An Effective Algorithm for Globally Solving Sum of Linear Ratios Problems

In this study, we propose an effective algorithm for globally solving the sum of linear ratios problems. Firstly, by introducing new variables, we transform the initial problem into an equivalent nonconvex programming problem. Secondly, by utilizing direct relaxation, the linear relaxation programming problem of the equivalent problem can be constructed. Thirdly, in order to improve the computational efficiency of the algorithm, an out space pruning technique is derived, which offers a possibility of pruning a large part of the out space region which does not contain the optimal solution of the equivalent problem. Fourthly, based on out space partition, by combining bounding technique and pruning technique, a new out space branch-and-bound algorithm for globally solving the sum of linear ratios problems (SLRP) is designed. Finally, numerical experimental results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.


Introduction
Sum of linear ratios problems (SLRP) have broad wide applications in information technology, control science and engineering, transportation design, government plan, economy and finance [1][2][3][4][5], and so on.In these applications, especially, the number of linear ratios is usually less than four or five.In addition, due to the fact that the sum of linear ratios problems (SLRP) possess generally multiple local optimal solutions that are not globally optimal, the kinds of problems pose significant theoretical difficulty and computational complexity.Therefore, they have attracted interest of many researchers and practitioners.
In this paper, we shall present a new out space branchand-bound algorithm for the sum of linear ratios problems (SLRP) using out space pruning technique.Firstly, we transform the initial problem into the equivalent nonconvex programming problem by introducing new variables.Next, the linear relaxation programming problem of the equivalent problem can be constructed by utilizing direct relaxation.

Journal of Control Science and Engineering
Secondly, by making full use of the special structure of the equivalent problem and the branch-and-bound algorithm, an out space pruning technique is derived to improve the computational efficiency of the proposed algorithm, which offers a possibility of pruning a large part of the investigated out space region which does not contain the global optimal solution of the equivalent problem.Thirdly, by combining bounding technique and pruning technique, a new out space branch-and-bound algorithm is designed for globally solving the sum of linear ratios problems (SLR).Finally, numerical experimental results are presented to demonstrate both computational efficiency and solution quality of the proposed algorithm.
The remaining sections of this paper are organized as follows.At first, in Section 2, by introducing new variables the equivalent nonconvex programming problem of the initial problem is introduced.Next, by utilizing direct relaxation the linear relaxation programming problem of the equivalent problem is constructed.Second, an out space pruning technique is derived by making full use of the special structure of the equivalent problem and the branchand-bound algorithm in Section 3. Third, a new out space branch-and-bound algorithm and its global convergence are described in Section 4. Fourth, numerical experiments and their computational results are presented in Section 5. Finally, the concluding remarks of this paper are drawn.
EQ ( 0 ) : The key equivalence theorem for the problem (SLRP) and the EQ( 0 ) is described as follows.
Proof.The conclusion can be easily drawn, so the proof is omitted.
From Theorem 1, to solve the problem (SLRP), we may globally solve its equivalent nonconvex programming problem EQ( 0 ) instead.
From the above Theorem 2, for any   ⊆  0 , the corresponding linear relaxation programming problem LRP(  ) of the problem EQ(  ) can be established as follows, which can offer a reliable upper bound for the global optimum of EQ(  ).

Out Space Pruning Technique
For any box   ⊆  0 , we need to recognize whether or not   contains global optimum point of the problem (EQ).Thus, in this section, we shall construct an out space pruning technique for pruning a part of the investigated out space region which does not contain the global optimum point of the problem (SLRP) and use the technique to improve the computational efficiency of the proposed algorithm.The proposed new out space pruning technique aims at replacing the box   = [  ,   ] with a smaller box   = [  ,   ] without pruning any global optimum point of the problem (EQ).We assume, without loss of generality, that LB is a currently known lower bound of the global optimum value of the EQ( 0 ) and that V(  ) denote the maximum value of the problem  0 (, ) over   and  and set Proof.(i) Suppose that RUB  < LB; then we have therefore, there does not exist any global optimum point of the problem EQ(  ) over   .
(ii) Suppose that RUB  ≥ LB; then we can get the following several conclusions.
By the above Theorem 3, we can construct an out space pruning technique to prune a part of the investigated out space region which does not contain the global optimum point of the problem (EQ).Assume that a subbox will be pruned; then from the Theorem 3, the investigated box   can be renewed by a subbox

Algorithm and Its Convergence
In this section, by utilizing the above new pruning technique, we will present a new out space branch-and-bound algorithm for globally solving the problem (SLRP).In the algorithm, the branching operation is performed in out space   .Assume that  = { ∈   |   ≤   ≤   ,  = 1, 2, . . ., } is  0 or a subbox of it, which will be partitioned, the branching rule is selected as follows. Set Obviously, from [26] we can get that this branching rule is exhaustive; that is, if {  } represents a nested subsequence of boxes (i.e.,  +1 ⊆   , for all ) formed by partitioning process, then there exists a unique point  * ∈   satisfying ⋂    = { * }.

4.1.
Out Space Branch-and-Bound Algorithm.Based upon the above linear relaxation bounding problem, the new out space pruning technique, and bisection method, an out space branch-and-bound algorithm is proposed for globally solving the (SLRP) as follows.
Step 3. Partition  −1 into two subboxes  ,1 ,  ,2 ⊆  −1 using the bisection technique.Let the new partitioned subboxes set be   .Set  =  ∪ { −1 }.For each  , ∈   , solve the (LRP) using simplex method to get UB( and by terminating step of the algorithm, the updating operations of the lower bound and upper bound, and Theorems 1 and 2, we can easily follow that By combining the above all inequalities and equality, we can easily get that Therefore, if the algorithm stops finitely at  th iteration, then   is the global -optimal solution of the (SLRP).
If the above algorithm generates an infinite sequence {  } of incumbent solutions by solving the LRP(  ), let   (28) Hence,  * is the global optimum point of the (SLRP); the conclusion is proved.

Numerical Example.
In this subsection, some wellknown test problems are implemented on Intel(R) Core(TM)2 Duo CPU microcomputer to verify the performance of the proposed out space branch-and-bound algorithm.The proposed out space branch-and-bound algorithm is coded in C++ program, and the simplex method is used to solve each linear relaxation programming problem.