Linear programming

In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear.

Algorithms

The simplex algorithm solves LP problems by constructing an admissible solution at a vertex of the polyhedron, and then walking along edges of the polyhedron to vertices with successively higher values of the objective function until the optimum is reached. Although this algorithm is quite efficient in practice, and can be guaranteed to find the global optimum if certain precautions against cycling are taken, it has poor worst-case behavior: it is possible to construct a linear programming problem for which the simplex method takes a number of steps exponential in the problem size. In fact for some time it was not known whether the linear programming problem was NP-complete or solvable in polynomial time.

Related Topics:
Simplex algorithm - Algorithm - Linear programming - NP-complete - Polynomial time

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The first worst-case polynomial-time algorithm for the linear programming problem was proposed by Leonid Khachiyan in 1979. It was based on the ellipsoid method in nonlinear optimization by Naum Shor, which is the generalization of the ellipsoid method in convex optimization by Arkadi Nemirovski, a 2003 John von Neumann Theory Prize winner, and D. Yudin.

Related Topics:
Leonid Khachiyan - 1979 - Ellipsoid method - Nonlinear optimization - Naum Shor - Convex optimization - Arkadi Nemirovski - 2003 - John von Neumann Theory Prize - D. Yudin

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However, the practical performance of Khachiyan's algorithm is disappointing: generally, the simplex method is more efficient. Its main importance is that it encouraged the research of interior point methods. In contrast to the simplex algorithm, which only progresses along points on the boundary of the feasible region, interior point methods can move through the interior of the feasible region.

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In 1984, N. Karmarkar proposed the projective method. This is the first algorithm performing well both in theory and in practice: Its worst-case complexity is polynomial and experiments on practical problems show that it is reasonably efficient compared to the simplex algorithm. Since then, many interior point methods have been proposed and analysed. A popular interior point method is the Mehrotra predictor-corrector method, which performs very well in practice even though little is known about it theoretically.

Related Topics:
1984 - N. Karmarkar - Projective method - Mehrotra predictor-corrector method

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The current opinion is that the efficiency of good implementations of simplex-based methods and interior point methods is similar for routine applications of linear programming.

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LP solvers are in widespread use for optimization of various problems in industry, such as optimization of flow in transportation networks, many of which can be transformed into linear programming problems only with some difficulty.

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