Column generation in infeasible predictor-corrector methods for solving linear programs.

摘要

Primal-dual interior-point methods (IPMs) are distinguished for their exceptional theoretical properties and computational behavior in solving linear programming (LP) problems. Consider solving the primal-dual LP pair using an IPM such as a primal-dual Affine-Scaling method, Mehrotra's Predictor-Corrector method (the most commonly used IPM to date), or Potra's Predictor-Corrector method. The bulk of the computation in the process stems from the formation of the normal equation matrix, AD2 AT, where A ∈ reals mxn and D2 = S-1 X is a diagonal matrix. In cases when n m, we propose to reduce this cost by incorporating a column generation scheme into existing infeasible IPMs for solving LPs. In particular, we solve an LP problem based on an iterative approach where we select a "small" subset of the constraints at each iteration with the aim of achieving both feasibility and optimality. Rather than n constraints, we work with k = |Q| ∈ [m, n] constraints at each iteration, where Q is an index set consisting of the k most nearly active constraints at the current iterate. The cost of the formation of the matrix, AQD2QATQ , reduces from theta(m2n) to theta(m2k) operations, where k is relatively small compared to n. Although numerical results show an occasional increase in the number of iterations, the total operation count and time to solve the LP using our algorithms is, in most cases, small compared to other "reduced" LP algorithms.