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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 <= Um

and 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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