Proximal methods for nonsmooth and nonconvex fractional programs: when sparse optimization meets fractional programs

Abstract

Nonsmooth and nonconvex fractional programs are ubiquitous and also highly challenging. It includes the composite optimization problems studied extensively lately, and encompasses many important modern optimization problems arising from diverse areas such as the recent proposed scale invariant sparse signal reconstruction problem in signal processing, the robust Sharpe ratio optimization problems in finance and the sparse generalized eigenvalue problem in discrimination analysis.

In this talk, we will introduce extrapolated proximal methods for solving nonsmooth and nonconvex fractional programs and analyse their convergence behaviour. Interestingly, we will show that the proposed algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction model, and the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraints. This is achieved by identifying the explicit desingularization function of the Kurdyka- Lojasiewicz inequality for the merit function of the fractional optimization models. Finally, if time permits, we will present some preliminary encouraging numerical results for the proposed methods for sparse signal reconstruction and sparse Fisher discriminant analysis

The talk is Based on joint work with R.I. Bot, M. Dao, T.K. Pong and P. Yu.

Biography

Guoyin Li received his Ph.D. in December 2007 from The Chinese University of Hong Kong. His research focuses on optimization, variational analysis, and multilinear algebra. Following three years of postdoctoral training at The University of New South Wales (UNSW Sydney), Australia, he joined UNSW as a lecturer in 2011. He is currently a Professor and Research Director in the School of Mathematics and Statistics.

Dr. Li has authored over 80 peer-reviewed journal articles, which have been published in high quality academic journals, including Foundations of Computational Mathematics, SIAM Journal on Optimization, Mathematical Programming, Mathematics of Operations Research, Numerische Mathematik, Mathematics of Computations, and Journal of Functional Analysis.  He currently serves on the editorial board of the following journals: SIAM Journal on Optimization, Journal of Optimization Theory and Applications, Optimization Letters, and Mathematical Methods of Operations Research.

He was awarded the 2022 Australian Mathematical Society Medal and is a Fellow of the Australian Mathematical Society. His work has received several best paper awards, such as the Marguerite Frank Award for Best Paper in EURO Journal on Computational Optimization (2024), the 2019 International Consortium of Chinese Mathematicians (ICCM) Best Paper Award, the 2019 Journal of Global Optimization Best Paper Award, and the 2015 OPTL Best Paper Award from Optimization Letters. Additionally, he was awarded a mid-career Future Fellowship from the Australian Research Council (2014–2018) and an Invited Visiting Fellowship at the Isaac Newton Institute, University of Cambridge, in August 2013.