New PDF release: Algorithms of informatics, vol. 3

By Ivanyi A. (ed.)

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Extra resources for Algorithms of informatics, vol. 3

Example text

3. Mixed integer programming with bounded variables 1243 variables are fixed to zero is equal to the current value of dp0 . Thus it is possible to define the new branches according to the values of xp . 61) which is defined by xp , has the form −fp ≥ j ∈ I ∩ IN dpj − µj ≥ 0 (dpj − µj )(−xj ) + j ∈ I ∩ IN dpj − µj < 0 fp (µj − dpj )(−xj ) 1 − fp dpj (−xj ) + + j ∈ IN \ I dpj < 0 j ∈ IN \ I dpj > 0 fp (−dpj )(−xj ) . 1 − fp The appropriate quantities Plp and Pup are as follows: Plp = min{a, b} , where a = min and d0j fp | j ∈ I ∩ IN , dpj − µj > 0 dpj − µj b = min d0j fp | j ∈ IN \ I, dpj > 0 dpj further Pup = min{c, d} , where c = min and d0j (1 − fp )2 | j ∈ I ∩ IN , dpj − µj < 0 (µj − dpj )fp d = min − d0j (1 − fp )2 | j ∈ IN \ I, dpj < 0 fp dpj .

E. it is the set conv(P ∩ Z n ) . 2. Thus the description of the integer hull as a polyhedral set is the inequality system: x1 ≥ 0, x1 + x2 ≤ 3, x1 ≤ 1, x1 − x2 ≤ 0 . Under the conditions of the definition the integer hull is a polyhedral set, too. It is a non-trivial statement and in the case of irrational coefficients it can be not true. e. a set of linear inequalities defining exactly the integer hull polyhedral set is known, then the integer programming problem can be reduced to a linear programming problem.

It will contain the only negative element in the first column that is the optimization in the lower branch starts by pivoting in this row. 53) can be reformulated according to the signs of the coefficients as s = −fp + (−dpj )(−xj ) . 54) j∈J − j∈J + The pivot element must be negative, thus it is one of −dpj ’s with j ∈ J + . 55) . 52) is equivalent to xp − dp0 = fp + j∈IN dpj (−xj ) ≥ 1 . Again the nonnegative slack variable s should be introduced. Then the row which can be added to the simplex tableau is s = (fp − 1) + dpj (−xj ) .

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Algorithms of informatics, vol. 3 by Ivanyi A. (ed.)


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