ILOG CPLEX 11.0 User's Manual

Stating a MIP ProblemA mixed integer programming (MIP) problem may contain both integer and continuousvariables. If the problem contains an objective function with no quadratic term, (a linearobjective), then the problem is termed a Mixed Integer Linear Program (MILP). If there is aquadratic term in the objective function, the problem is termed a Mixed Integer QuadraticProgram (MIQP). If the model has any constraints containing a quadratic term, regardless ofthe objective function, the problem is termed a Mixed Integer Quadratically ConstrainedProgram (MIQCP).In ILOG CPLEX documentation, if the discussion pertains specifically to the MILP, MIQP,or MIQCP case, then that term is used. For the majority of topics that pertain equally toMILP, MIQP, and MIQCP, the comprehensive term MIP is used.Integer variables may be restricted to the values 0 (zero) and 1 (one), in which case they arereferred to as binary variables. Or they may take on any integer values, in which case theyare referred to as general integer variables. A variable of any MIP that may take either thevalue 0 (zero) or a value between a lower and an upper bound is referred to assemi-continuous. A semi-continuous variable that is restricted to integer values is referred toas semi-integer. Using Semi-Continuous Variables: a Rates Example on page 333 says a bitmore about semi-continuous variables later in this manual. Special Ordered Sets (SOS) arediscussed in Using Special Ordered Sets (SOS) on page 329. Continuous variables in a MIPproblem are those which are not restricted in any of these ways, and are thus permitted totake any solution value within their (possibly infinite) lower and upper bounds.In ILOG CPLEX documentation, the comprehensive term integer variable means any of thevarious types just mentioned except for continuous or SOS. The presence or absence of aquadratic term in the objective function or among the constraints for a given variable has nobearing on its being classified as continuous or integer.The following formulation illustrates a mixed integer programming problem, which issolved in the example program ilomipex1.cpp / mipex1.c, discussed later in thischapter:Maximize x 1 + 2x 2 + 3x 3 + x 4subject to - x 1 + x 2 + x 3 + 10x 4 ≤ 20x 1 - 3x 2 + x 3 ≤ 30x 2 - 3.5x 4 = 0with these bounds 0 ≤ x 1 ≤ 400 ≤ x 2 ≤ +∞0 ≤ x 3 ≤ +∞2 ≤ x 4 ≤ 3integerx 4256 ILOG CPLEX 11.0 — USER’ S MANUAL