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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2201.00193 |
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| _version_ | 1866917990742622208 |
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| author | Yang, Yaguang |
| author_facet | Yang, Yaguang |
| contents | The Hirsch Conjecture stated that any $d$-dimensional polytope with n facets has a diameter at most equal to $n - d$. This conjecture was disproved by Santos (A counterexample to the Hirsch Conjecture, Annals of Mathematics, 172(1) 383-412, 2012). The implication of Santos' work is that all {\it vertex} pivot algorithms cannot solve the linear programming problem in the worst case in $n - d$ vertex pivot iterations.
In the first part of this series of papers, we proposed a {\it facet} pivot method. In this paper, we show that the proposed facet pivot method can solve the canonical linear programming problem in the worst case in at most $n-d$ facet pivot iterations. This work was inspired by Smale's Problem 9 (Mathematical problems for the next century, In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics: frontiers and perspectives, American Mathematical Society, 271-294, 1999). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_00193 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the facet pivot simplex method for linear programming II: a linear iteration bound Yang, Yaguang Optimization and Control The Hirsch Conjecture stated that any $d$-dimensional polytope with n facets has a diameter at most equal to $n - d$. This conjecture was disproved by Santos (A counterexample to the Hirsch Conjecture, Annals of Mathematics, 172(1) 383-412, 2012). The implication of Santos' work is that all {\it vertex} pivot algorithms cannot solve the linear programming problem in the worst case in $n - d$ vertex pivot iterations. In the first part of this series of papers, we proposed a {\it facet} pivot method. In this paper, we show that the proposed facet pivot method can solve the canonical linear programming problem in the worst case in at most $n-d$ facet pivot iterations. This work was inspired by Smale's Problem 9 (Mathematical problems for the next century, In Arnold, V. I.; Atiyah, M.; Lax, P.; Mazur, B. Mathematics: frontiers and perspectives, American Mathematical Society, 271-294, 1999). |
| title | On the facet pivot simplex method for linear programming II: a linear iteration bound |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2201.00193 |