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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).
The configuration linear program (configuration-LP) is a linear programming technique used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock problem. [1] [2] Later, it has been applied to the bin packing [3] [4] and job scheduling problems.
Suppose we have the linear program: Maximize c T x subject to Ax ≤ b, x ≥ 0. We would like to construct an upper bound on the solution. So we create a linear combination of the constraints, with positive coefficients, such that the coefficients of x in the constraints are at least c T. This linear combination gives us an upper bound on the ...
The tableau is a representation of the linear program where the basic variables are expressed in terms of the non-basic ones: [1]: 65 = + = + where is the vector of m basic variables, is the vector of n non-basic variables, and is the maximization objective.
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 large linear-programming problems A is typically a sparse matrix and, when the resulting sparsity of B is exploited when maintaining its invertible representation, the revised simplex algorithm is much more efficient than the standard simplex method. Commercial simplex solvers are based on the revised simplex algorithm.
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 ...
If is a subset of a vector space then a linear sub-variety (that is, an affine subspace) of the vector space is called a support variety if meets (that is, is not empty) and every open segment whose interior meets is necessarily a subset of . [3] A 0-dimensional support variety is called an extreme point of . [3]