IMAGE SPACE BRANCH-REDUCTION-BOUND ALGORITHM FOR GLOBALLY MINIMIZING A CLASS OF MULTIPLICATIVE PROBLEMS

. This paper presents an image space branch-reduction-bound algorithm for solving a class of multiplicative problems (MP). First of all, by introducing auxiliary variables and taking the logarithm of the objective function, an equivalent problem (EP) of the problem (MP) is obtained. Next, by using a new linear relaxation technique, the parametric linear relaxation programming (PLRP) of the equivalence problem (EP) can be established for acquiring the lower bound of the optimal value to the problem (EP). Based on the characteristics of the objective function of the equivalent problem and the structure of the branch-and-bound algorithm, some region reduction techniques are constructed for improving the convergence speed of the algorithm. Finally, the global convergence of the algorithm is proved and its computational complexity is estimated, and numerical experiments are reported to indicate the higher computational performance of the algorithm.


Introduction
This paper considers the following a class of multiplicative problems: (MP) : where   ,   , and   ,  = 1, . . ., ,  = 1, . . ., , are all arbitrary real numbers;  ∈ R × ,  ∈ R  ;  is a nonempty bounded set, and for any  ∈ , ∑︀  =1     +   > 0,  = 1, 2, . . ., .The problem (MP) and its special form are worth studying, and which have attracted much attention from scholars.The first reason is that the problem (MP) and its special form have a wide of practical applications, such as network flows [3,4], VLISI chip design [6], robust optimization [26], decision tree optimization [1], financial optimization [19,24], and so on [10,14,15].The second reason is that the problem (MP) and its special form are all non-convex programming problem, which generally contain multiple local optimal solutions that are not global optimal, so that there are many computational difficulties.
In the past several decades, some algorithms have been proposed for solving such problems, which can be classified into the following categories: branch-and-bound algorithms [7,9,12,23,25,29,[35][36][37][38], level set algorithm [22], outer-approximation [21], cutting-plane method [2], finite algorithms [20,27], etc.In recent years, Shen et al. [30] proposed an outer space branch-and-bound algorithm for linear multiplicative programming problem by combining the linear relaxation method with the rectangular branching technique and the outer space region reduction technique; by using the decomposability of the problem, Shen and Huang [28] proposed a decomposition branch-and-bound algorithm for linear multiplicative problem; Jiao et al. [13] presented an efficient outer space branch-and-bound algorithm for generalized linear multiplicative programming problem based on the outer space search and the branch-and-bound framework; Zhang et al. [39] presented a new relaxation bounding method based on the search of the output space; based on the characteristics of the initial problem, Shen et al. [31] proposed a branch-and-bound algorithm for globally solving the linear multiplicative problem.Zhang et al. [40] proposed an efficient polynomial time algorithm for a class of generalized linear multiplicative programs with positive exponents by utilizing a new two-stage acceleration technique; Jiao and Shang [11] gave a two-Level linear relaxation method for generalized linear fractional programming problem, which includes linear multiplicative problem; by using new affine relaxed technique, Jiao et al. [16] formulated a novel branch-and-bound algorithm for solving generalized polynomial problem.However, although many scholars have proposed some algorithms, these algorithms either can only solve a certain special form of the problem (MP), or it is difficult to solve the problem (MP) with large-size variables.Therefore, it is still very necessary to propose a new effective algorithm for globally solving the general form of the problem (MP).
In this paper, an image space branch-reduction-bound algorithm is proposed for globally solving the problem (MP).In the algorithm, first of all, by introducing new auxiliary variables and taking the logarithm of the objective function, an equivalent problem (EP) of the problem (MP) is constructed.Next, by using the firstorder differential median theorem, we present a new linear relaxation technique, which can be used to construct the parametric linear relaxation programming problem of the problem (EP).By subsequently refining the initial image space rectangle, and by solving a series of parametric linear relaxation programming problems, the optimal solution of the parametric linear relaxation programming problem (PLRP) can infinitely approximate the optimal solution of the problem (EP).Finally, the numerical results show that the proposed algorithm can effectively find the globally optimal solutions of all test instances with the given tolerance .
Compared with the known methods, the main advantages of the proposed algorithm are given as follows.Firstly, the branching search takes place in the image space R  of the affine function ∑︀  =1     +   , rather than in the variable dimension space R  , since  usually far exceeds , which mitigates the required computational efforts.Secondly, some image region reduction techniques are proposed based on the characteristics of the objective function of the equivalent problem (EP) and the structure of the branch-and-bound algorithm, which provide a possibility to compress the investigated region, so that we can improve the computational efficiency of the algorithm.Thirdly, by analysing the computational complexity of the algorithm, we estimate the maximum iterations of the proposed algorithm for the first time.Finally, numerical comparison demonstrates the superiority of the algorithm.
The paper is organized as follows.In Section 2, the equivalence problem (EP) of the problem (MP) and its parametric linear relaxation programming problem (PLRP) are established.In Section 3, some image region reduction techniques are presented for improving the convergence speed of the algorithm.In Section 4, we propose an image space branch-reduction-bound algorithm, the global convergence of the algorithm is proved, and the complexity of the algorithm is analyzed.Numerical comparisons of some test examples in recent literatures are reported in Section 5. Finally, some conclusions are presented in Section 6.

