Practical finite pivoting rules for the simplex method

Pan, Ping

doi:10.1007/bf01721801

Cited by 67 publications

(11 citation statements)

References 9 publications

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“…Pan proposed this method in [29] and further examined it computationally in [30]. The basic idea is to remove at least one dual infeasibility at every iteration without giving any guarantee that no new infeasibilities are created.…”

Section: Pan's Methodsmentioning

confidence: 99%

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

Koberstein

Suhl

2007

Comput Optim Appl

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The dual simplex algorithm has become a strong contender in solving large scale LP problems. One key problem of any dual simplex algorithm is to obtain a dual feasible basis as a starting point. We give an overview of methods which have been proposed in the literature and present new stable and efficient ways to combine them within a state-of-the-art optimization system for solving real world linear and mixed integer programs. Furthermore, we address implementation aspects and the connection between dual feasibility and LP-preprocessing. Computational results are given for a large set of large scale LP problems, which show our dual simplex implementation to be superior to the best existing research and open-source codes and competitive to the leading commercial code on many of our most difficult problem instances.

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Section: Pan's Methodsmentioning

confidence: 99%

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

Koberstein

Suhl

2007

Comput Optim Appl

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“…Let us discuss the second one. From the canonical tableau of (1), we have xp + ~ djxj = bp, (12) jEJN where bp < 0 and min{dj I J = 1,...,n-ml} >_ 0.…”

Section: Most-obtuse-angle Column Rulementioning

confidence: 99%

“…Under the spirit of Pan's geometrical characterization of an optimal basis (or nonbasis) [12], therefore, we know that Zkq is eligible to be used as an optimal nonbasic variable for the dual problem. Thus, xkq is an optimal basic variable with respect to the primal problem according to the complementary slackness conditions.…”

Section: Comments On the Pivoting Rulementioning

confidence: 99%

A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule

Guerrero-Garcı́a

Santos-Palomo

2006

Computers & Mathematics with Applications

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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. (~)

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“…When there is a tie, those which bear higher L, subscri~ts have ~rioritv. Without any changes, the preceding is a!so app!icable to h e a r programs involving equa!ity constraints (see PAN, 1990). In the sequel.…”

Section: --mentioning

confidence: 98%

A simplex-like method with bisection for linear programming¹

Pan

1991

Optimization

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Practical finite pivoting rules for the simplex method

Cited by 67 publications

References 9 publications

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule

A simplex-like method with bisection for linear programming¹

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Practical finite pivoting rules for the simplex method

Cited by 67 publications

References 9 publications

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

Progress in the dual simplex method for large scale LP problems: practical dual phase 1 algorithms

A basis-deficiency-allowing primal phase-I algorithm using the most-obtuse-angle column rule

A simplex-like method with bisection for linear programming1

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A simplex-like method with bisection for linear programming¹