# Projection-Free Functional Constrained Optimization for Risk Aversion and Sparsity Control

arXiv: https://arxiv.org/abs/2210.05108 · HTML: https://mlsanity.com/paper/2210.05108/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 55 references that are in the map.
- First posted on arXiv 11 Oct 2022.

## Builds on (most cited first, 14 of 14)

- [Stochastic First-order Methods for Convex and Nonconvex Functional Constrained Optimization](https://mlsanity.com/paper/1908.02734/) (2019)
- [Quadratically Regularized Subgradient Methods for Weakly Convex Optimization with Weakly Convex Constraints](https://mlsanity.com/paper/1908.01871/) (2019)
- [Frank-Wolfe Algorithms for Saddle Point Problems](https://mlsanity.com/paper/1610.07797/) (2016)
- [Complexity of a quadratic penalty accelerated inexact proximal point method for solving linearly constrained nonconvex composite programs](https://mlsanity.com/paper/1802.03504/) (2018)
- [Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis](https://mlsanity.com/paper/1605.02408/) (2016)
- [Non-convex Conditional Gradient Sliding](https://mlsanity.com/paper/1708.04783/) (2017)
- [Level Constrained First Order Methods for Function Constrained Optimization](https://mlsanity.com/paper/2205.08011/) (2022)
- [CCCP is Frank-Wolfe in disguise](https://mlsanity.com/paper/2206.12014/) (2022)
- [Efficient Projection-Free Algorithms for Saddle Point Problems](https://mlsanity.com/paper/2010.11737/) (2020)
- [Level-set methods for convex optimization](https://mlsanity.com/paper/1602.01506/) (2016)
- [Conditional Accelerated Lazy Stochastic Gradient Descent](https://mlsanity.com/paper/1703.05840/) (2017)
- [Conditional Gradient Methods for Convex Optimization with General Affine and Nonlinear Constraints](https://mlsanity.com/paper/2007.00153/) (2020)
- [A Feasible Level Proximal Point Method for Nonconvex Sparse Constrained Optimization](https://mlsanity.com/paper/2010.12169/) (2020)
- [An Average Curvature Accelerated Composite Gradient Method for Nonconvex Smooth Composite Optimization Problems](https://mlsanity.com/paper/1909.04248/) (2019)

## Built on by (most cited first, 2 of 2)

- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [Nonsmooth Projection-Free Optimization with Functional Constraints](https://mlsanity.com/paper/2311.11180/) (2023)

## Abstract (arXiv)

We study projection-free methods for functional constrained optimization with convex or smooth nonconvex objectives. Such problems arise in applications such as portfolio optimization and radiation therapy planning, where risk-aware criteria and sparsity frequently appear together. For the convex setting, we propose a Level Conditional Gradient (LCG) method that combines a level-set outer loop with a conditional gradient oracle for saddle-point subproblems, and we show an iteration complexity of Obig(ε-2log(ε-1)big) for smooth and nonsmooth cases without dependence on the magnitude of an optimal dual Lagrange multiplier. For the nonconvex setting, we propose the Inexact Proximal Point LCG (IPP-LCG) method, which solves a sequence of convex subproblems by LCG and attains Obig(ε-3log(ε-1)big) complexity for computing an ((ε,ε))-near-KKT point. Numerical results on portfolio selection and IMRT illustrate the practical sparsity/risk trade-offs of the proposed methods.