Equivalent problem and its parametric linear relaxation programming
In order to globally solve the problem (MP), we first convert the problem (MP) into an equivalent problem (EP).Without losing generality, for each  = 1, 2, . . ., , let  )︀ is a global optimal solution of the problem (EP).
In the following, for globally solving the problem (MP), we can solve its equivalent problem (EP) instead.Next, to design the branch-and-bound algorithm for solving the problem (EP), we need to construct the parametric linear relaxation programming of the problem (EP), which can provide a reliable lower bound for the global optimal value of the problem (EP).
Without losing generality, for any Therefore, by the above discussions, and introduce parameters   ,  = 1, 2, . . ., , we can construct the parametric linear relaxation programming problem (PLRP) of the problem (EP) as follows: (PLRP) : It is obvious that the optimal value of the parametric linear relaxation programming problem (PLRP) is less than or equal to the optimal value of the problem (EP).

Image space region reduction techniques
In order to improve the convergence speed of the algorithm, we propose some image space region reduction techniques, which provide a possibility to reduce the current rectangle to a smaller rectangle or delete the rectangle without losing the global optimal solution of the problem (EP).Thus, without losing generality, we assume that UB is a best currently known upper bound of the global optimal value of the problem (EP), for any The specific region reduction process is given by the following theorem.)︁ ≤   , then there exists no globally optimal solution of the problem (EP) over ̂︀ Λ 2 , where Therefore, there exists no globally optimal solution of the problem (EP) over Λ.
since   > 0, and taking the logarithm of both sides of the above formula, we can follow that Also because we can follow that min Thus, there exists no globally optimal solution of the problem (EP) over ̂︀ Λ 1 .
For any )︁ , Thus, we can obtain since   < 0, and taking the logarithm of both sides of the above formula, we can follow that Also because ln   , Thus, we can follow that min Thus, there exists no globally optimal solution of the problem (EP) over ̂︀ Λ 2 , and the proof is complete.
According to Theorem 3.1, we can construct some image space region reduction techniques, which can provide the possibility for deleting the whole investigated rectangle Λ or part of it, in which there is no globally optimal solution of the problem (EP).

Algorithm and its computational complexity
In this section, we firstly present a fundamental branching rule.Next, we propose an image space branchreduction-bound algorithm for solving the problem (EP) based on the former parametric linear relaxation programming problem.Furthermore, the global convergence of the algorithm is proved, and the computational complexity of the algorithm is analysed.

Algorithm statement
Let LB(Λ) and ((Λ), (Λ)) be the optimal value and the optimal solution of the (PLRP) over the subrectangle Λ, respectively.Next, the basic steps of the proposed algorithm can be summarized as follows.
Step 2. Use the proposed branching rule to subdivide Λ  into two new sub-rectangles, and denote the set of new partitioning sub-rectangles as Λ  .For each Λ ∈ Λ  , use the proposed image space region reduction techniques to reduce the sub-rectangle Λ, and still denote the remaining sub-rectangles and their set as Λ and Λ  , respectively.
Remark.In fact, the objective function of the problem (PLRP) is always less than or equal to the currently known upper bound, when we add this condition to the problem (PLRP) as a linear constraint, we can get an improved parameter linear relaxation programming problem (IPLRP).

