In our talk we present an algorithm for determining initial bound for the Branch and Bound (B&B) method. The idea of the algorithm is based on the use of the "ray" introduced in the "ray-method" developed for solving integer programming problems [13], [14]. Instead of solving a common integer programming problem we use the main idea of the ray-method to find an integer feasible solution of
integer linear programming (ILP) problem along the ray as close to optimal solution of relaxation problem, as possible. Objective value obtained in this way may be used as an initial bound for B&B
method. The algorithm has been implemented in the frame of educational package WinGULF [3] for linear and linear-fractional programming and has been tested on various ILP problems. Then inspired by the results obtained we implemented the method using the so-called callable library of CPLEX package by IBM. omputational experiments with the algorithm proposed show that such preprocessing procedure in many cases results an integer feasible solution very close to the solution of relaxation problem. Initial bound for branch and bound method determined in this way often can significantly decrease the size of the binary tree to be searched and in this manner can improve performance of the B&B method

Anett Rácz

Erik Bajalinov

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Croatian Operational Research Review (CRORR), Vol. 3, 2012      137 THE RAY-METHOD: THEORETICAL BACKGROUND AND COMPUTATIONAL RESULTS   Erik Bajalinov University College of Nyíregyháza, Institute of Mathematics and Informatics H-4400 Nyíregyháza, Sóstói út 31/b, Hungary E-mail: Bajalinov@NyF.Hu  Anett Rácz Debrecen University, Faculty of Informatics 4028 Debrecen, Kassai u. 26, Hungary E-mail: Racz.Anett@Inf.UniDeb.Hu  Abstract In our talk we present an algorithm for determining initial bound for the Branch and Bound (B&B) method. The idea of the algorithm is based on the use of the "ray" introduced in the "ray-method" developed for solving integer programming problems [13], [14]. Instead of solving a common integer programming problem we use the main idea of the ray-method to find an integer feasible solution of integer linear programming (ILP) problem along the ray as close to optimal solution of relaxation problem, as possible. Objective value obtained in this way may be used as an initial bound for B&B method. The algorithm has been implemented in the frame of educational package WinGULF [3] for linear and linear-fractional programming and has been tested on various ILP problems. Then inspired by the results obtained we implemented the method using the so-called callable library of CPLEX package by IBM. Computational experiments with the algorithm proposed show that such preprocessing procedure in many cases results an integer feasible solution very close to the solution of relaxation problem. Initial bound for branch and bound method determined in this way often can significantly decrease the size of the binary tree to be searched and in this manner can improve performance of the B&B method.  Key words: Branch and bound, Integer programming   1. INTRODUCTION  The Branch and Bound (B&B) algorithm, first proposed by Land and Doig [16], is a well known and efficient algorithm for solving Integer Programming (IP) or Mixed Integer Programming (MIP) problems, it is used by all commercial solvers. Croatian Operational Research Review (CRORR), Vol. 3, 2012      138It is well known that the performance of the B&B method mainly depends on the following three main factors:  the rule used for choosing the branching variable,   the strategy used for generating binary search tree and  the value of the initial bound. Numerous efforts have been made in past decades to investigate general properties and behavior of B&B method, e.g. [5], [8], [9], [15], [17], [18], [19], [20], to improve its computational efficiency, e.g. [6], [10], [11], [12], to maximize its performance in different computational environments, see for example [7], [21], etc. Generally speaking, while branching variable and searching strategy determine the size of binary tree to be generated, getting a "good" bound as soon as possible can dramatically reduce the size of the tree to be considered, since the bound is used to prune those parts of the tree where the value of objective function cannot be better than the current bound. Our aim was to construct such a general algorithm which