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The linear programming problem was first shown to be solvable in polynomial time by Leonid Khachiyan in 1979, [9] but a larger theoretical and practical breakthrough in the field came in 1984 when Narendra Karmarkar introduced a new interior-point method for solving linear-programming problems. [10]
Generalized assignment problem; Integer programming. The variant where variables are required to be 0 or 1, called zero-one linear programming, and several other variants are also NP-complete [2] [3]: MP1 Some problems related to Job-shop scheduling; Knapsack problem, quadratic knapsack problem, and several variants [2] [3]: MP9
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.
"The linear complementarity problem, sufficient matrices, and the criss-cross method" (PDF). Linear Algebra and Its Applications. 187: 1– 14. doi: 10.1016/0024-3795(93)90124-7. Murty, Katta G. (January 1972). "On the number of solutions to the complementarity problem and spanning properties of complementary cones" (PDF). Linear Algebra and ...
A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities. [6] The list of constraints is a system of linear inequalities.
Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework. Since the following is valid:
Otherwise, it is called unbalanced assignment. [1] If the total cost of the assignment for all tasks is equal to the sum of the costs for each agent (or the sum of the costs for each task, which is the same thing in this case), then the problem is called linear assignment.
Basic Linear Algebra Subprograms (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, dot products, linear combinations, and matrix multiplication.