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The set of solutions of a real linear inequality constitutes a half-space of the 'n'-dimensional real space, one of the two defined by the corresponding linear equation. The set of solutions of a system of linear inequalities corresponds to the intersection of the half-spaces defined by individual inequalities.
A solution of a linear system is an assignment of values to the variables ,, …, such that each of the equations is satisfied. The set of all possible solutions is called the solution set. [5] A linear system may behave in any one of three possible ways: The system has infinitely many solutions.
Generalizations of the Farkas' lemma are about the solvability theorem for convex inequalities, [4] i.e., infinite system of linear inequalities. Farkas' lemma belongs to a class of statements called "theorems of the alternative": a theorem stating that exactly one of two systems has a solution. [5]
Relaxation methods were developed for solving large sparse linear systems, which arose as finite-difference discretizations of differential equations. [2] [3] They are also used for the solution of linear equations for linear least-squares problems [4] and also for systems of linear inequalities, such as those arising in linear programming.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: A half-space is a polyhedron defined by a single linear inequality, a 1 T x ≤ b 1.; A hyperplane is a polyhedron defined by two inequalities, a 1 T x ≤ b 1 and a 1 T x ≥ b 1 (which is equivalent to -a 1 T x ≤ -b 1).
Redundant constraint can be identified by solving a linear program as follows. Given a linear constraints system, if the -th inequality is satisfied for any solution of all other inequalities, then it is redundant. Similarly, STIs refers to inequalities that are implied by the non-negativity of information theoretic measures and basic ...
More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope , which is a set defined as the intersection of finitely many half spaces , each of which is defined by a linear inequality.