could provide a bound before we start the tree-building. Below we describe a procedure which can define a feasible integer solution aspiring to the best possible objective value using minimal computational effort and time.   2. RAY METHOD  Let us consider the following integer linear programming problem (ILP):  nj jj 1f (x) c x min   (1) st.  nij j ij 1a x b , i 1,2, ,m     (2)  jx 0,  integer, j 1,2, ,n  (3) Here and in what follows we assume that the corresponding relaxation problem of (1)-(3) is solvable (i.e. has a non-empty feasible set S and objective function f(x) over the feasible set has a finite lower bound) and vector  min min min min1 2 nx (x ,x , , x )  Croatian Operational Research Review (CRORR), Vol. 3, 2012      139 is its relaxation non-integer solution.  Furthermore, we suppose that the corresponding maximization relaxation problem is solvable too, and vector   max max max max1 2 nx (x ,x , , x )  is its optimal solution. These two assumptions play a very important role in the new algorithm we proposed since points minx  and maxx  are used to determine the ray for indicating the direction of the search. Moreover we assume that  min maxx x   2.1. Main steps   Using notation given above, the new algorithm proposed can be described in the following way.  1. Initial point. Let us denote point minx  by 0x ,  and then let us enter vector 1 2 nl = (l , l ,  , l ) , where    max minj j jl x x , j 1,2, ,n .   Figure 1:  The ray L. Croatian Operational Research Review (CRORR), Vol. 3, 2012      140 2. Ray. Define the ray in the following way:  0 max 0L x (x x ), 0 1.   Note that since feasible set S is convex, it means that all points of ray L are elements of set S. 3. Constructing set0O(x ) . Let 0J  be a set of indexes of integer components of vector 0x , i.e.  0 0 0j jJ {j J | x [x ]},  where J {1,2, ,n}.  Define set 0O(x )  as the set of such points x which satisfy the following constraints:  0 0 0j j j0 0 0j j j j0 0 0j j j j0 0j j j[x ] x [x ] 1,  if j J[x ] x [x ] 1,  if j J  and l 0,[x ] 1 x [x ],  if j J  and l 0,x [x ],  if j J  and l 0, (4)   Figure 2: The first unit-cube. Before starting the iterations we have to define point 0x : x  and calculate the first perforation point  act 1 2 nP (p ,p , , p )  solving the following optimization problem:  max  (5) st. Croatian Operational Research Review (CRORR), Vol. 3, 2012      141  0j j jp x l j 1,2, ,n,  (6)  0 0 0j j j0 0 0j j j j0 0 0j j j j0 0j j j[x ] p [x ] 1,  if j J ,[x ] p [x ] 1,  if j J  and l 0 ,[x ] 1 p [x ],  if j J  and l 0 ,p [x ],  if j J  and l 0 . (7) 4. Shifting. Now we enter new variables j j jy x [x ], j 1,2, ,n,  and construct new feasible set S'  in the following way  nij j ij 1S : a y b , i 1,2, ,m ,  where  ni i ij jj 1b b a [x ], i 1,2, ,m .    Figure 3: The shifted problem.  Obviously, set S' is the intersection of the current set O(x') and feasible set S shifted to point 0. Then we solve the following 0-1 LP problem  nj j 0j 1f (y) c y c max  (8) st.  nij j ij 1a y b , i 1,2, ,m,  (9)  jy 0 /1, j 1,2, ,n,  (10) where Croatian Operational Research Review (CRORR), Vol. 3, 2012      142 n0 j jj 1c c [x ] ,  using Balas' additive algorithm (implicit enumeration) [4]. If problem (8)-(10) has 0-1 optimal solution *y , then we determine vector * * * *1 2 nx (x ,x , , x ) , where  * *j j jx y [x ], j 1,2, ,n ,  and use value * *f (x ) f (y )  as an initial bound for the branch and bound method. Stop. Otherwise, 5. Perforation point. We determine point nextP  where the ray "perforates" the hull of the next unit-cube along the ray solving the following optimization problem:  max  (11) st.  0j j jx x l j 1,2, ,n ,  (12)  j j jj j j jj j j jj j j[p ] x [p ] 1, j J ,[p ] x [p ] 1, j J  and l 0 ,[p ] 1 x [p ], j J  and l 0 ,x [p ], j J  and l 0 , (13) where j jJ {j J | p [p ]} . Here constraints (12) and (13) provide nextP L and next actP O(P )  correspondingly. Obviously, this problem is solvable, i.e. has a non-empty    feasible set and its objective function is bounded from above. Let next 1 2 nP (p ,p , , p )  solve    problem (11)-(13) and  be the maximal value of objective function (11). Let us define middle point x' of the line segment act next[P ; P ] :  j jjp px j 1,2, ,n .2 (14) Furthermore, we do not need point actP   anymore, so we overwrite it with the value of nextP , i.e. act nextP : P . 