ILOG CPLEX 11.0 User's Manual

Restricting Presolve ReductionsAs mentioned in Introduction to Presolve on page 474, some presolve reductions areinvalidated when a problem is modified. The advanced presolve interface therefore allows auser to tell presolve what sort of modifications will be made in the future, so presolve canavoid possibly invalid reductions. These considerations only apply to linear programs. Anymodifications of QP or QCP models will cause ILOG CPLEX to discard the presolvedmodel. The following sections document special considerations about presolve reductions.◆ Adding Constraints to the First Solution on page 476◆ Primal and Dual Considerations in Presolve Reductions on page 477◆ Cuts and Presolve Reductions on page 477◆ Infeasibility or Unboundedness in Presolve Reductions on page 478◆ Protected Variables in Presolve Reductions on page 478Adding Constraints to the First SolutionConsider adding a constraint to a problem after solving it. Imagine that you want to optimizea linear program:Primal:Dual:max -x1, + x2 + x3 min 6y1, + 5y2,st x1, + x2 + 2x3 ≤ 6 st y1, ≥ -1x2 + x3 ≤ 5 y1, + y2, ≥ 10 2y1, + y2, ≥ 1x1, x2, x3 ≥ 0 y1, y2, y3 ≥ 0Note that the first constraint in the dual (y1 ≥ -1) is redundant. Presolve can use thisinformation about the dual problem (and complementary slackness) to conclude thatvariable x1 can be fixed to 0 and removed from the presolved problem. While it may beintuitively obvious from inspection of the primal problem that x1 can be fixed to 0, it isimportant to note that dual information (redundancy of the first dual constraint) is used toprove it formally.Now consider the addition of a new constraint x2 ≤ 5x1:Primal:Dual:max -x1, + x2 + x3 min 6y1, + 5y2,st x1, + x2 + 2x3 ≤ 6 st y1, − 5y3 ≥ -1x2 + x3 ≤ 5 y1, + y2, + y3 ≥ 1476 ILOG CPLEX 11.0 — USER’ S MANUAL