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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.
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
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.
The weak duality theorem says that, for each feasible solution x of the primal and each feasible solution y of the dual: c T x ≤ b T y. In other words, the objective value in each feasible solution of the dual is an upper-bound on the objective value of the primal, and objective value in each feasible solution of the primal is a lower-bound ...
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 ...
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 possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called infeasible. In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point.
Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special ... Starting with a feasible solution to the reduced master program ...