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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.11786 |
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| _version_ | 1866908914914689024 |
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| author | Ghezzi, Andrea Van Roy, Wim Sager, Sebastian Diehl, Moritz |
| author_facet | Ghezzi, Andrea Van Roy, Wim Sager, Sebastian Diehl, Moritz |
| contents | Sequential quadratic programming and sequential convex programming efficiently solve nonlinear programs (NLPs) by linearizing inner nonlinearities while preserving the outer convex structure. This paper introduces a sequential mixed-integer quadratic programming (MIQP) algorithm to extend this methodology to mixed-integer nonlinear problems (MINLPs), leveraging the efficiency of modern MIQP solvers. The algorithm uses a three-step iterative process. First, the MINLP is linearized around the current iterate. Second, an MIQP is formulated and solved, with its feasible region restricted to a specific area around the linearization point. This region is defined using objective values and derivatives from previous iterations, drawing on concepts from generalized Benders' decomposition. Third, the integer variables from the MIQP solution are fixed, and an NLP involving only the continuous variables is solved. The best solution among all iterates becomes the linearization point for the next iteration. A fallback strategy based on a mixed-integer linear program (MILP) is used when MIQP progress stalls. This guarantees convergence to the global optimal solution for convex MINLPs. For nonconvex problems, the algorithm functions as a heuristic without global optimality guarantees. Numerical experiments show its competitiveness with other MINLP solvers on benchmark problems. In addition, the algorithm was successfully applied to mixed-integer optimal control problems, demonstrating its effectiveness in handling challenging nonlinear equality constraints. The proposed algorithm is publicly available at https://github.com/minlp-toolbox/CAMINO with the name s-b-miqp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Sequential Benders-based Mixed-Integer Quadratic Programming Algorithm and Its Implementation in the CAMINO Toolbox Ghezzi, Andrea Van Roy, Wim Sager, Sebastian Diehl, Moritz Optimization and Control Sequential quadratic programming and sequential convex programming efficiently solve nonlinear programs (NLPs) by linearizing inner nonlinearities while preserving the outer convex structure. This paper introduces a sequential mixed-integer quadratic programming (MIQP) algorithm to extend this methodology to mixed-integer nonlinear problems (MINLPs), leveraging the efficiency of modern MIQP solvers. The algorithm uses a three-step iterative process. First, the MINLP is linearized around the current iterate. Second, an MIQP is formulated and solved, with its feasible region restricted to a specific area around the linearization point. This region is defined using objective values and derivatives from previous iterations, drawing on concepts from generalized Benders' decomposition. Third, the integer variables from the MIQP solution are fixed, and an NLP involving only the continuous variables is solved. The best solution among all iterates becomes the linearization point for the next iteration. A fallback strategy based on a mixed-integer linear program (MILP) is used when MIQP progress stalls. This guarantees convergence to the global optimal solution for convex MINLPs. For nonconvex problems, the algorithm functions as a heuristic without global optimality guarantees. Numerical experiments show its competitiveness with other MINLP solvers on benchmark problems. In addition, the algorithm was successfully applied to mixed-integer optimal control problems, demonstrating its effectiveness in handling challenging nonlinear equality constraints. The proposed algorithm is publicly available at https://github.com/minlp-toolbox/CAMINO with the name s-b-miqp. |
| title | A Sequential Benders-based Mixed-Integer Quadratic Programming Algorithm and Its Implementation in the CAMINO Toolbox |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2404.11786 |