Novel update techniques for the revised simplex method

Abstract

This paper introduces three novel techniques for updating the invertible representation of the basis matrix when solving practical sparse linear programming problems using a high performance implementation of the dual revised simplex method, being of particular value when suboptimization is used. Two are variants of the product form update and the other permits multiple Forrest-Tomlin updates to be performed. Computational results show that one of the product form variants is significantly more efficient than the traditional approach, with its performance approaching that of the Forrest-Tomlin update for some problems. The other is less efficient, but valuable in the context of the dual revised simplex method with suboptimization. Results show that the multiple Forrest-Tomlin updates are performed with no loss of serial efficiency.

Analisis Sensitivitas dalam Optimalisasi Produksi UD Mekar Sari Karang Klesem Menggunakan Program Linier dengan Metode Simpleks Diperbaiki

“…Jika persoalan linier program cukup besar maka sangatlah tidak efisien apabila membawa semua elemen pada tabel berikutnya. Metode simpleks yang diperbaiki atau metode simpleks multiplier menjadi cara atau metode yang lebih efisien yang dapat digunakan agar tidak semua elemen terbawa ke tabel berikutnya [13], [14].…”

Permutations in the Factorization of Simplex Bases

“…representations of the basis inverse (e.g. for the product-form inverse [11]). As a result of his computational experience with CPLEX [25] and Gurobi [8], Bixby recommended the use of "some variant" of the Forrest-Tomlin update in 2009 [2].…”

Parallelizing the dual revised simplex method

“…The standard choices are the relatively simple product form (PF) update [18] or the efficient Forrest-Tomlin (FT) update [5]. A comprehensive report on simplex update techniques is given by Elble and Sahinidis [3] and novel techniques, some motivated by the design and development of pami, are described by Huangfu and Hall [14]. For the purpose of this report, the features of all relevant update methods are summarised as follows.…”

“…• The Forrest-Tomlin (FT) update assumes B k = L k U k and uses both the partial ftran resultã q = L −1 k a q and partial btran resultẽ T p = e T p U −1 k to modify U k and augment L k . • The alternate product form (APF) update [14] uses the btran result e T p so that…”

“…• Following suboptimization, the collective Forrest-Tomlin (CFT) update [14] updates B −1 k to B −1 k+t directly, using partial results obtained with B −1 k which are required for simplex iterations.…”