ILOG CPLEX 11.0 User's Manual

When a linear optimization problem is stated in that conventional form, its coefficients andvalues are customarily referred to by these terms:objective function coefficients c 1 , ..., c nconstraint coefficients a 11 , ..., a mnrighthand side b 1 , ..., b mupper bounds u 1 , ..., u nlower bounds l 1 , ..., l nvariables or unknowns x 1 , ..., x nIn the most basic linear optimization problem, the variables of the objective function arecontinuous in the mathematical sense, with no gaps between real values. To solve such linearprogramming problems, ILOG CPLEX implements optimizers based on the simplexalgorithms (both primal and dual simplex) as well as primal-dual logarithmic barrieralgorithms and a sifting algorithm. These alternatives are explained more fully in SolvingLPs: Simplex Optimizers on page 171.ILOG CPLEX can also handle certain problems in which the objective function is not linearbut quadratic. Such problems are known as quadratic programs or QPs. Solving Problemswith a Quadratic Objective (QP) on page 227, covers those kinds of problems.ILOG CPLEX also solves certain kinds of quadratically constrained problems. Suchproblems are known as quadratically constrained programs or QCPs. Solving Problems withQuadratic Constraints (QCP) on page 239, tells you more about the kinds of quadraticallyconstrained problems that ILOG CPLEX solves, including the special case of second ordercone programming (SOCP) problems.ILOG CPLEX is also a tool for solving mathematical programming problems in which someor all of the variables must assume integer values in the solution. Such problems are knownas mixed integer programs or MIPs because they may combine continuous and discrete (forexample, integer) variables in the objective function and constraints. MIPs with linearobjectives are referred to as mixed integer linear programs or MILPs, and MIPs withquadratic objective terms are referred to as mixed integer quadratic programs or MIQPs.Likewise, MIPs that are also quadratically constrained in the sense of QCP are known asmixed integer quadratically constrained programs or MIQCPs.Within the category of mixed integer programs, there are two kinds of discrete integervariables: if the integer values of the discrete variables must be either 0 (zero) or 1 (one),then they are known as binary; if the integer values are not restricted in that way, they areknown as general integer variables. This manual explains more about the mixed integeroptimizer in Solving Mixed Integer Programming Problems (MIP) on page 255.ILOG CPLEX also offers a Network Optimizer aimed at a special class of linear problemwith network structures. ILOG CPLEX can optimize such problems as ordinary linearprograms, but if ILOG CPLEX can extract all or part of the problem as a network, then itILOG CPLEX 11.0 — USER’ S MANUAL 29