A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| author | Fiaschi, Lorenzo Cococcioni, Marco |
| author_facet | Fiaschi, Lorenzo Cococcioni, Marco |
| contents | This work presents a generalized implementation of the infeasible primal-dual Interior Point Method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint) NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_15658 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems Fiaschi, Lorenzo Cococcioni, Marco Optimization and Control 90C20, 90C51, 90-08, 03H10 This work presents a generalized implementation of the infeasible primal-dual Interior Point Method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here non-Archimedean IPM (NA-IPM), is proved to converge in polynomial time to a global optimum and to be able to manage infeasibility and unboundedness transparently, i.e., without considering them as corner cases: by means of a mild embedding (addition of two variables and one constraint) NA-IPM implicitly and transparently manages their possible presence. Moreover, the new algorithm is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. To support the theoretical properties of NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool, paving the way for its use in real-world problems in the near future. |
| title | A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems |
| topic | Optimization and Control 90C20, 90C51, 90-08, 03H10 |
| url | https://arxiv.org/abs/2110.15658 |