ILOG CPLEX is a tool for solving, first of all, linear optimization problems. Such problems are conventionally written like this:

Minimize (or maximize)	c1x1 + c2x2 + . . . + cnxn
subject to	a11x1 + a12x2 + . . . + a1nxn	~	b1
	a21x1 + a22x2 + . . . + a2nxn	~	b2
	. . .
	am1x1 + am2x2 + . . . + amnxn	~	bm
with these bounds

where the relation ~ may be greater than or equal to, less than or equal to, or simply equal to, and the upper bounds ui and lower bounds li may be positive infinity, negative infinity, or any real number.

When a linear optimization problem is stated in that conventional form, its coefficients and values are customarily referred to by these terms:

ILOG CPLEX is also a tool for solving mathematical programming problems in which some or all of the variables must assume integer values in the solution. Such problems are known as mixed integer programs or MIPs because they may combine continuous and discrete (for example, integer) variables in the objective function and constraints. MIPs with linear objectives are referred to as mixed integer linear programs or MILPs, and MIPs with quadratic 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 as mixed integer quadratically constrained programs or MIQCPs.