<p>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 (<InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>=</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$p=2$</EquationSource> </InlineEquation>) is replaced by a <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation>-order regularizer with <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$p > 1$</EquationSource> </InlineEquation>. After establishing several basic properties of HOPE and HOME, we study the differentiability and weak smoothness of HOME under <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>q</mi> </math></EquationSource> <EquationSource Format="TEX">$q$</EquationSource> </InlineEquation>-prox-regularity with <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi>q</mi> <mo>≥</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$q \geq 2$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <mi>p</mi> </math></EquationSource> <EquationSource Format="TEX">$p$</EquationSource> </InlineEquation>-calmness for <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$p \in (1,2]$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <mn>2</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi>q</mi> </math></EquationSource> <EquationSource Format="TEX">$2 \leq p \leq q$</EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq9"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$p>1$</EquationSource> </InlineEquation> 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.</p>

Moreau Envelope and Proximal-Point Methods Under the Lens of High-Order Regularization

期刊论文
发表日期：

2025年12月12日

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Alireza Kabgani,
Masoud Ahookhosh

摘要

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 ( $p = 2$ $p=2$ ) is replaced by a $p$ $p$ -order regularizer with $p > 1$ $p > 1$ . After establishing several basic properties of HOPE and HOME, we study the differentiability and weak smoothness of HOME under $q$ $q$ -prox-regularity with $q \geq 2$ $q \geq 2$ and $p$ $p$ -calmness for $p \in (1, 2]$ $p \in (1,2]$ and $2 \leq p \leq q$ $2 \leq p \leq q$ . 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 $p > 1$ $p>1$ 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.