An Out Space Accelerating Algorithm for Generalized Affine Multiplicative Programs Problem

This paper presents an out space branch-and-bound algorithm for solving generalized affine multiplicative programs problem. Firstly, by introducing new variables and constraints, we transform the original problem into an equivalent nonconvex programs problem. Secondly, by utilizing new linear relaxation technique, we establish the linear relaxation programs problem of the equivalent problem.Thirdly, based on the out space partition and the linear relaxation programs problem, we construct an out space branch-and-bound algorithm. Fourthly, to improve the computational efficiency of the algorithm, an out space reduction operation is employed as an accelerating device for deleting a large part of the investigated out space region. Finally, the global convergence of the algorithm is proved, and numerical results demonstrate the feasibility and effectiveness of the proposed algorithm.


Introduction
The generalized affine multiplicative programs problem (GAMP) arises naturally in many practical applications including management science, engineering optimization design, optimal control, Euclidean geometry, economic planning, production planning, and combinatorial mathematics [1,2].The problem (GAMP) usually contains many local optimum points which are not global optimum solution, so that important theoretical difficulties and computational complexities exist.Therefore, it is very necessary to establish an effective algorithm for globally solving the problem (GAMP).

Equivalent Problem and Its Linear Relaxation
In this section, we will convert the problem (GAMP) into an equivalent nonconvex problem (EQ).Since ∑  =1     +   and ∑  =1     +   are all finite affine functions defined in , there exist positive constant numbers   and   such that ∑  =1     +   +   > 0 and ∑  =1     +   +   > 0 for any  ∈ .Therefore, in the following, without loss of generality, we can assume that ∑  =1     +   ≥ 0 and ∑  =1     +   ≥ 0 for all  ∈ .
Without loss of generality, we let (EQ) Obviously, if ( * ,  * ) is a global optimal solution for (EQ), then  * is a global optimal solution for (GAMP), and  *  = ∑  =1    *  +   ,  = 1, . . ., .If  * is a global optimal solution for (GAMP), then ( * ,  * ) is a global optimal solution for (EQ), where  *  = ∑  =1    *  +   ,  = 1, . . ., .In the following, the main computation is to globally solve (EQ).Next, we will construct the linear relaxation programs problem of (EQ), which can provide the upper bounds for (EQ) in the proposed branch-and-bound algorithm.Assume that  = { ∈   |   ≤   ≤   ,  = 1, . . ., } ⊆  0 .And without loss of generality, let ( Then the linear relaxation programs (  ) of (EQ) in  can be given as follows: where (EQ) in  is defined by Since (, ) is a global optimal solution for (EQ  ), it follows that Since  is a feasible solution for (  ), By the above discussions, we have we have the following conclusions.
If (  ) is infeasible, then we can easily prove that (  ) is also infeasible; the conclusion is obvious.
If (  ) is feasible, since  is compact set, then   is finite.If (  ) is infeasible, then   = −∞ <   .Thus, in the following, we assume that (  ) is feasible, then   is finite, and there exists a point  ∈  such that   ≤ ∑  =1     +   ,  = 1, . . ., .And Since  is a feasible solution of (  ), we have By the above discussions, we have   ≥   , and the proof is completed.
By Theorem 1, (  ) can provide the valid upper bound for (EQ  ).

Algorithm and Its Convergence
In this section, we shall present a branch-and-bound algorithm for solving (EQ).The critical operation in guaranteeing the global convergence of the proposed algorithm is the selection of a suitable branching technique.In this paper, we choose a simple bisection method.For any selected rectangle Ẑ = { ∈   | L ≤   ≤ Û ,  = 1, . . ., } ⊆  0 , this branching rule is given as follows: (1) Let Û − L = max{ Û − L |  = 1, . . ., }.
(2) Let 3.1.New Region Reduction Operation.For any rectangle   ⊆  0 , during the process of iteration, we want to recognize whether or not   contains a global optimal solution of (EQ).
The proposed new reduction operation aims at replacing the rectangle   = [  ,   ] with a smaller rectangle   = [  ,   ] without deleting any global optimal solution of (EQ).
Without loss of generality, we can assume that   is a known lower bound of the optimal value for EQ( 0 ) and that V() is the maximum value of (, ) over  and , and let Theorem 2. For any subrectangle   = (   ) ×1 = [   ,     ] ×1 ⊆  0 , the following conclusions hold: (i) If   <   , then there is no global optimal solution of EQ( 0 ) over   .
(ii) If   ≥   , then, for each  ∈ {1, 2, . . ., }, there is no global optimal solution of the EQ( 0 ) over   , where Proof.(i) If   <   , then we have Therefore, there exists no global optimal solution of the problem EQ( 0 ) over   .(ii) If  k ≥   , then we have the following conclusions.
By Theorem 2, we can construct a new reduction operation to cut away a part of region in which the global optimal solution of (EQ) does not exist.Assuming that a subrectangle will be compressed, then, according to Theorem 2, the original rectangle   should be replaced with a new subrectangle: Let (  ) be the optimal value of (  ) in   ; the basic steps of the proposed algorithm are summarized as follows.
If  0 −  0 ≤ , then the algorithm stops with (  ,   ) and   as global optimal solutions of (EQ) and (GAMP), respectively.Otherwise, proceed to next step.
Step 1. Use the proposed reduction operation to compress the investigated rectangle   , and still represent remaining part of the rectangle as   .
Step 2. Subdivide   into two subrectangles; denote by Θ  its partitioning set.

