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The variables y of the dual LP are the coefficients of this linear combination. The dual LP tries to find such coefficients that minimize the resulting upper bound. This gives the following LP: [1]: 81–83 Minimize b T y subject to A T y ≥ c, y ≥ 0 . This LP is called the dual of the original LP.
According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem. Von Neumann noted that he was using information from his game theory , and conjectured that two person zero sum matrix game was equivalent to linear programming.
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 ...
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 theorem of linear programming duality says that we can reduce the above minimization problem to the search problem: find x,y s.t. Ax ≤ b ; A T y = c ; y ≤ 0 ; c T x=b T y. The first problem is solvable iff the second problem is solvable; in case the problem is solvable, the x -components of the solution to the second problem are an ...
These added costs are used instead of the strict inequality constraints in the optimization. In practice, this relaxed problem can often be solved more easily than the original problem. The problem of maximizing the Lagrangian function of the dual variables (the Lagrangian multipliers) is the Lagrangian dual problem.
Similarly, an integer program (consisting of a collection of linear constraints and a linear objective function, as in a linear program, but with the additional restriction that the variables must take on only integer values) satisfies both the monotonicity and locality properties of an LP-type problem, with the same general position ...
A basis B of the LP is called dual-optimal if the solution = is an optimal solution to the dual linear program, that is, it minimizes . In general, a primal-optimal basis is not necessarily dual-optimal, and a dual-optimal basis is not necessarily primal-optimal (in fact, the solution of a primal-optimal basis may even be unfeasible for the ...