On Hoffman's celebrated cycling LP example

Abstract

Sincerely dedicated to Alan J. Hoffman on the 50th birthday of his example. AbstractIn this short note we answer two questions that naturally arise while dealing with Hoffman's celebrated 50-years-old cycling example for the primal simplex method to solve linear programs, where an angle θ and a scaling factor ω are adjustable parameters in his example. In particular, we determine what conditions have to be imposed on ω for cycling to occur with θ = 2π/5, and what on θ with |ω| = tan(θ). Using this notation, Hoffman's example can be stated as follows:Note that we have used the following unsymmetric primal-dual pair of linear programs with a non-standard notation (and that we have deliberately exchanged the usual roles of b and c, x and y, n and m, and (P ) and (D), as e.g. in [9, §2]):where A ∈ R n×m with m ≥ n and rank(A) = n. We denote with F and G the feasible region of (P ) and (D), respectively. The notation used here iswhere a j is the jth column of A. As usual, B is the ordered index set of basic variables and N that of the non-basic ones, and the current iteration is noted by a superindex (k) when it is not clear from context. We assume that the reader is capable of applying to problem (D) the primal simplex algorithm (starting from a vertex y ∈ G such that |B| = n) and we do not repeat it here.As Alan recently confirmed to us, his original report [6] had not been published Lee's paper [7, p. 99], but the key point is that θ = 2π/5 and ω > (1−c)/(1−2c)is the unique requisite in all its published occurrences. We are interested in particularizing Hoffman's example to the case |ω| = t, because it is the only value of ω ∈ R such that a j 2 is constant for all j ∈ 1: m, an important feature to design a cycling example for certain sparse linear programming algorithm in which we have been involved for several years [5].Let us tell how our misunderstanding arose, because it constitutes the main motivation of this work. In Hoffman's example we have to apply the following

Cycling problems in linear programming

“…Problem 1 The first cycling example was given in [13,9,10] The optimal solution is x 1 = 1 and x j = 0 (j = 2 . .…”

“…The original paper[13] was not published. A detailed description of this problem was given in[9]. The form presented here was given in[10].…”