Global convergence analysis
In the subsection, we will discuss the global convergence of the above algorithm.Proof.If the presented algorithm is finite, by steps of the presented algorithm, then it terminates at some step  ≥ 0. According to the termination condition of the algorithm, we have that By Steps 1 and 4 of the presented algorithm, a better feasible solution (︀   ,   )︀ for the problem (EP) can be found, and we have the following inequality: Let  be the global optimal value of the problem (EP), we have that LB  ≤ .

Since
(︀   ,   )︀ is a feasible solution for the problem (EP), we can obtain that Taken together above, it implies that is a global -optimal solution for the problem (EP) in the sense that Hence, when the presented algorithm finitely terminates after  iterations, it follows that   is a global -optimal solution for the problem (MP).
If the presented algorithm creates an infinite sequence {(  ,   )} of solution which is obtained by solving the problem (PLRP) over Λ  , then it is obvious that {(  ,   )} is a sequence of feasible solution for the problem (EP) over Λ

Computational complexity of the algorithm
In this subsection, we will analyze the computational complexity of the presented algorithm for estimating the maximum iterations of the algorithm.To this end, we define the size (Λ) of a rectangle and the proof of the theorem is completed.
Theorem 4.3.For any given termination tolerance  ∈ (0, 1), the presented algorithm can obtain a global -optimal solution in at most Proof.Without losing generality, we assume that, at each iteration, the rectangle partition is performed in the selected sub-rectangle Λ at step 2 of the algorithm.After  •  iterations, we have that we can get that |() − ()| ≤ .Therefore, after at most where ( * ,  * ) is the global optimal solution for the problem (EP).From Step 3 of the proposed algorithm, (  ,   ) is the current best feasible point for the problem (EP), and by Theorem 4.2, we know that {︀  (︀   )︀}︀ is a decreasing sequence and it satisfies  (︀   )︀ ≤ ().Therefore, we have which shows that, when the presented algorithm terminates, (  ,   ) is a global -optimal solution for the problem (EP).Therefore,   is a global -optimal solution for the problem (MP), and the proof of the theorem is completed.

Numerical experiments
In this section, we numerically compare our algorithm with the software BARON [18] and the known branchand-bound algorithms [5,8,22,32,34,39].All numerical tests are implemented in MATLAB R2014a and run on a microcomputer with Intel(R) Core(TM) i5-7200U CPU @2.50 GHz processor and 16 GB RAM.In all numerical tests, the maximum CPU time limit is set to 3600 s.
First of all, with the given convergence tolerance 10 −6 , we tested some small-size examples, see Appendices A.1-A.8 for details, compared with the existing branch-and-bound algorithms [5,8,22,32,34,39] and the software BARON, and the corresponding numerical comparisons are reported in Table 1.
Next, with the given convergence tolerance 10 −6 , we tested some large-size examples, see Problems 5.1-5.3 for details, which are generated randomly to further verify the robustness and effectiveness of our algorithm, and the corresponding numerical comparisons are reported in Tables 2 and 4-7.Moreover, with the given convergence tolerance 10 −2 , we further tested Problem 5.1 with the larger , and the numerical comparison results are shown in Table 3.
Some notations have been used for column headers in Tables 1-7: Iter: the number of iteration for the algorithm; CPU Time: the execution CPU time of algorithm in seconds; Avg.N: the average number of iterations; Std.N: the standard deviations for number of iteration; Avg.T: the average execution CPU time of the algorithm in seconds; Std.T: the standard deviation of the execution CPU time of the algorithm; and "−" denotes the situation that some of ten random instances failed to be solved in 3600 s.
According to the numerical results in Table 1, for small-size examples A.1-A.8 in appendix, it can be observed that, with almost the same computational efficiency, our algorithm can obtain the same optimal solutions and optimal values as the existing branch-and-bound algorithms [5,8,22,32,34,39] and the software BARON [18].At the same time, we can also observe that, when the variable  ≤ 5, the software BARON is more effective for solving the problem (MP).
For large-size Problems 5.1-5.3,we solved ten independently generated instances and recorded their average number of iteration and average CPU time.For Problems 5.1-5.3, when  > 20, the software BARON failed to solve some of ten independently generated instances in 3600 s, so that we only numerically compared our algorithm with the algorithm of Jiao et al. [17] in Table 3, and we only numerically compared our algorithm with the algorithm of Zhang et al. [39] in Table 5, and we only reported the numerical results of our algorithm in Tables 4 and 7.Meanwhile, we use graphs to show the numerical comparison results of Tables 3 and 5 respectively, and see Figures 1-3 for details.
According to the numerical results in Table 2, it can be understood that, compared with the algorithm of Wang and Liu [33], our algorithm takes less average time and average number of iteration to solve Problem 5.1 with large-size variables, and the software BARON is only effective for Problem 5.1 with small-size variables, such as  = 2,  ≤ 10 and  ≤ 20.When  = 2,  ≥ 20, and  ≥ 20, the software BARON failed to    solve some of them in 3600 s.But, our algorithm can solve Problem 5.1 with large-size variables and has better computational performance.
According to the numerical results in Table 3, and from Figures 1 and 2, it can be seen that our algorithm takes less average time and average number of iteration than that of Jiao et al. [17] for solving Problem 5.1 with the larger , so our algorithm is superior than the algorithm of Jiao et al. [17].
According to the numerical results in Tables 4 and 7, we can see that our algorithm can solve Problems 5.2 and 5.3 with the large-size variables, but the software BARON failed to solve some of them in 3600 s, so this indicates the effectiveness and robustness of our algorithm.According to the numerical results in Table 5, and from Figure 3, when  ≥ 4 and  ≥ 2000, compared with the algorithm of Zhang et al. [39], it can be known that our algorithm spend less average time and average number of iterations for solving Problem 5.2 with large-size variables, so our algorithm outperforms that of Zhang et al. [39].
According to the numerical results in Table 6, for Problem 5.3, it can be obtained that, when  = 2,  ≥ 20 and  ≥ 20, the software BARON failed to solve some of them in 3600 s, but our algorithm can solve such a problem with large-size variables, so that our algorithm has strong robustness.
According to the numerical results in Tables 1-7, we can obtain that the software BARON is most effective for small-size problems, but the software BARON is difficult to solve large-size Problems 5.1-5.3.In addition, our algorithm outperforms the existing branch-and-bound algorithms [5,8,22,32,34,39] and the software BARON [18] for globally solving large-size Problems 5.1-5.3.

