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A problem with five linear constraints (in blue, including the non-negativity constraints). In the absence of integer constraints the feasible set is the entire region bounded by blue, but with integer constraints it is the set of red dots. A closed feasible region of a linear programming problem with three variables is a convex polyhedron.
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.
In the theory of linear programming, a basic feasible solution (BFS) is a solution with a minimal set of non-zero variables. Geometrically, each BFS corresponds to a vertex of the polyhedron of feasible solutions. If there exists an optimal solution, then there exists an optimal BFS.
An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...
The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming.
In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners.
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).
In linear programming, a discipline within applied mathematics, a basic solution is any solution of a linear programming problem satisfying certain specified technical conditions. For a polyhedron P {\displaystyle P} and a vector x ∗ ∈ R n {\displaystyle \mathbf {x} ^{*}\in \mathbb {R} ^{n}} , x ∗ {\displaystyle \mathbf {x} ^{*}} is a ...