Presolving Mixed-Integer Linear Programs - COR@L

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PresolvingMixedInteger LinearPrograms 3 is infeasible because of this constraint. If we have a singleton row, namely, a ik ≠ 0,a ij = 0 ∀j ≠ k,j ∈ {1,2,...,n}, for some i ∈ {1,2,...,m} and some k ∈ {1,2,...,n}, then the ith constraint can be removed,and the bounds ofvariablex k can be tightenedif necessary. Removing variables If we have a ij = 0 ∀i ∈ {1,2,...,m} for some j ∈ {1,2,...,n}, then we can fix the variable x j to l j if c j ≥ 0 or to u j if c j < 0. After fixing the variable, we may have an additional constant term in the objective function. If we have a singleton column x j , and if j ∉ I, a ij > 0, l j = −∞, u j = ∞, c j ≤ 0 then we can drop the constraint i and substitute x j = b i− ∑ k:k≠j a ikx k a ij intheobjectivefunction. Afterthesubstitution,wehaveonelessvariableand one less constraint. The number of nonzeros in A is also reduced. The number of nonzeros in the objectivefunction,however,may increase. Rearrangements The variables and constraints of the instance can be rearranged depending on howthemethodsofsolvingareimplemented. Forexample,itmaybebeneficialtohaveallbinary variablesstoredintheendofthearrayofvariablessothatdeletingthem(incasewefixthem)from the sparse representation of the matrix A is faster. Similarly, the constraints can be rearranged so that “constraints of interest” appear together in the arrays in which they are stored. If it is known thataninterior-pointmethodwillinitiallybeusedforsolvingtherelaxationof(P),thenrowsand columns may be rearranged so as to reduce the “fill in”. Rothberg and Hendrickson (1998) study the effectsof varioussuchpermutation techniqueson interior-pointmethods. 2.2 Granularity Givenan instance ofform(P) and avector a ∈ Q n ,wedefine granularityof aas g(a) = min x 1 ,x 2 ∈P ⎧ ⎨ n∑ ⎩ a ij x 1 j − ∣ j=1 n∑ a ij x 2 n∑ j ∣ : a ij x 1 j ≠ j=1 j=1 ⎫ n∑ ⎬ a ij x 2 j ⎭ , whereP isthesetofallpointssatisfyingtheconstraintsof(P). Granularityreferstotheminimum difference (greater than zero) between the values of activity that a constraint or objective function may have at two different feasible points. For a constraint that has only integer-constrained variables, then granularity is at least the greatest common divisor (GCD) of all the coefficients. If any of the coefficients in such a constraint are not integers, we can find a multiplier 10 K , where K ∈ Z + , to make all coefficients integers. The GCD can be divided by 10 K to get granularity. Hoffman and Padberg (1991) call this procedure Eucledean reduction. The problem can be declaredinfeasibleifthereisanequalityconstraintwhoseright-hand sideisnotanintegermultiple of the of the GCD. For an inequality constraint i ∈ {1,2,...,m}, if b i is not an integer multiple of the granularity g(a i ), then b i can be reduced to the nearest multiple of g i less than b i . So the constraint 3x 1 +6x 2 −3x 3 ≤ 5.3 j=1
4 Ashutosh Mahajan can be tightenedto 3x 1 +6x 2 −3x 3 ≤ 3, ifx 1 ,x 2 ,x 3 ∈ Z. TheboundtighteninginthiscaseisaspecialcaseofaChvátal-Gomoryinequality introduced by Gomory (1958). Similarly, one can find the granularity of the objective vector c when only integer-constrained variables have nonzero coefficients in the objective function. The differencebetweenupperboundobtainedfromafeasiblesolutionandtheoptimalsolutionvalue will always be at least g(c). If the difference between an upper bound and a lower bound falls below the granularity, we can stop the branch-and-bound or cutting-plane algorithm and claim that the optimal solutionhas beenfound. 2.3 Constraintand Variable Duplication Sometimesaninstancemayhaveduplicateconstraintsthatareexactcopiesofeachother. Atother times,wemayhaveconstraintsthatareidenticalexceptforascalarmultiple. Theobviousbenefit ofdetectingsuchconstraintsisthatwecanreducethememoryneededforstoringandsolvingthe instance. ItalsohelpsimprovethenumericalstabilityofsolutionoftheassociatedLPrelaxations. Bixby and Wagner (1987), and Tomlinand Welch (1986) describe efficient ways to detect from a column-ordered format of A, whether two constraints are identical except for scalar multiples. When constraints with duplicate entries are found, we can, based on the values of the right hand sides and the scalar multiples, declare the problem infeasible or delete one of the constraints or combine the twoconstraints into one. Similarly,wecan detectwhether twocolumns arealikeexcept forascalarmultiple. However, the possible integrality constraint on one or both of these variables can make any substitution complicated. If A j = αA i and if c j = αc i for some i,j ∈ {1,2,...,n} \ I,α ∈ Q, then we can replacethetwocolumnsA i ,A j byasinglecolumnA k thatisidenticaltoA i . Thelowerandupper bounds of the new variable x k are l k = l i + αl j ,u k = u i + αu j if α > 0 and l k = l i + αu j ,u k = u i + αl j otherwise. If one or both variables are integer constrained, we can use the substitution x k = x i + αx j if we can find feasible values of x i ,x j for each x k ∈ [l k ,u k ]. We can also add a constraint x k = x i +αx j to avoidsuchcomplications. 2.4 ConstraintDomination One way to identify a redundant constraint is to check whether it can be termwise dominated by another constraint. If b k ≥ b i for i,k ∈ {1,2,...,m}, each variable in constraints i,k is either non-negative or non-positive, a kj ≤ a ij for j such that x j is non-negative and a kj ≥ a ij for j such that x j is non-positive, then constraint k can be deleted because it is redundant in the presence of constraint i and the bounds onvariables.
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Presolving Mixed-Integer Linear Programs - COR@L

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