{"ID":2859217,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.05690","arxiv_id":"2510.05690","title":"On Implicit Concave Structures in Half-Quadratic Methods for Signal Reconstruction","abstract":"In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by leveraging Fenchel conjugate theory to construct an augmented optimization problem. This reformulation yields a one-to-one correspondence between the stationary points and local minima of the original and augmented problems. Crucially, the augmented problem admits a natural variable splitting that reveals convexity with respect to at least one block, and, in some cases, leading to a biconvex structure that is more amenable to optimization. This enables the use of efficient block coordinate descent algorithms for solving otherwise non-convex problems. As a representative application, we show how this framework applies to half-quadratic regularization in signal reconstruction and image processing. We demonstrate that common edge-preserving regularizers fall within the proposed class, and that their corresponding augmented problems are biconvex and bounded from below. Our results offer both a theoretical foundation and a practical pathway for solving a broad class of structured non-convex problems.","short_abstract":"In this work, we introduce a new class of non-convex functions, called implicit concave functions, which are compositions of a concave function with a continuously differentiable mapping. We analyze the properties of their minimization by leveraging Fenchel conjugate theory to construct an augmented optimization proble...","url_abs":"https://arxiv.org/abs/2510.05690","url_pdf":"https://arxiv.org/pdf/2510.05690v1","authors":"[\"Vittorio Latorre\"]","published":"2025-10-07T08:47:42Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}