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By Ivanyi A. (ed.)

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Example text

In the case of pure binary problem an explicit objective function constraint can give a lot of consequences as well. 2. Preprocessing Preprocessing means to obtain information on variables and constraints based on algebraic constraints and integrality. 36) are summed up then the inequality 6x1 ≤ 15 is obtained implying that x1 ≤ 2. 16). Many tests can be based on the following two easy observations: 1. 73) then the constraint is redundant. 2. e. 16) has no feasible solution. If under some further restriction the second observation is true then the restriction in question can be excluded.

E. in binary optimization, it is possible to modify it such that it is valid in the original problem, too. For practical reasons the type of the generated cut can be restricted. It is the case in TSP as the subtour elimination constraints can be found relatively easily. 1260 24. 7. Branch and Price The Branch and Price method is the dual of B&C in a certain sense. If a problem has very large number of variables then it is often possible not to work explicitely with all of them but generate only those which may enter the basis of the LP relaxation.

3. 8. It is point C on the figure. Now branching is possible according only to variable x2 . Branches 4 and 5 are generated by the cuts x2 ≤ 1 and x2 ≥ 2, respectively. The feasible regions of the relaxed problems are OHG of Branch 4, and AEF of Branch 5. The method continues with the investigation of Branch 5. The reason will be given in the next subsection when the quickly calculable upper bounds are discussed. e. 3. The algebraic details discussed in the next subsection serve to realize the decisions in higher dimensions what is possible to see in 2-dimension.

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