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

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 smalland medium-sized problems.


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: min  0 () =    0  + ( 0 ) +  0 , s.t.  () =      + (  )   +   ≤ 0,  = 1, 2, . . ., ,  ∈  = { ∈   :  ≤  ≤ } , (𝑄𝑃) where   = (   1  2 ) × are -dimension symmetric matrices,   = (  1 ,   2 , . . .,    )  ∈   ,  ∈   ,  ∈   ,   ∈ , and  = 0, 1, . . ., .
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][9][10][11][12][13][14][15].
The rest of this paper is organized as follows.In Section 2, we give the linear relaxation programming problem () of the problem ().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.

Linear Relaxation Programming
In this section, we construct a linear relaxation programming problem of the original problem.
We have Suppose that    and    are the th indicators of   and   , respectively.We know that, for each  ∈ {1, 2, . . ., }, a linear lower function of − where We can obtain the following two theorems.
Proof.From the formula (1) and the definitions of the functions    () and     (), we have Hence, the conclusion is established.
Therefore, from Theorem 1, we obtain the linear relaxation programming problem of () on the rectangle   : Solving the problem ( (  )), its optimal value is obtained, which is a lower bound of the global optimum of the problem () on the rectangle   .

The Subdivision and Reduction of the Rectangle
In this section, we give the bisection and reduction methods of the rectangle.Let   = {  ≤  ≤   } be a rectangle on   , and   ∈   .

The Subdivision of the Rectangle.
The method of the subdivision of the rectangle is described as follows.

The Algorithm Description and Convergence Analysis
Next, we can describe a branch and bound reduced algorithm of problem () as follows.
Suppose when the iteration proceeds in step , the feasible region of the problem () is denoted by ,  represents the feasible set at present,   represents the divided rectangle soon, the set of remained rectangle after pruning is denoted by , and the current lower bound and upper bound of the global optimal value of the problem () are denoted by   and   , respectively.
Step 3 (selection rule).Select a rectangle which has a minimum lower bound in the rectangle set ; that is,   = arg min   .
Step 4 (subdivision rule).Using the subdivision method in the former section, then the rectangle   can be divided into subrectangles  1 and  2 , and int  1 ∩ int  2 = 0.
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   ,  ∈ Γ, where Γ is the index set of the rectangles after reduction.
(b) If the algorithm is infinite, then it produces a solution sequence {  } of problem (), where for each  ≥ 1,   is obtained by solving problem ( (  )).For each   ⊆ , for the optimal solution   ∈   ⊆ , the sequence {  } constitute a solution sequence of problem (); from the iteration of the algorithm, we have Because the series {  } do not decrease and have an upper bound, and {  } do not increase and have a lower bound, then the series {  } and {  } are both convergent.Taking the limits on both sides of ( 14), we have lim Let lim  → ∞   = , lim  → ∞   = ; then the formula (15) converts into Without loss of generality, assume that the sequence of rectangle {  = [  ,   ]} satisfy   ∈   and  +1 ⊂   .
In our algorithm, the rectangles are divided into two equal parts continuously; then ⋂ ∞ =1  +1 = {  }, and because of the continuity of function  0 (), So any accumulation of {  } is a global optimal solution of problem ().

Numerical Experiment
Several experiments are given to turn out the feasibility and effectiveness of our algorithm.
After that, based on our selection rule, select the rectangle with the minimum lower bound  1 to divide; then  1 is divided into two subrectangles  1,1 = [ 2.0000 3.5000 1.0000 3.0000 ] and  1,2 = [ 3.5000 5.0000 1.0000 3.0000 ] 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  2 =  1,1 = [ 2.0000 3.5000 1.0000 3.0000 ].Solving the linear relaxation programming problem  on the rectangle  2 , its optimal value is  2 = ( 1,1 ) = 4.9996; then the lower bound of the original problem is not updated, also being 4.9996.Next, we choose  2 to divide, until ⋅ ⋅ ⋅ the 15th iteration,  14 = [ 2.0000 2.0408 1.6538 1.7019 ]; solve the linear relaxation programming problem ( 14 ), 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  = (2.0000;1.6778); here the lower bound of the optimal value is also approximate optimal value, where the accuracy is  = 0.1.
Table 2 shows the different results of Example 1 under different accuracy.
The optimal value is 118.3838.
The optimal value is 6.7778.
The optimal value is 1.0000.
The optimal value is −1.1629.
The optimal value is −31.8878.
We choose  = 1.0 − 4; then the approximate optimal value satisfying accuracy and the CPU running time are obtained; the results are shown in Table 1.

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.
8. Set  =  + 1; go to Step 2. If the algorithm terminates in limited steps, then   is a -global optimal solution of problem ().(b) For each  ≥ 1, let   be the solution after step .If the algorithm is infinite, then {  } is a feasible solution sequence of problem (), and any accumulation is a global optimal solution of problem (), and lim  → ∞

Table 1 Example
The optimal solution within accuracy or one solution among solutions

Table 2 :
Different results of Example 1 under different accuracy.