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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 theory behind linear programming drastically reduces the number of possible optimal solutions that must be checked. In 1963, Dantzig's Linear Programming and Extensions was published by Princeton University Press. The book quickly became a standard text in linear programming.
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:
FICO Xpress – solver for linear and quadratic programming with continuous or integer variables (MIP). FortMP – linear and quadratic programming. FortSP – stochastic programming. GAMS – General Algebraic Modeling System. Gurobi Optimizer – solver for linear and quadratic programming with continuous or integer variables (MIP).
This is an integer linear program. However, we can solve it without the integrality constraints (i.e., drop the last constraint), using standard methods for solving continuous linear programs. While this formulation allows also fractional variable values, in this special case, the LP always has an optimal solution where the variables take ...
Pages in category "Linear programming" The following 58 pages are in this category, out of 58 total. This list may not reflect recent changes. ...
Cutting plane methods for MILP work by solving a non-integer linear program, the linear relaxation of the given integer program. The theory of Linear Programming dictates that under mild assumptions (if the linear program has an optimal solution, and if the feasible region does not contain a line), one can always find an extreme point or a ...
In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners.