Moreau envelope and proximal-point methods under the lens of high-order regularization
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| Format: | Preprint |
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2025
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| _version_ | 1866911311874490368 |
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| author | Kabgani, Alireza Ahookhosh, Masoud |
| author_facet | Kabgani, Alireza Ahookhosh, Masoud |
| contents | This paper is devoted to investigating the fundamental properties of the high-order proximal operator (HOPE) and the high-order Moreau envelope (HOME) in the nonconvex setting, where the quadratic regularization ($p=2$) is replaced by a $p$-order regularizer with $p > 1$. After establishing several basic properties of HOPE and HOME, we study the differentiability and weak smoothness of HOME under $q$-prox-regularity with $q \geq 2$ and $p$-calmness for $p \in (1,2]$ and $2 \leq p \leq q$. Furthermore, we propose a high-order proximal-point algorithm (HiPPA) and analyze the convergence of the generated sequence to proximal fixed points. Our results pave the way for the development of a high-order smoothing theory with $p>1$ that can lead to new algorithmic advances in the nonconvex setting. To illustrate this potential for nonsmooth and nonconvex optimization, we apply HiPPA to the Nesterov-Chebyshev-Rosenbrock functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_04577 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moreau envelope and proximal-point methods under the lens of high-order regularization Kabgani, Alireza Ahookhosh, Masoud Optimization and Control 49J52, 65K10, 90C26, 90C56 This paper is devoted to investigating the fundamental properties of the high-order proximal operator (HOPE) and the high-order Moreau envelope (HOME) in the nonconvex setting, where the quadratic regularization ($p=2$) is replaced by a $p$-order regularizer with $p > 1$. After establishing several basic properties of HOPE and HOME, we study the differentiability and weak smoothness of HOME under $q$-prox-regularity with $q \geq 2$ and $p$-calmness for $p \in (1,2]$ and $2 \leq p \leq q$. Furthermore, we propose a high-order proximal-point algorithm (HiPPA) and analyze the convergence of the generated sequence to proximal fixed points. Our results pave the way for the development of a high-order smoothing theory with $p>1$ that can lead to new algorithmic advances in the nonconvex setting. To illustrate this potential for nonsmooth and nonconvex optimization, we apply HiPPA to the Nesterov-Chebyshev-Rosenbrock functions. |
| title | Moreau envelope and proximal-point methods under the lens of high-order regularization |
| topic | Optimization and Control 49J52, 65K10, 90C26, 90C56 |
| url | https://arxiv.org/abs/2503.04577 |