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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 ...
A closed feasible region of a linear programming problem with three variables is a convex polyhedron. In mathematical optimization and computer science , a feasible region, feasible set, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints ...
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).
A LP can also be unbounded or infeasible. Duality theory tells us that: If the primal is unbounded, then the dual is infeasible; If the dual is unbounded, then the primal is infeasible. However, it is possible for both the dual and the primal to be infeasible. Here is an example:
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 ...
Linear programming relaxation is a standard technique for designing approximation algorithms for hard optimization problems. In this application, an important concept is the integrality gap, the maximum ratio between the solution quality of the integer program and of its relaxation.
In some cases, infeasible problems are handled by minimizing a sum of feasibility violations. Some special cases of nonlinear programming have specialized solution methods: If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex , then the program is called convex and ...
Leukapheresis (/ ˌ l u ˈ k ʌ f ɜːr iː s ɪ s / ⓘ) is a laboratory procedure in which white blood cells are separated from a sample of blood. It is a specific type of apheresis, the more general term for separating out one particular constituent of blood and returning the remainder to the circulation.