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1.1 Linear programming problem
In MathProg it is assumed that the linear programming (LP) problem has the following statement:
Minimize (or maximize)
z = c1 x1 + c2 x2 + … + cn xn + c0
subject to linear constraints
L1 <= a11 x1 + a12 x2 + … + a1n xn <= U1
L2 <= a21 x1 + a22 x2 + … + a2n xn <= U2
. . . . .
Lm <= am1 x1 + am2 x2 + … + amn xn <= Umand bounds of variables
l1 <= x1 <= u1
l2 <= x2 <= u2
. . . . .
ln <= xn <= un
where: | |
x1, x2, …, xn | are variables; |
z | is the objective function; |
c1, c2, …, cn | are coefficients of the objective function; |
c0 | is the constant term (“shift”) of the objective function; |
a11, a12, …, amn | are constraint coefficients; |
L1, L2, …, Lm | are lower constraint bounds; |
U1, U2, …, Um | are upper constraint bounds; |
l1, l2, …, ln | are lower bounds of variables; |
u1, u2, …, un | are upper bounds of variables. |
Bounds of variables and constraint bounds can be finite as well as infinite. Besides, lower bounds can be equal to corresponding upper bounds. Thus, the following types of variables and constraints are allowed:
-inf < x < +inf
Free (unbounded) variable
x >= l
Variable with lower bound
x <= u
Variable with upper bound
l <= x <= u
Double-bounded variable
x = l (= u)
Fixed variable
-inf < sum aj xj < +inf
Free (unbounded) linear form
sum aj xj >= L
Inequality constraint “greater than or equal to”
sum aj xj <= U
Inequality constraint “less than or equal to”
L <= sum aj xj <= U
Double-bounded inequality constraint
sum aj xj = L (= U)
Equality constraint
In addition to pure LP problems MathProg allows mixed integer linear programming (MIP) problems, where some (or all) structural variables are restricted to be integer.
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