑘≤𝑣+𝜀.(13) Therefore, 𝑥^𝑘 is a 𝜀-global optimal solution of problem (𝑄𝑃).
(b) If the algorithm is infinite, then it produces a solution sequence {𝑥^𝑘} of problem (𝑄𝑃), where for each 𝑘≥1,  𝑥^𝑘 is obtained by solving problem 𝐿𝑃𝑆^𝑘. For each 𝑆^𝑘⊆𝑆, for the optimal solution 𝑥^𝑘∈𝑆^𝑘⊆𝑆, the sequence {𝑥^𝑘} constitute a solution sequence of problem (𝑄𝑃); from the iteration of the algorithm, we have 𝛽_𝑘≤𝑣≤𝛼_𝑘=𝑓⁰𝑥^𝑘,𝑘=1,2,….(14) Because the series {𝛽_𝑘} do not decrease and have an upper bound, and {𝛼_𝑘} do not increase and have a lower bound, then the series {𝛽_𝑘} and {𝛼_𝑘} are both convergent. Taking the limits on both sides of (14), we have lim_𝑘→∞𝛽_𝑘≤𝑣≤lim_𝑘→∞𝛼_𝑘=lim_𝑘→∞𝑓⁰𝑥^𝑘.(15) Let lim_𝑘→∞𝛽_𝑘=𝛽,  lim_𝑘→∞𝛼_𝑘=𝛼; then the formula (15) converts into 𝛽≤𝑣≤lim_𝑘→∞𝑓⁰𝑥^𝑘=𝛼.(16) Without loss of generality, assume that the sequence of rectangle {𝑆^𝑘=[𝑙^𝑘,𝑢^𝑘]} satisfy 𝑥^𝑘∈𝑆^𝑘 and 𝑆^𝑘+1⊂𝑆^𝑘. In our algorithm, the rectangles are divided into two equal parts continuously; then ⋂^∞_𝑘=1𝑆^𝑘+1={𝑥^𝑘}, and because of the continuity of function 𝑓⁰(𝑥), 𝛽=𝑣=𝛼=lim_𝑘→∞𝑓⁰𝑥^𝑘=𝑓⁰𝑥^∗.(17) So any accumulation of {𝑥^𝑘} is a global optimal solution of problem (𝑄𝑃).

5. Numerical Experiment

Several experiments are given to turn out the feasibility and effectiveness of our algorithm.

Example 1. min𝑥²₁+𝑥²₂s.t.0.3𝑥₁𝑥₂≥1,2≤𝑥₁≤5,1≤𝑥₂≤3.(18)
From the algorithm, the initial rectangle is 𝑆¹=^2.00005.0000_1.00003.0000; first, we solve the problem 𝐿𝑃(𝑆¹), its optimal solution is 𝑥¹=(2.0000;3.0000), and optimal value is 𝛽₁=𝛽(𝑆¹)=4.9996; then 4.9996 is a lower bound of the global optimal value of problem (𝑄𝑃). Because 𝑥¹=(2.0000;3.0000) is feasible, then 𝑄=[2.0000;3.0000] is a set of current feasible solutions, and the current upper bound is 𝛼₁=𝑓⁰(𝑥¹)=13.0000; the current optimal solution is 𝑥^∗=𝑥¹=(2.0000;3.0000).
After that, based on our selection rule, select the rectangle with the minimum lower bound 𝑆¹ to divide; then 𝑆¹ is divided into two subrectangles 𝑆^1,1=^2.00003.5000_1.00003.0000 and 𝑆^1,2=^3.50005.0000_1.00003.0000 from the dividing method in Section 3.1; then reduce the rectangles using the reduction technique in Section 3.2, and the new rectangle after reduction is denoted by 𝑆²=𝑆^1,1=^2.00003.5000_1.00003.0000. Solving the linear relaxation programming problem 𝐿𝑃 on the rectangle 𝑆², its optimal value is 𝛽₂=𝛽(𝑆^1,1)=4.9996; then the lower bound of the original problem is not updated, also being 4.9996. Next, we choose 𝑆² to divide, until ⋯ the 15th iteration, 𝑆¹⁴=^2.00002.0408_1.65381.7019; solve the linear relaxation programming problem 𝐿𝑃(𝑆¹⁴), its optimal solution is (2.0000;1.6665), and optimal value is 6.7765; while, the current upper bound is 6.8151, the current optimal solution is (2.0000;1.6778). Because |6.8151−6.7765|<0.1, it satisfies our termination rule; then the optimal value of the original problem is 6.8151, the lower bound of the optimal value is 6.7765, and the optimal solution is 𝑥=(2.0000;1.6778); here the lower bound of the optimal value is also approximate optimal value, where the accuracy is 𝜀=0.1.
Table 2 shows the different results of Example 1 under different accuracy.

