@MISC{n.n._interiorpoint, author = {n.n.}, title = { Interior Point Methods }, year = {} }

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Abstract

The interior point method for linear programming was introduced by Karmakar in 1984. It runs in polynomial time and is a practical method. For many problems it is competitive or superior to the simplex method. Many LP packages, e.g., CPLEX, offer a simplex as well as an interior point method. Links to implementations can be found on Michael Trick’s web page. We give a brief account and refer the readers to [GT89] for more details. We describe the approach for problems in the form maximize c T x subject to Ax ≤ b. Let f0(x) = c T x and let fi(x) = bi − a T i x where aT i is the i-th row of A, 1 ≤ i ≤ m. Let P = {x; Ax ≤ b} be the feasible set, let x ∗ be an optimal solution, and let p ∗ = c T x ∗ be the optimal objective value. The simplex method walks along the boundary of the feasible set, the interior point method walks through the interior. It replaces the constraint fi(x) ≥ 0 by a penalty term in the objective function. For real parameter t ≥ 1, consider the function The second term gt(x) = f0(x)+ 1 t ∑ ln fi(x).