Abstract
In general if a linear program has an optimal solution, then a primal and dual optimal solution is a certificate of the solvable status. Furthermore, it is well known that in the solvable case, then the linear program always has an optimal basic solution. Similarly, when a linear program is primal or dual infeasible then by Farkas's Lemma a certificate of the infeasible status exists. However, in the primal or dual infeasible case then there is not an uniform definition of what a suitable basis certificate of the infeasible status is.
In this work we present a definition of a basis certificate and develop a strongly polynomial algorithm which given a Farkas type certificate of infeasibility computes a basis certificate of infeasibility. This result is relevant for the recently developed interior-point methods because they do not compute a basis certificate of infeasibility in general. However, our result demonstrates that a basis certificate can be obtained at a moderate computational cost.
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References
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Andersen, E.D. Certificates of Primal or Dual Infeasibility in Linear Programming. Computational Optimization and Applications 20, 171–183 (2001). https://doi.org/10.1023/A:1011259103627
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DOI: https://doi.org/10.1023/A:1011259103627