Second-Order Optimization via Quiescence
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910644079427584 |
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| author | Agarwal, Aayushya Pileggi, Larry Rohrer, Ronald |
| author_facet | Agarwal, Aayushya Pileggi, Larry Rohrer, Ronald |
| contents | Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence of highly nonlinear objective functions with large Lipschitz constants, increasingly small step-sizes become a bottleneck to fast convergence. We propose a second-order optimization method that utilizes a dynamic system model to represent the trajectory of optimization variables as an ODE. We then follow the quasi-steady state trajectory by forcing variables with the fastest rise time into a state known as quiescence. This optimization via quiescence allows us to adaptively select large step-sizes that sequentially follow each optimization variable to a quasi-steady state until all state variables reach the actual steady state, coinciding with the optimum. The result is a second-order method that utilizes large step-sizes and does not require a monotonically decreasing objective function to reach a critical point. Experimentally, we demonstrate the fast convergence of this approach for optimizing nonconvex problems in power systems and compare them to existing state-of-the-art second-order methods, including damped Newton-Raphson, BFGS, and SR1. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_08033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Second-Order Optimization via Quiescence Agarwal, Aayushya Pileggi, Larry Rohrer, Ronald Optimization and Control Systems and Control Second-order optimization methods exhibit fast convergence to critical points, however, in nonconvex optimization, these methods often require restrictive step-sizes to ensure a monotonically decreasing objective function. In the presence of highly nonlinear objective functions with large Lipschitz constants, increasingly small step-sizes become a bottleneck to fast convergence. We propose a second-order optimization method that utilizes a dynamic system model to represent the trajectory of optimization variables as an ODE. We then follow the quasi-steady state trajectory by forcing variables with the fastest rise time into a state known as quiescence. This optimization via quiescence allows us to adaptively select large step-sizes that sequentially follow each optimization variable to a quasi-steady state until all state variables reach the actual steady state, coinciding with the optimum. The result is a second-order method that utilizes large step-sizes and does not require a monotonically decreasing objective function to reach a critical point. Experimentally, we demonstrate the fast convergence of this approach for optimizing nonconvex problems in power systems and compare them to existing state-of-the-art second-order methods, including damped Newton-Raphson, BFGS, and SR1. |
| title | Second-Order Optimization via Quiescence |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2410.08033 |