Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912298898030592 |
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| author | Luo, Dayou Yu, Yue Fazel, Maryam Açıkmeşe, Behçet |
| author_facet | Luo, Dayou Yu, Yue Fazel, Maryam Açıkmeşe, Behçet |
| contents | We propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, enabling the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance efficiency, we design a specialized matrix factorization technique that leverages the typical sparsity of optimal control problems in such systems. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly when feasibility is easily guaranteed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22131 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control Luo, Dayou Yu, Yue Fazel, Maryam Açıkmeşe, Behçet Optimization and Control 49N10, 49M15, 90C20 We propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, enabling the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance efficiency, we design a specialized matrix factorization technique that leverages the typical sparsity of optimal control problems in such systems. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly when feasibility is easily guaranteed. |
| title | Newton-PIPG: A Fast Hybrid Algorithm for Quadratic Programs in Optimal Control |
| topic | Optimization and Control 49N10, 49M15, 90C20 |
| url | https://arxiv.org/abs/2503.22131 |