Letter to the Editor—Computational Results of an Integer Programming Algorithm

Abstract

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Classical Cuts for Mixed-Integer Programming and Branch-and-Cut

“…Secondly, there were plenty of early attempts to implement cutting plane theory numerically nevertheless, c.f. Fleischmann (1970), Fiorot and Gondran (1969), Haldi and Isaacson (1965), Krolak (1969), Knuth (1961), Martin (1963), Miliotis (1978), Story andWagner (1963), Trauth Jr andWoolsey (1969), Wagner, Giglio, and Glaser (1964) and probably more. Most of the pertaining computational studies used the fractional cut (GP = ) or its "all-integer" variant (Glover, 1968;Gondran, 1973;Wilson, 1967;Young, 1968), whereas the stronger mixed-integer cut (GM # ) was studied to a lesser degree; probably because it makes a "mixed" integer program even more "mixed" as the surplus variable in (GM # ) is in general a flow variable.…”

“…(Krolak, 1969;Knuth, 1961;Story and Wagner, 1963) and Chapter 13 (written by R. Woolsey) in Salkin (1975). The only known exception to the rule was G. Martin's work (Martin, 1963) on set covering and traveling salesman problems of small to medium size; see also (Miliotis, 1978) where a discussion of implementational difficulties of Gomory's fractional cuts can be found.…”

Classical cuts for mixed-integer programming and branch-and-cut

“…Secondly, there were plenty of early attempts to implement cutting plane theory numerically nevertheless, c.f. Fleischmann (1970), Fiorot and Gondran (1969), Haldi and Isaacson (1965), Krolak (1969), Knuth (1961), Martin (1963), Miliotis (1978), Story andWagner (1963), Trauth Jr andWoolsey (1969), Wagner, Giglio, and Glaser (1964) and probably more. Most of the pertaining computational studies used the fractional cut (GP = ) or its "all-integer" variant (Glover, 1968;Gondran, 1973;Wilson, 1967;Young, 1968), whereas the stronger mixed-integer cut (GM # ) was studied to a lesser degree; probably because it makes a "mixed" integer program even more "mixed" as the surplus variable in (GM # ) is in general a flow variable.…”

“…None of the studies reported in the early literature on the subject showed significant numerical success, indeed some reported outright disastrous results, c.f. (Krolak, 1969;Knuth, 1961;Story and Wagner, 1963) and Chapter 13 (written by R. Woolsey) in Salkin (1975). The only known exception to the rule was G. Martin's work (Martin, 1963) on set covering and traveling salesman problems of small to medium size; see also (Miliotis, 1978) where a discussion of implementational difficulties of Gomory's fractional cuts can be found.…”