Results

**1 - 5**of**5**### Geometric random edge

, 2014

"... We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx: Ax 6 b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanne ..."

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We show that a variant of the random-edge pivoting rule results in a strongly polynomial time simplex algorithm for linear programs max{cTx: Ax 6 b}, whose constraint matrix A satisfies a geometric property introduced by Brunsch and Röglin: The sine of the angle of a row of A to a hyperplane spanned by n − 1 other rows of A is at least δ. This property is a geometric generalization of A being integral and all sub-determinants of A being bounded by ∆ in absolute value (since δ> 1/(∆2n)). In particular, linear programs defined by totally unimodular matrices are captured in this famework (δ> 1/n) for which Dyer and Frieze previously described a strongly polynomial-time randomized algorithm. The number of pivots of the simplex algorithm is polynomial in the dimension and 1/δ and independent of the number of constraints of the linear program. Our main result can be viewed as an algorithmic realization of the proof of small diameter for such polytopes by Bonifas et al.

### OPTIMA Mathematical Optimization Society Newsletter

, 2011

"... with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful mid-year meet ..."

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with goodies central to our field. After the summer months most of us are now back to our more usual occupations and our research activities in optimization. I truly hope that you share my anticipation of its moments of collaborative inspiration. One thing is sure: after the successful mid-year meetings, we are now heading towards the high point of 2012: the ISMP in Berlin. I hear from good sources that preparations are progressing well, and that all augurs are favourable. As you all know, several prizes will be awarded at the ISMP opening ceremony, recognizing the contributions or both younger and more senior colleagues. You undoubtedly have seen the various calls for nominations for the Dantzig, Lagrange, Fulkerson, Beale-Orchard-Hays and Tucker prizes as well as that for the Paul Tseng lectureship. I encourage you to seriously consider nominating one or more optimization researchers for these prizes. These awards and the high scientific standards of their recipients not only recognize

### THE RANDOM EDGE SIMPLEX ALGORITHM ON DUAL CYCLIC 4-POLYTOPES

, 2006

"... The simplex algorithm using the random edge pivot-rule on any realization of a dual cyclic 4-polytope with n facets does not take more than O(n) pivot-steps. This even holds for general abstract objective functions (AOF) / acyclic unique sink orientations (AUSO). The methods can be used to show a ..."

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The simplex algorithm using the random edge pivot-rule on any realization of a dual cyclic 4-polytope with n facets does not take more than O(n) pivot-steps. This even holds for general abstract objective functions (AOF) / acyclic unique sink orientations (AUSO). The methods can be used to show analogous results for products of two polygons. In contrast, we show that the random facet pivot-rule is slow on dual cyclic 4-polytopes, i.e. there are AUSOs on which random facet takes at least Ω(n²) steps.

### An improved version of the Random-Facet pivoting rule for the simplex algorithm

, 2015

"... The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using th ..."

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The Random-Facet pivoting rule of Kalai and of Matoušek, Sharir and Welzl is an elegant randomized pivoting rule for the simplex algorithm, the classical combinatorial algorithm for solving linear programs (LPs). The expected number of pivoting steps performed by the simplex algorithm when using this rule, on any linear program involving n inequalities in d variables, is