Conclusion
This paper studies a class of multiplicative problems (MP) and presents an image space branch-reductionbound algorithm.In this algorithm, we proposed a novel linear relaxation technique for constructing the parametric linear relaxation programming problem of the equivalent problem.In contrast to the existing branch-and-bound algorithms, the branching search takes place in the image space R  of the affine function ∑︀  =1     +  ,  = 1, . . ., , which mitigates the required computational efforts and the computational complexity of the algorithm.Our algorithm can find an -approximate optimal solution in at most  • Appendix A.
A.1.Reference [34] and the proof is complete.Next, let () denote the underestimating function of the function (), by Theorem 2.1, we have

Theorem 4 . 1 .
The presented algorithm either terminates finitely at the global optimum solution of the problem (MP) or generates an infinite sequence {  } of which any gathering point of the sequence {  } is a global optimal solution for the problem (MP).

Figure 1 .
Figure 1.Numerical comparisons between our algorithm and that of Jiao et al. [17] on the average number of iteration (left) and the average CPU time (right) for Problem 5.1 with the fixed  = 10 and  = 10.

Figure 2 .
Figure 2. Numerical comparisons between our algorithm and that of Jiao et al. [17] on the average number of iteration (left) and the average CPU time (right) for Problem 5.1 with the fixed  = 2 and  = 10.

Figure 3 .
Figure 3. Numerical comparisons between our algorithm and that of Zhang et al. [39] on the average number of iteration (left) and the average CPU time (right) for Problem 5.2 with the fixed  = 10 and  = 2000.

2 𝑝𝜇𝜎𝛿 2 (
Λ 0 )  ]︂iterations.Numerical comparisons show the superiority and efficiency of our algorithm.The future work is to extend our algorithm to solve generalized convex (concave) multiplicative programming problems.

Table 4 .
Numerical computational results of our algorithm for Problem 5.2.

Table 5 .
[39]rical comparisons between the algorithm of Zhang et al.[39]and our algorithm on Problem 5.2.

Table 6 .
Numerical comparisons between the software BARON and our algorithm on Problem 5.3.

Table 7 .
Numerical computational results of our algorithm for Problem 5.3.