{"ID":2850260,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.22223","arxiv_id":"2510.22223","title":"Partial Envelope for Optimization Problem with Nonconvex Constraints","abstract":"In this paper, we consider the nonlinear constrained optimization problem (NCP) with constraint set $\\{x \\in \\mathcal{X}: c(x) = 0\\}$, where $\\mathcal{X}$ is a closed convex subset of $\\mathbb{R}^n$. Building upon the forward-backward envelope framework for optimization over $\\mathcal{X}$, we propose a forward-backward semi-envelope (FBSE) approach for solving (NCP). In the proposed semi-envelope approach, we eliminate the constraint $x \\in \\mathcal{X}$ through a specifically designed envelope scheme while preserving the constraint $x \\in \\mathcal{M} := \\{x \\in \\mathbb{R}^n: c(x) = 0\\}$. We establish that the forward-backward semi-envelope for (NCP) is well-defined and locally Lipschitz smooth over a neighborhood of $\\mathcal{M}$. Furthermore, we prove that (NCP) and its corresponding forward-backward semi-envelope have the same first-order stationary points within a neighborhood of $\\mathcal{X} \\cap \\mathcal{M}$. Consequently, our proposed forward-backward semi-envelope approach enables direct application of optimization methods over $\\mathcal{M}$ while inheriting their convergence properties for (NCP). Additionally, we develop an inexact projected gradient descent method for minimizing the forward-backward semi-envelope over $\\mathcal{M}$ and establish its global convergence. Preliminary numerical experiments demonstrate the practical efficiency and potential of our proposed approach.","short_abstract":"In this paper, we consider the nonlinear constrained optimization problem (NCP) with constraint set $\\{x \\in \\mathcal{X}: c(x) = 0\\}$, where $\\mathcal{X}$ is a closed convex subset of $\\mathbb{R}^n$. Building upon the forward-backward envelope framework for optimization over $\\mathcal{X}$, we propose a forward-backward...","url_abs":"https://arxiv.org/abs/2510.22223","url_pdf":"https://arxiv.org/pdf/2510.22223v1","authors":"[\"Xiaoyin Hu\",\"Xin Liu\",\"Kim-Chuan Toh\",\"Nachuan Xiao\"]","published":"2025-10-25T09:04:28Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}