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An Optimal First-Order Active-Hull Method for Convex Composite Hölder Smooth Optimization
Abstract
We consider the problem (P): minx∈ ℝn f(x)+g(x) under the Euclidean geometry, where f is convex and (M,ν)-Hölder smooth on ℝn, and g is proper, closed and convex on ℝn, admits a generalized linear minimization oracle (GLMO), and has bounded domain with diameter D. An first-order active-hull (AH) method has access to both the first-order oracle (FO) of f and the GLMO of g, and outputs a point that lies in the convex hull of all the past GLMO returns. We propose a new first-order AH method for solving (P), which is based on our newly developed ``recentered homotopy smoothing framework''. We show that in the high-dimensional regime, this method achieves the optimal FO and GLMO complexties simultaneously for all ν∈[0,1].
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BibTeX
@misc{zhao2026optimal,
title = {{An Optimal First-Order Active-Hull Method for Convex Composite Hölder Smooth Optimization}},
author = {Renbo Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10916},
eprint = {2611.10916},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10916},
}
RIS
TY - UNPB
TI - An Optimal First-Order Active-Hull Method for Convex Composite Hölder Smooth Optimization
AU - Renbo Zhao
PY - 2026
DA - 2026/11/07
PB - arXiv
DO - 10.48550/arxiv.2611.10916
UR - https://arxiv.org/abs/2611.10916
AB - We consider the problem (P): min_x∈ ℝ^n f(x)+g(x) under the Euclidean geometry, where f is convex and (M,ν)-Hölder smooth on ℝ^n, and g is proper, closed and convex on ℝ^n, admits a generalized linear minimization oracle (GLMO), and has bounded domain with diameter D. An first-order active-hull (AH) method has access to both the first-order oracle (FO) of f and the GLMO of g, and outputs a point that lies in the convex hull of all the past GLMO returns. We propose a new first-order AH method for solving (P), which is based on our newly developed ``recentered homotopy smoothing framework''. We show that in the high-dimensional regime, this method achieves the optimal FO and GLMO complexties simultaneously for all ν∈[0,1].
ER -