# ADMM for Structured Fractional Minimization

arXiv: https://arxiv.org/abs/2411.07496 · HTML: https://mlsanity.com/paper/2411.07496/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 54 references that are in the map.
- First posted on arXiv 12 Nov 2024.

## Builds on (most cited first, 21 of 21)

- [Convergence Analysis of Alternating Direction Method of Multipliers for a Family of Nonconvex Problems](https://mlsanity.com/paper/1410.1390/) (2014)
- [Efficiency of minimizing compositions of convex functions and smooth maps](https://mlsanity.com/paper/1605.00125/) (2016)
- [A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems](https://mlsanity.com/paper/2010.15768/) (2020)
- [Global convergence of splitting methods for nonconvex composite optimization](https://mlsanity.com/paper/1407.0753/) (2014)
- [A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming](https://mlsanity.com/paper/1804.08544/) (2018)
- [Momentum Accelerates the Convergence of Stochastic AUPRC Maximization](https://mlsanity.com/paper/2107.01173/) (2021)
- [The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates](https://mlsanity.com/paper/1801.01994/) (2018)
- [Variable Smoothing for Weakly Convex Composite Functions](https://mlsanity.com/paper/2003.07612/) (2020)
- [Limited-angle CT reconstruction via the L1/L2 minimization](https://mlsanity.com/paper/2006.00601/) (2020)
- [Moreau Envelope Augmented Lagrangian Method for Nonconvex Optimization with Linear Constraints](https://mlsanity.com/paper/2101.08519/) (2021)
- [Generalized Conditional Gradient with Augmented Lagrangian for Composite Minimization](https://mlsanity.com/paper/1901.01287/) (2019)
- [Smoothing Proximal Gradient Methods for Nonsmooth Sparsity Constrained Optimization: Optimality Conditions and Global Convergence](https://mlsanity.com/paper/2104.13782/) (2021)
- [Exact penalty decomposition method for zero-norm minimization based on MPEC formulation](https://mlsanity.com/paper/1412.4452/) (2014)
- [A Riemannian ADMM](https://mlsanity.com/paper/2211.02163/) (2022)
- [A proximal minimization algorithm for structured nonconvex and nonsmooth problems](https://mlsanity.com/paper/1805.11056/) (2018)
- [First-order algorithms for a class of fractional optimization problems](https://mlsanity.com/paper/2005.06207/) (2020)
- [Coordinate Descent Methods for Fractional Minimization](https://mlsanity.com/paper/2201.12691/) (2022)
- [Error bounds for rank constrained optimization problems and applications](https://mlsanity.com/paper/1603.07070/) (2016)
- [Inertial Proximal Block Coordinate Method for a Class of Nonsmooth Sum-of-Ratios Optimization Problems](https://mlsanity.com/paper/2011.09782/) (2020)
- [A full splitting algorithm for fractional programs with structured numerators and denominators](https://mlsanity.com/paper/2312.14341/) (2023)
- [ADMM for Nonsmooth Composite Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2405.15129/) (2024)

## Abstract (arXiv)

This paper considers a class of structured fractional minimization problems. The numerator consists of a differentiable function, a simple nonconvex nonsmooth function, a concave nonsmooth function, and a convex nonsmooth function composed with a linear operator. The denominator is a continuous function that is either weakly convex or has a weakly convex square root. These problems are prevalent in various important applications in machine learning and data science. Existing methods, primarily based on subgradient methods and smoothing proximal gradient methods, often suffer from slow convergence and numerical stability issues. In this paper, we introduce sf FADMM, the first Alternating Direction Method of Multipliers tailored for this class of problems. sf FADMM decouples the original problem into linearized proximal subproblems, featuring two variants: one using Dinkelbach's parametric method (sf FADMM-D) and the other using the quadratic transform method (sf FADMM-Q). By introducing a novel Lyapunov function, we establish that sf FADMM converges to ε-approximate critical points of the problem within an oracle complexity of O(1/ε3). Extensive experiments on synthetic and real-world datasets, including sparse Fisher discriminant analysis, robust Sharpe ratio minimization, and robust sparse recovery, demonstrate the effectiveness of our approach. Keywords: Fractional Minimization, Nonconvex Optimization, Proximal Linearized ADMM, Nonsmooth Optimization, Convergence Analysis