Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibility
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912247851253760 |
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| author | Dinh, Thi Lan Bennecke, Wiebke Jansen, G. S. Matthijs Luke, D. Russell Mathias, Stefan |
| author_facet | Dinh, Thi Lan Bennecke, Wiebke Jansen, G. S. Matthijs Luke, D. Russell Mathias, Stefan |
| contents | The main challenge of nonconvex optimization is to find a global optimum, or at least to avoid ``bad'' local minima and meaningless stationary points. We study here the extent to which algorithms, as opposed to optimization models and regularization, can be tuned to accomplish this goal. The model we consider is a nonconvex, inconsistent feasibility problem with many local minima, where these are points at which the gaps between the sets are smallest on neighborhoods of these points. The algorithms that we compare are all projection-based algorithms, specifically cyclic projections, the cyclic relaxed Douglas-Rachford algorithm, and relaxed Douglas-Rachford splitting on the product space. The local convergence and fixed points of these algorithms have already been characterized in pervious theoretical studies. We demonstrate the theory for these algorithms in the context of orbital tomographic imaging from angle-resolved photon emission spectroscopy (ARPES) measurements, both synthetically generated and experimental. Our results show that, while the cyclic projections and cyclic relaxed Douglas-Rachford algorithms generally converge the fastest, the method of relaxed Douglas-Rachford splitting on the product space does move away from bad local minima of the other two algorithms, settling eventually on clusters of local minima corresponding to globally optimal critical points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19052 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibility Dinh, Thi Lan Bennecke, Wiebke Jansen, G. S. Matthijs Luke, D. Russell Mathias, Stefan Optimization and Control Numerical Analysis 65K10, 65K05, 90C26, 49M27, 49J53, 49K40, 49M05 The main challenge of nonconvex optimization is to find a global optimum, or at least to avoid ``bad'' local minima and meaningless stationary points. We study here the extent to which algorithms, as opposed to optimization models and regularization, can be tuned to accomplish this goal. The model we consider is a nonconvex, inconsistent feasibility problem with many local minima, where these are points at which the gaps between the sets are smallest on neighborhoods of these points. The algorithms that we compare are all projection-based algorithms, specifically cyclic projections, the cyclic relaxed Douglas-Rachford algorithm, and relaxed Douglas-Rachford splitting on the product space. The local convergence and fixed points of these algorithms have already been characterized in pervious theoretical studies. We demonstrate the theory for these algorithms in the context of orbital tomographic imaging from angle-resolved photon emission spectroscopy (ARPES) measurements, both synthetically generated and experimental. Our results show that, while the cyclic projections and cyclic relaxed Douglas-Rachford algorithms generally converge the fastest, the method of relaxed Douglas-Rachford splitting on the product space does move away from bad local minima of the other two algorithms, settling eventually on clusters of local minima corresponding to globally optimal critical points. |
| title | Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibility |
| topic | Optimization and Control Numerical Analysis 65K10, 65K05, 90C26, 49M27, 49J53, 49K40, 49M05 |
| url | https://arxiv.org/abs/2502.19052 |