Year: 2025
Type: article
Abstract: . We consider a convex minimization problem for which the objective is the sum of a homogeneous polynomial of degree four and a linear term. Such a task arises as a subproblem in algorithms for quadratic inverse problems with a difference-of-convex structure. We design a first-order method called homogenized gradient, along with an accelerated version, which enjoy fast convergence rates of, respectively, [Formula: see text] and [Formula: see text] in relative accuracy, where [Formula: see text] is the iteration counter. The constant [Formula: see text] is the quartic condition number of the problem. Then, we show that for a certain class of problems, it is possible to compute a preconditioner for which this condition number is [Formula: see text], where [Formula: see text] is the problem dimension. To establish this, we study the more general problem of finding the best quadratic approximation of an [Formula: see text] norm composed with a quadratic map. Our construction involves a generalization of the so-called Lewis weights.
Source: SIAM Journal on Optimization
DOI: 10.1137/23m1583363
Authors: Radu-Alexandru Dragomir, Yurii Nesterov
Institutions: École Polytechnique Fédérale de Lausanne, UCLouvain, KU Leuven
Language: English
Subfield: Numerical Analysis
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4409173086