Convergence of the Algorithm.
In this subsection, the convergence of the algorithm is given.Theorem 3. If the algorithm is finite, upon termination, we can obtain the -global optimal solutions for the problems (EQ) and (GAMP), respectively.If the algorithm is infinite, for each  ≥ 0, denote by   the incumbent solution of (GAMP), which is found at the end of iteration .Then, lim →∞   =  * will a global optimal solution for (GAMP).
Proof.If the algorithm is finite, then the algorithm terminates at iteration ,  ≥ 0. By the termination condition of the algorithm and the updating method of the upper bound, it can follow easily that the conclusion is obvious.
If the algorithm is infinite, then, it generates a sequence of incumbent solutions for problem (EQ), which may be denoted by {(  ,   )}.For each  ≥ 1, (  ,   ) is found by solving problem (   ), for some rectangle   ⊆  0 , for an optimal solution   ∈ , and setting    = ∑  =1      +   ,  = 1, . . ., .Therefore, the sequence {  } consists of feasible solutions for problem (GAMP).Let  * be an accumulation point of {  }, and without loss of generality we assume that lim →∞   =  * .Then, since  is a compact set,  * ∈ .Furthermore, since {  } is infinite, we may assume without loss of generality that, for each ,  +1 ⊆   .Since the rectangles   ,  ≥ 1, are formed by rectangular bisection, this implies that, for some point  * ∈   , we have lim Let  * = { * } and, for each , let   be given by For each , from Step 2 of the algorithm, Since  +1 ⊂   ⊆  0 , for each , by Theorem 1 and Step 4, this implies that {(  )} is a nonincreasing sequence bounded by V. Therefore, lim →∞ (  ) is a finite number and satisfies For each , from Step 1, (  ) is equal to the optimal value of problem (   ) given by max From ( 22) and ( 23), since (  ) =   , we have lim →∞   = V = ( * ,  * ).Therefore, ( * ,  * ) is a global optimal solution for (EQ).By Theorem 1, this implies that  * is a global optimal solution for (GAMP).The proof is complete.

Numerical Experiments
To verify the performance of the proposed algorithm, some test problems are implemented on an Intel(R) Core(TM)2 Duo CPU (1.58 GHZ) microcomputer.The algorithm is coded in C++, and each linear programs problem is solved by simplex method in our experiments.These test problems are given in the following, and their numerical results are listed in Tables 1 and 2.
Example 1 (see [14]).One has In Table 1, the notations have been used for column headers: iteration, number of iterations; time: execution time(s) of the algorithm in seconds.
Numerical results for Example 3 are given in Table 2, where the convergence tolerance is set to  = 10 −6 .
From Tables 1 and 2, it is seen that the proposed algorithm in this paper can be used to globally solve the problem (GAMP) with the effectiveness and robustness.

Concluding Remarks
In this article, an out space accelerating branch-and-bound algorithm is presented for globally solving the generalized affine multiplicative programs problem (GAMP).First of all, we transform the (GAMP) into an equivalent problem (EQ).By utilizing the new linear relaxation technique, we systematically convert (EQ) into a series of linear relaxation programs problems, which can infinitely approximate the global optimal solution of (EQ) by a successive partition.To improve the computational efficiency of the algorithm, we introduce an out space reduction operation, which offers a theoretical possibility of reducing a large part of the investigated out space region and which can be seen as an accelerating device for improving the convergent speed of the proposed algorithm.By combining the established linear relaxed programs problem with the new reduction operation in a branch-and-bound framework, we design an out space accelerating algorithm for effectively solving (GAMP).By subsequently dividing the initial rectangle and subsequently solving a series of linear relaxed programs problems, the presented algorithm is convergent to the global minimum of (GAMP).Compared with the known algorithms, numerical results demonstrate that the proposed algorithm has higher computational efficiency.

Table 1 :
Computational results of Examples 1 and 2.

Table 2 :
Numerical results for Example 3.