2025/03/06 by Alireza Kabgani, Masoud Ahookhosh, Kabgani, Alireza +1 · 3 citations
Computer Science · Engineering · #49J52 #65K10 #90C26 #90C56 #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2503.04577
openalex publication_date 2025/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≥ 2 and p-calmness for p ∈ (1,2] and 2 ≤ p ≤ 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.