The set of solutions of a system of linear inequalities corresponds to the intersection of the half-spaces defined by individual inequalities.
12.
The " Birkhoff von Neumann theorem " states that this polytope can be described by two types of linear inequality or equality.
13.
Usually there are, depending on the technology used, additional restrictions that can be modeled as a linear inequalities with integer or binary variables.
14.
Integer programming is the problem of optimizing a linear or nonlinear objective function over the set of integer points satisfying a system of linear inequalities.
15.
Hence, a "'closed convex polytope "'may be regarded as the set of solutions to the system of linear inequalities:
16.
If it is not, there is guaranteed to exist a linear inequality that " separates " the optimum from the convex hull of the true feasible set.
17.
The coefficients of each row of " A " and " b " correspond with the coefficients of the linear inequality defining the respective half-space.
18.
It turns out that any linear programming problem can be reduced to a linear feasibility problem ( e . g . minimize the zero function subject to some linear inequality and equality constraints ).
19.
The above definition requires well-defined operations of addition, multiplication and comparison, therefore the notion of a linear inequality may be extended to ordered rings, in particular, to ordered fields.
20.
"' Sergei Nikolaevich Chernikov "'( 11 May 1912 23 January 1987; ) was a Russian mathematician who contributed significantly to the development of infinite group theory and linear inequalities.
