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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.
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization).
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 ...
In linear programming problems with n variables, a necessary but insufficient condition for the feasible set to be bounded is that the number of constraints be at least n + 1 (as illustrated by the above example). If the feasible set is unbounded, there may or may not be an optimum, depending on the specifics of the objective function.
There is a close connection between linear programming problems, eigenequations, and von Neumann's general equilibrium model. The solution to a linear programming problem can be regarded as a generalized eigenvector. The eigenequations of a square matrix are as follows:
In the study of algorithms, an LP-type problem (also called a generalized linear program) is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms.
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 ...
However, there is a fractional solution in which each set is assigned the weight 1/2, and for which the total value of the objective function is 3/2. Thus, in this example, the linear programming relaxation has a value differing from that of the unrelaxed 0–1 integer program.