Example 2. min6𝑥²₁+4𝑥²₂+5𝑥₁𝑥₂s.t.−6𝑥₁𝑥₂≤−48,0≤𝑥₁,𝑥₂≤10.(19) The optimal value is 118.3838.

Example 3. min𝑥²₁+𝑥²₂s.t.−0.3𝑥₁𝑥₂≤−1,−𝑥₁−𝑥₂≤1,𝑥∈𝑋⁰=2≤𝑥₁≤5,1≤𝑥₂≤3.(20) The optimal value is 6.7778.

Example 4. min𝑥₁1s.t.4𝑥₁+12𝑥₂−16𝑥²₂−16𝑥²₁1≤1,𝑥14²¹+1𝑥14²²−37𝑥₁−37𝑥₂≤1,1≤𝑥₁≤5.5,1≤𝑥₂≤5.5.(21) The optimal value is 1.0000.

Example 5. min−𝑥₁+𝑥₁𝑥₂^0.5−𝑥₂s.t.−6𝑥₁+8𝑥₂≤3,3𝑥₁−𝑥₂≤3,𝑥∈𝑋⁰=𝑥∣0≤𝑥_𝑖≤1.5,𝑖=1,2.(22) The optimal value is −1.1629.

Example 6. min6𝑥²₁+4𝑥²₂+2.5𝑥₁+𝑥₂²−2.510𝑥₁+10𝑥₂s.t.3𝑥₁−𝑥₂²−310𝑥₁+10𝑥₂≤−48,0≤𝑥₁,𝑥₂≤10.(23) The optimal value is −31.8878.

Example 7. min21𝑥²₁+34𝑥₁𝑥₂−24𝑥²₂+2𝑥₁−14𝑥₂s.t.2𝑥²₁+4𝑥₁𝑥₂+2𝑥²₂+8𝑥₁+6𝑥₂−9≤0,−5𝑥²₁−8𝑥₁𝑥₂−5𝑥²₂−4𝑥₁+4𝑥₂𝑥+4≤0,₁+2𝑥₂≤2,𝑥∈[0,1]².(24) The optimal value is −3.3205.

Example 8. min5.3578𝑥²₃+0.8357𝑥₁𝑥₅+37.2392𝑥₁s.t.2.584×10⁻⁵𝑥₃𝑥₅−6.663×10⁻³𝑥₂𝑥₅−7.34×10⁻⁵𝑥₁𝑥₄≤1,8.53007×10⁻⁴𝑥₂𝑥₅+9.395×10⁻⁵𝑥₁𝑥₄−3.3.85×10⁻⁴𝑥₃𝑥₅≤1,−𝑥₂𝑥₅−0.42𝑥₁𝑥₂−0.30586𝑥²₃≤−1.3303294×10³,−𝑥₃𝑥₅−0.2668𝑥₁𝑥₃−0.40584𝑥₃𝑥₄≤−2.2751327×10³,2.4186×10⁻⁴𝑥₂𝑥₅+1.0159×10⁻⁴𝑥₁𝑥₂+7.379×10⁻⁵𝑥²₃≤1,2.9955×10⁻⁴𝑥₃𝑥₅+7.992×10⁻⁵𝑥₁𝑥₃+1.2157×10⁻⁴𝑥₃𝑥₄≤1,𝑥∈𝑋⁰=𝑥∣78≤𝑥₁≤102,33≤𝑥₂≤45,27≤𝑥_𝑖≤45,𝑖=3,4,5.(25) The optimal value is 1.0128×10⁴.

We choose 𝜀=1.0𝜀−4; then the approximate optimal value satisfying accuracy and the CPU running time are obtained; the results are shown in Table 1.

Table 1 

Table 2 

Different results of Example 1 under different accuracy.

6. Conclusion

In this paper, we presented a branch and bound reduced algorithm for solving the quadratic programming problems with quadratic constraints. By constructing a linear relaxation programming problem, the lower bound of the optimal value of original problem can be obtained. Meanwhile, we used a rectangle reduction technique to improve the degree of approximation and the convergence rate of acceleration. Numerical experiments show the effectiveness of our algorithm.

Acknowledgment

The work is supported by the Foundation of National Natural Science of China under Grant no. 11161001.