An Effective Branch and Bound Algorithm for Minimax Linear Fractional Programming

An effective branch andbound algorithm is proposed for globally solvingminimax linear fractional programming problem (MLFP). In this algorithm, the lower bounds are computed during the branch and bound search by solving a sequence of linear relaxation programming problems (LRP) of the problem (MLFP), which can be derived by using a new linear relaxation bounding technique, and which can be effectively solved by the simplex method. The proposed branch and bound algorithm is convergent to the global optimal solution of the problem (MLFP) through the successive refinement of the feasible region and solutions of a series of the LRP. Numerical results for several test problems are reported to show the feasibility and effectiveness of the proposed algorithm.

The minimax linear fractional programming problems (MLFP) pose significant theoretical and computational challenges.This is mainly because the problems (MLFP) possess multiple local optima that are not globally optimal.Therefore, it is necessary to put forward an effective global optimization algorithm for solving the minimax linear fractional programming problem (MLFP).During the past years, some algorithms have been proposed for solving the minimax linear fractional programming problem (MLFP), for instance, dual methods [2,6,7], parametric programming methods [8,9], interior-point algorithms [10,11], monotonic optimization approach [12], exact method [13], approximation algorithm [14], cutting plane algorithm [15], method of centers [16], inexact proximal point method [17], interval-type algorithm [18], and so on.Recently, based on the linear relaxation and branch and bound scheme, a solution algorithm has been developed for solving globally the minimax linear fractional programming problem (MLFP) [19].Up to now, although there has been significant progress in the development of algorithms for solving the minimax linear fractional programming, to our knowledge, less work has been still done for globally solving the minimax linear fractional programming problem (MLFP).
The purpose of this paper is to present a new global optimization algorithm for solving the minimax linear fractional programming problem (MLFP), and the goals of this research are threefold.First, we present a transformation of the problem; thus, the original problem (MLFP) is reformulated as an equivalent problem (EP).Second, in order to design an effective branch and bound algorithm for the equivalent problem (EP), a new linear relaxation bounding technique is presented, and, by utilizing this technique, the nonconvex programming problem (EP) is reduced to a sequence of linear relaxation programming problems, which can provide reliable lower bounds for the optimal value of the problem (EP) and are embedded into the branch and bound framework.The main computational operation in the algorithm only involves solving a sequence of linear relaxation programming subproblems that do not grow in size from iteration to iteration.Third, compare our algorithm (using new linear relaxation bounding technique) with the known algorithms (recent literatures) with respect to robustness (finding the optimum) and efficiencies (number of function evaluations), and the numerical experimental results show that the proposed algorithm is robust and effective.
This paper is organized as follows.The next section firstly converts the problem (MLFP) into an equivalent problem (EP); a new linear relaxation bound method is presented, and then the linear relaxation programming of the problem (EP) is established.Section 3 a branch and bound algorithm is proposed for globally solving the problem (MLFP), and the convergence property of the algorithm is given.In Section 4, we report the numerical results for solving some examples with the proposed algorithm.Finally, a few concluding remarks are given in Section 5.

Linear Relaxation Programming
In order to globally solve the problem (MLFP), we first compute the initial lower bound   = min ∈   and upper bound   = max ∈   of each variable   and denote the initial rectangle by Next, we can convert the problem (MLFP) into the following equivalent problem (EP), which has the same global optimal solution and optimal value as the problem (MLFP): In the following, we only consider solving the equivalent problem (EP); the important step in the construction of a solution procedure for globally solving the problem (EP) is the establishment of a linear relaxation programming for computing the lower bounds of the optimal value for this problem.Here, we only need to construct linear lower bounding function of Ψ  () in constraint function.For each  ∈ {1, . . ., }, we can let The detailed new linear relaxation bounding technique can be given as follows.
One has where  and  are constant vectors, which satisfy By the above discussion, it is obvious that the conclusion is followed.

Algorithm and Its Convergence
In this section, based on the former linear relaxation method, we will present a branch and bound algorithm for globally solving problem (EP).There are three fundamental operations in the proposed algorithm: a branching operation, an updating upper bounds operation, and an updating lower bounds operation.The first fundamental operation iteratively subdivides the investigated rectangle  into two subrectangles.During the process of iteration of the algorithm, the branching operation produces a more refined partition that cannot yet be excluded from further consideration in finding the global optimum for the problem (EP).In this paper, we choose a simple and standard branching rule.This branching rule is enough to ensure the convergence of the algorithm since it drives the intervals shrinking to a singleton for all variables along any infinite branch of the branch and bound tree.Consider any node subproblem identified by the hyperrectangle  = [, ] ⊆  0 .This branching rule is as follows.

(b) Let
By this branching rule, the rectangle  is subdivided into two subrectangles X1 and X2 .
The second fundamental operation is to update the lower bounds of the optimal value of the problem (EP).This main computation needs to solve a sequence of linear relaxation programming problems, which can be easily solved by using the simplex method.
The third fundamental operation is to update the upper bounds of the optimal value of the problem (EP).The upper bounds can be updated by computing the objective function values of the original problem (MLFP) and the equivalent problem (EP) which corresponds to optimal solution of each linear relaxation programming problem, respectively.
The set  in the algorithm is the set of fathomed subrectangles  of  0 .Let LB(  ) refer to the optimal objective function value of the problem (LRP) on the subhyper-rectangles   and   = (  ) refer to an element of corresponding argmin.The basic steps of the proposed algorithm are summarized as follows.
Step 5.For the given feasible tolerance  1 , if ( , , set and continue.
The convergence properties of the proposed algorithm are given as follows.
Theorem 2. If the proposed algorithm terminates in finite steps, then a global optimal solution of the problem (MLFP) is obtained when the algorithm is terminated.

Numerical Experiments
To verify the performance of our algorithm, several test examples in recent literatures are implemented on on a Intel(R) Core(TM)2 Duo CPU (1.58 GHZ) microcompute; the algorithm program is coded in C++, and each linear relaxation programming problem is solved by using simplex method, and the convergence tolerance is set to  = 5 × 10 −8 in our experiment.For the test problems, numerical results are illustrated in Table 1.For Examples 1-5, feasible errors  1 are set by 0.005, 0.005, 0, 0.005, and 0.001, respectively.
In Tables 1 and 2, the notations have been used for column headers: Iter: number of algorithm iteration;  max : the maximal number of algorithm active nodes necessary; time: execution time of algorithm in seconds.
In Table 1, optimal value is denoted by objective function value of the optimal solution in computational procedure of [19] and this paper, respectively.

Table 1 :
Computational results for test Examples 1-4.Assume that the algorithm is terminated finitely at   .Obviously, LB  =   when it is terminated at the  ℎ iteration; therefore,   is a global optimal solution of the problem (MLFP).If the algorithm generates an infinite sequence {  }, then every accumulation point  * of this sequence is a global optimal solution of the problem (MLFP).Proof.Let  * be an accumulation point of the sequence {  }, and let {   } be a subsequence {   } of the sequence {  } which is convergent to  * .Obviously, in the proposed algorithm, the lower bound sequence {LB  } is monotonic increasing and the upper bound sequence {  } is monotonic decreasing, so that {LB  } and {  } are convergent and Without loss of generality, we assume that    is the solution of the problem (LRP) on    which satisfies   +1 ⊆    ,  = 1, 2, ....Because the proposed rectangle partition is exhaustive; that is, lim  → ∞    =  * , and, fromTheorem1, we have 0 ≤    −    ≤ max {      Ψ 1 (   ) −   1 (   )      , . . .,      Ψ  (   ) −    (   )      } → 0 as  → ∞.