Presolving Mixed-Integer Linear Programs - COR@L

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PresolvingMixedInteger LinearPrograms 5 2.5 BoundImprovement Bound improvement is one of the most important steps of presolving. If we can improve the boundsonvariablesorreduceb i ,thenwecantightentheLPrelaxationof(P). Thetightestbounds onaconstraint can be obtained(Savelsbergh,1994) by solvingthe followingproblems, n∑ L i = min a T i x j=1 s.t. A i x ≤ b i , (2.1) l j ≤ x j ≤ u j , j = 1,2,...,n x j ∈ Z, j ∈ I, and n∑ U i = max a T i x j=1 s.t. A i x ≤ b i , (2.2) l j ≤ x j ≤ u j , j = 1,2,...,n x j ∈ Z, j ∈ I, where the set of constraints A i x ≤ b i is obtained by deleting the ith constraint ∑ n j=1 a ijx j ≤ b i from (P). However, solving the bound improvement problems (2.1) or (2.2) can be as difficult as solving (P). Thus, there is a trade-off between the quality of a good bound and the time spent in findingit. ThemostcommonmethodoftighteningtheboundistocalculateboundsonL i andU i : ˆL i = ∑ a ij l j + ∑ a ij u j , j:a ij >0 Û i = ∑ j:a ij >0 j:a ij
6 Ashutosh Mahajan and U k = maxx k s.t. Ax ≤ b, (2.5) l j ≤ x j ≤ u j , j = 1,2,...,n x j ∈ Z, j ∈ I. Again,solvingtheproblems(2.4)and(2.5)canbeasdifficultastheoriginalproblem(P). Asimilar trade-off then occurs on the quality of the bound and the time spent in obtaining it. The bounds L i and U i onconstraints can alsobe usedto improve the bounds on variables. If a ij > 0,thenlet Û ik = b i a ik − L i −a ik l k a ik . Clearly, Ûik (≥ U k ) is an upper bound on the feasible values of x k . If U k < u k (or Ûk < u k ), then update the bound on variable x k to U k . If U k < l k , then the problem is infeasible. If U k = l k , then the variablex k can be fixed to the valuel k . Similarly,ifa ik < 0, thenlet ˆL ik = b i a ik − U i −a ik l k a ik . Then ˆL ik (≤ L k ) is also a lower bound on feasible values of x k . If L k < l k , then update the lower bound on variable x k to L ik . If L ik > u k , then the problem is infeasible. If L ik = u k , then the variable x k can be fixed to the value u k . Once L i and U i (or their bounds ˆL i ,Ûi) are known, then the bounds ˆL,Û canbe calculatedin O(nz) steps. Iftheboundsaretightenedusingthecompletemodels(2.1)and(2.2),thenconstraintdomination is just a special case of bound tightening. This is no longer true, however, when the bounds areestimatedfromeachrow. Inthelattercase,weneedonlyO(nz)calculations,wherenz denotes the number ofnonzeros inthe Amatrix. If the bounds of a variable x j are implied by a constraint, then the variable can be declared “free”,and we cantry substituting the variableoutas describedinSection2.1. 2.6 Dual/Reduced-CostImprovement If,foravariablex j ,j ∈ {1,2,...,n},a ij ≥ 0 ∀i ∈ {1,2,...,m},andc j ≥ 0,thenthevariablex j can be fixed to its lower bound l j . Similarly, if a ij ≤ 0 ∀i ∈ {1,2,...,m} and c j ≤ 0, then the variable x j canbe fixed to its upper bound u j . Sometimes, an upper bound on the optimal value of solution of (P) is known (it may be provided by the user or may be obtained from some heuristics) before presolving. In such a case, we have an additional inequality ∑ n j=1 c jx j ≤ z u , where z u is the best-known upper bound. The basic presolving techniques applicable to other constraints can be applied to this constraint as well. Valid inequalities of the form ∑ n j=1ĉjx j ≥ z l , where z l ∈ Q, can also be obtained by linear combinations of the objective function with other constraints. Such inequalities are automatically
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Presolving Mixed-Integer Linear Programs - COR@L

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