ILOG CPLEX 11.0 User's Manual

◆ Parameters Affecting Cuts on page 274Clique CutsA clique is a relationship among a group of binary variables such that at most one variable inthe group can be positive in any integer feasible solution. Before optimization starts,ILOG CPLEX constructs a graph representing these relationships and finds maximal cliquesin the graph.Cover CutsIf a constraint takes the form of a knapsack constraint (that is, a sum of binary variables withnonnegative coefficients less than or equal to a nonnegative righthand side), then there is aminimal cover associated with the constraint. A minimal cover is a subset of the variables ofthe inequality such that if all the subset variables were set to one, the knapsack constraintwould be violated, but if any one subset variable were excluded, the constraint would besatisfied. ILOG CPLEX can generate a constraint corresponding to this condition, and thiscut is called a cover cut.Disjunctive CutsA MIP problem can be divided into two subproblems with disjunctive feasible regions oftheir LP relaxations by branching on an integer variable. Disjunctive cuts are inequalitiesvalid for the feasible regions of LP relaxations of the subproblems, but not valid for thefeasible region of LP relaxation of the MIP problem.Flow Cover CutsFlow covers are generated from constraints that contain continuous variables, where thecontinuous variables have variable upper bounds that are zero or positive depending on thesetting of associated binary variables. The idea of a flow cover comes from considering theconstraint containing the continuous variables as defining a single node in a network wherethe continuous variables are in-flows and out-flows. The flows will be on or off dependingon the settings of the associated binary variables for the variable upper bounds. The flowsand the demand at the single node imply a knapsack constraint. That knapsack constraint isthen used to generate a cover cut on the flows (that is, on the continuous variables and theirvariable upper bounds).Flow Path CutsFlow path cuts are generated by considering a set of constraints containing the continuousvariables that define a path structure in a network, where the constraints are nodes and thecontinuous variables are in-flows and out-flows. The flows will be on or off depending onthe settings of the associated binary variables.Gomory Fractional CutsGomory fractional cuts are generated by applying integer rounding on a pivot row in theoptimal LP tableau for a (basic) integer variable with a fractional solution value.272 ILOG CPLEX 11.0 — USER’ S MANUAL