The dual Phase-I algorithm using the most-obtuse-angle row pivot rule is very efficient for providing a dual feasible basis, in either the classical or the basis-deficiency-allowing context. In this paper, we establish a basis-deficiency-allowing Phase-I algorithm using the so-called most-obtuseangle column pivot rule to produce a primal (deficient or full) basis. Our computational experiments with the smallest test problems from the standard NETLIB set show that a dense projected-gradient implementation largely outperforms that of the variation of the primal simplex method from the commercial code MATLAB LINPROG vl.17, and that a sparse projected-gradient implementation of a normalized revised version of the proposed algorithm runs 34°~ faster than the sparse implementation of the primal simplex method included in the commercial code TOMLAB LPSOLVE V3.0. (~)

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

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An interesting question for linear programming (LP) algorithms is how to deal with solutions in which the number of nonzero variables is less than the number of rows of the matrix in standard form. An approach is that of basis deficiency-allowing (BDA) simplex variations, which work with a subset of independent columns of the coefficient matrix in standard form, wherein the basis is not necessarily represented by a square matrix. We describe one such algorithm with several variants. The research question deals with studying the computational behaviour by using small, extreme cases. For these instances, we must wonder which parameter setting or variants are more appropriate. We compare the setting of two nonsimplex active-set methods with Holmström’s TomLab LpSimplex v3.0 commercial sparse primal simplex commercial implementation. All of them update a sparse QR factorization in Matlab. The first two implementations require fewer iterations and provide better solution quality and running time.