6. Next unit-cube. Having point x' we can determine the next unit-cube along the ray using the following rule:  j j jj j j jj j j jj j j[x ] x [x ] 1,  if j J ,[x ] x [x ] 1,  if j J  and l 0 ,[x ] 1 x [x ],  if j J  and l 0 ,x [x ],  if j J  and l 0 , (15) Croatian Operational Research Review (CRORR), Vol. 3, 2012      143 where j jJ {j J | x [x ]}. Go to step 3. Since the number of unit-cubes "perforated" by the ray is finite, the process will terminate in a finite number of iterations. It may occur that when determining next perforation point x' we obtain 1 . It means that unit-cubes constructed along the ray do not contain any integer feasible point for original problem (1)-(3). It means the method fails.  3. NUMERIC EXAMPLE  Here the main steps of the method proposed using a small numerical example are illustrated. The method proposed was partially implemented in the frame of the educational linear and linear-fractional package WinGULF [3]. The package has numerous options for the B&B method - we can choose the direction of the search (first left node and then right one or vice versa), different rules for selecting a branching variable (for example, "fractional part most close to 0.5", "smallest fractional part", "biggest fractional part", "smallest value", "biggest value", etc.), user defined initial bound, etc. When testing the method proposed, most of the options built in were used. Consider the following numerical example:  1 2 3f (x) 20x 21x 18x min  st.  1 2 31 2 31 2 31 2 39x 1.5x 7x 1350 ,5.5x 1x 9x 1250 ,4.5x 10x 2.5x 1050 ,x , x , x  integer. Solving both (minimization and maximization) relaxation problems we obtain the following:  minmaxx ( 65.042, 97.799, 88.274) ,x ( 0.000, 523.076, 80.769) ,L ( 65.042, 425.277, 7.504) . Let us denote minx  with 0x  and construct set 0O(x ), i.e. the following unit-cube: Croatian Operational Research Review (CRORR), Vol. 3, 2012      144 102365 x 66 ,O(x ) : 97 x 98 ,88 x 89 . So the first shifted problem will be as follows:  1 2 31 2 31 21 2 331 2 3f (y) 20y 21y 18y 4921 maxst.9y 1.5y 7y 3.5 ,5.5y 1y 9y 3.5 ,4.5y 10yy ,y ,y -2.5y 7.5 , 0/1 Since this problem is infeasible, we have to determine the next unit-cube along the ray. In order to obtain the next unit-cube first we solve the following problem (see (12)-(13)):  max  st.  123x 65.042 ( 65.042) ,x 97.799 (425.277) ,x 88.274 ( 7.504) ,  12365 x ,x 98 ,88 x  and obtain the first perforation point 1P :  10.00047, P (65.011, 98, 88.270) .  To find the next perforation point 2P  we have to solve the following problem:  max  st.  123x 65.042 ( 65.042) ,x 97.799 (425.277) ,x 88.274 ( 7.504) , Croatian Operational Research Review (CRORR), Vol. 3, 2012      145  12365 x ,x 99 ,88 x , so we obtain   20.00064, P (65, 98.07, 88.26) .  Using these points 1P  and 2P  we find the middle point  x'=(65.006, 98.03, 88.26).  This point allows us to construct the next shifted problem:  1 2 31 2 31131322 32f (y) 20y 21y 18y 4942 maxst.9y 1.5y 7y 2 ,5.5y 1y 9y 2.5 ,4.5y 10y 2.5y 2.5 ,y ,y ,y  - 0/1. This problem has no feasible solution. Proceeding to the next perforation point 3P , we obtain the following optimization problem to solve  max  st.  123x 65.042 ( 65.042) ,x 97.799 (425.277) ,x 88.274 ( 7.504) ,  12364 x ,x 99 ,88 x . We obtain 0.0028  and the next perforation point 3P (64.85, 99, 88.25) . Therefore the next middle point is x (64.93, 98.53, 88.26)  and the shifted problem is as follows: Croatian Operational Research Review (CRORR), Vol. 3, 2012      146 1 2 31 2 31 2 31 2 31 2 3f (y) 20y 21y 18y 4922 maxst.9y 1.5y 7y 11,5.5y 1y 9y 8 ,4.5y 10y 2.5y 2 ,y , y , y 0 /1. The optimal solution of this problem is *y (0,1,1)  and *f (y ) 4961. This value may be used as an initial bound. The corresponding integer point is *x (64, 99, 89) . Note that the initial bound obtained (4961) is very close to the optimal value (after solving the problem we obtain 4959). Below is presented the table with results obtained from WinGULF 11F1 after running B&B method on this numerical example using different strategies (from left to right and vice versa) and different branching rules. "Wo.I.B." means "without an initial bound" and "W.I.B." means "with an initial bound".  Table 1:     In this numeric example the average reduction of the number of nodes to be processed in the binary tree is equal to 32.4%, while the best result is 90.5%.  