Augmented Lagrangian method for large-scale linear programming problems

Abstract

The augmented Lagrangian and Newton methods are used to simultaneously solve the primal and dual linear programming problems. The proposed approach is applied to the primal linear programming problem with a very large number (≈10 6 ) of nonnegative variables and a moderate (≈10 3 ) number of equality-type constraints. Computation results such as the solution of a linear programme with 10 million primal variables are presented.

Augmented Lagrangian method within L-shaped method for stochastic linear programs

“…In each iteration of the modified L-shaped algorithm, master problem (26) or (27) and subproblems (8) are solved. Although, if the current point is feasible, N concave, piecewise quadratic, unconstrained maximization problems (28) have to be solved as well.…”

“…(14) It can be shown that if β is sufficiently large, solving the inner maximization problem in (14) gives the unique solution of the problem (9). Therefore, instead of problem (14), we propose to solve the unconstrained maximization problem [8] max p∈R m 2 S(p, β,ŷ) (15) in which β,ŷ are constants and the function S(p, β,ŷ) is defined by…”

“…[8] Assume that the solution set Y * is non-empty. For each β > 0 andŷ ∈ Y * , π * = p(β ) β is one exact solution for the dual of the problem (7) where p(β) is the solution for the problem (15).…”

The ‘Idiot’ crash quadratic penalty algorithm for linear programming and its application to linearizations of quadratic assignment problems

“…Although originally intended for nonlinear programming problems, the augmented Lagrangian method has also been applied to linear programming problems [3,7]. However, neither article assesses its performance on large scale practical LP problems.…”

Computing minimum norm solution of linear systems of equations by the generalized Newton method

“…Thus from (4), (5), and (6), it can be concluded that x * satisfies Karush-Kuhn-Tucker optimality conditions for problem (1).…”