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Linear programming is a mathematical method for determining a way to achieve the best outcome (such as maximum profit or lowest cost) in a given mathematical model for some list of requirements represented as linear relationships. Linear programming arose as a mathematical model developed during World War II to plan expenditures and returns in ...
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.
Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture ...
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.
Dantzig–Wolfe decomposition relies on delayed column generation for improving the tractability of large-scale linear programs. For most linear programs solved via the revised simplex algorithm, at each step, most columns (variables) are not in the basis. In such a scheme, a master problem containing at least the currently active columns (the ...
The Stigler diet is an optimization problem named for George Stigler, a 1982 Nobel laureate in economics, who posed the following problem:. For a moderately active man weighing 154 pounds, how much of each of 77 foods should be eaten on a daily basis so that the man’s intake of nine nutrients will be at least equal to the recommended dietary allowances (RDAs) suggested by the National ...
GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms.
Column generation or delayed column generation is an efficient algorithm for solving large linear programs. The overarching idea is that many linear programs are too large to consider all the variables explicitly. The idea is thus to start by solving the considered program with only a subset of its variables.