4. MIPLIB TESTS  Encouraged and inspired by this results we performed further numeric experiments and tests using the professional solver CPLEX (i.e. IBM ILOG CPLEX Optimizer) and official test files from MIPLIB collection (see for instance http://miplib.zib.de/). Therefore A. Rácz implemented the MIP adaptation of the method proposed using callable library of package CPLEX.                                                  1 Web site: http://zeus.nyf.hu/~bajalinov Croatian Operational Research Review (CRORR), Vol. 3, 2012      147 The steps of the test code developed are as follows: 1. Reading .mps or .lp input file with the test problem.  2. CPLEX solves the problem without our bound. 3. Ray method calculates an initial bound. 4. CPLEX solves the problem using the bound defined in the previous step. The following table shows the results obtained. Most of the files can be found in MIPLIB library (contributor A. Atamtürk, [1], [2]).   Table 2: Test results with MIPLIB problems   This table shows the corresponding information about the test problems in the first five column (name, number of rows and columns, number of integer variables and nonzero coefficients). The other columns present the test results. In column "Time without bound" you can find the time in sec which was needed for IBM CPLEX for solving the IP problem without our bound. In the column "Time to define bound" we see the time elapsed while method was calculating the initial bound. Finally, column "Time with bound" shows CPLEX's reduced run time while solving the same IP problem but using initial bound calculated by our ray-method. As we can see an overall about 20,9% average improvement on run time can be observed in case of these test files. So spending a very little time for calculating an initial bound using our method often leads to significant improvement in performance of branch and bound procedure.  5. CONCLUSIONS  Summarizing results presented above, we have to emphasize that when choosing a ray it is very important to find such a ray that is located in the most inner part of feasible set S, since it gives us a hope that the ray selected in such a way will result a unit-cube with at least one feasible vertex quite quickly.  Croatian Operational Research Review (CRORR), Vol. 3, 2012      148Originally the idea of using some sort of ray when solving IP problems was proposed by V.R. Khachaturov12F2 et al. [13], [14]. Instead of search an optimal integer solution we use the idea of the ray to find a feasible integer solution as soon as possible. Our experiments with our package WINGULF have shown that such type of procedure often results such a good initial bound that dramatically reduces the size of the binary tree to be searched. All these results inspired us for experiments with callable library of CPLEX. The experiments with CPLEX have shown that the main idea of the method proposed may be useful, so we intend to continue numeric experiments and check the method with using package GUROBI13F3.  REFERENCES  1 Atamtürk, A. (2002), “On Capacitated Network Design Cut-Set Polyhedra” Mathematical Programming, Vol. 92, pp. 425-437. 2 Atamtürk, A.,  Rajan D. (2002), “On Splittable and Unsplittable Capacitated Network Design Arc-Set Polyhedra” Mathematical Programming, Vol. 92, pp. 315-333. 3 Bajalinov, E. (2003), Linear-fractional programming: theory, methods, software and applications, Kluwer. 4 Balas, E. (1965), “An additive algorithm for solving linear programs with zero-one variables” Operations Research, Vol. 13, pp. 517-46. 5  Berliner, H. (1979), “The B tree search algorithm: A best-first proof procedure” Artificial Intell., Vol. 12, no. 1, pp. 23-40. 6 Borchers, B., Mitchell, J.E. 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(1966), “Branch-And-Bound Methods: A Survey” Operations Research, Vol. 14,  no. 4, pp. 699-719. 18 Mitten, L. (1970), “Branch and bound methods: General formulation and properties” Operation Research, Vol. 18, pp. 24-34. 19 Smith, D.R. (1984), “Random trees and the analysis of branch and bound procedures” Journal of the Association for computing machinery, Vol.31, no.1, pp. 163-188. 20 Yu, C.-F., Wah, B.W. (1985), “Stochastic modeling of branch-and-bound algorithms with best-first search” IEEE Transactions on Software Engineering, Vol. SE-11, no. 9, pp. 922-934. 21 Yu, C.-F., Wah, B.W. (1988), “Efficient Branch-and-Bound Algorithms on a Two-Level Memory System” IEEE Transactions on Software Engineering, Vol. 14, no. 9, pp. 1342-1356.      

THE RAY-METHOD: THEORETICAL BACKGROUND AND COMPUTATIONAL RESULTS

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