In this paper we present an algorithm for determining initial bound for the Branch and Bound (B&B) method. The idea of this algorithm is based on the use of "ray" as introduced in the "ray-method" developed for solving integer linear programming problems [11], [12]. Instead of solving an integer programming problem we use the main idea of the ray-method to find an integer feasible solution of an integer linear programming problem along the ray as close to an optimal solution of the relaxation problem as possible. The objective value obtained in this manner may be used as an initial bound for the B&B method. It is well known that getting a "good bound" as soon as possible can often significantly increase the performance of the B&B method

Rácz, Anett

Rácz Anett

University of Szeged

English

Determining initial bound by "Ray-method" in branch and bound procedure

Determining Initial Bound by "Ray-method" in Branch and BoundProcedureAnett RáczIt is very hard to solve an optimization problem if the problem contains integrality con-straints. Such problems are integer programming problems, which can be solved by variousalgorithms. The most widespread algorithm is the Branch and Bound algorithm, which sim-ilarly to other methods, builds a search tree. Following from the structure of the problem,sometimes these methods obtain a result only at the expense of building a large tree. It mayalso happen that it take a lot of time to recognise that the problem does not have an integersolution. Therefor, the demand to reduce the size of the tree built by the algorithms is under-standable. The paper aims to develop a reliable method, that can reduce the size of the searchtree. The idea is based on the results of the theoretical research of the Computing Centre of theRussian Academy of Sciences.[1] It is difficult to adapt the main theoretical results into a com-puter environment. As a result these methods in original form are not applicable yet. Basedon the theorems, that have been worked out recently, a new method, which can be relativelyeasily implemented has been developed, which has proved to be efficient on the basis of tests.By using the solution of the relaxation problem, the algorithm gives a bound for the value ofthe objective function of the problem with integer conditions, thereby creating an initial boundfor the solvers, wich build a search tree. The key to the efficiency of the algoritms based onthe ray-method is to create a feasible integer solution relatively quickly. One characteristic ofthe tree is that the optima get worse as we move down on its branches. If we find a feasibleinteger solution on one of the nodes to the original problem, we use the objective value of thissolution as a bound. If a node has to be branched, but its objective value is already worse thenour current bound we do not have to perform this operation, and we say this node is fathomed.However, we do not have such a bound at the starting node. The algorithms based on the ray-method aim to create a initial bound before we start to build the search tree. As a result, weget the opportunity to cut the unnecessary branches as soon as possible. This means that wedo not have to build a big part of the tree in practice. It is essential to be able to implementthese methods, since they serve to accelerate an applied method, namely Branch and Bound.It is obvious that, if calculating the initial bound takes more time then building the part of thetree, which can be cut with this bound, then we have not improved the efficiency. The methodpresented in this paper is relatively simple, easy to implement, which has proved to be efficientduring the tests so far.AcknowledgementsMy special thanks are due to my supervisor Dr. Erik Bajalinov for his invaluable comments. Ithank the Department of Applied Mathematics and Probability Theory for the technical back-ground and other useful support.References[1] Khachaturov, V.R., Combinatorial methods and algorithms for solving problems of discreteoptimization with large dimensionality, Nauka, 2000.51

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Determining initial bound by "Ray-method" in branch and bound procedure

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University of Szeged

University of Szeged