# Optimization on Pareto sets: On a theory of multi-objective optimization

arXiv: https://arxiv.org/abs/2308.02145 · HTML: https://mlsanity.com/paper/2308.02145/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 48 references that are in the map.
- First posted on arXiv 4 Aug 2023.

## Builds on (most cited first, 13 of 13)

- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [Pareto Multi-Task Learning](https://mlsanity.com/paper/1912.12854/) (2019)
- [Multi-task Self-Supervised Visual Learning](https://mlsanity.com/paper/1708.07860/) (2017)
- [Minimax Pareto Fairness: A Multi Objective Perspective](https://mlsanity.com/paper/2011.01821/) (2020)
- [Oracle Complexity in Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2104.06763/) (2021)
- [Deterministic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2302.08300/) (2023)
- [Understanding Notions of Stationarity in Non-Smooth Optimization](https://mlsanity.com/paper/2006.14901/) (2020)
- [Pareto Navigation Gradient Descent: a First-Order Algorithm for Optimization in Pareto Set](https://mlsanity.com/paper/2110.08713/) (2021)
- [Pareto Efficient Fairness in Supervised Learning: From Extraction to Tracing](https://mlsanity.com/paper/2104.01634/) (2021)
- [Optimizing fairness tradeoffs in machine learning with multiobjective meta-models](https://mlsanity.com/paper/2304.12190/) (2023)
- [Bezier Simplex Fitting: Describing Pareto Fronts of Simplicial Problems with Small Samples in Multi-objective Optimization](https://mlsanity.com/paper/1812.05222/) (2018)
- [Topology of Pareto sets of strongly convex problems](https://mlsanity.com/paper/1904.03615/) (2019)
- [Majorization-Minimization-Based Levenberg--Marquardt Method for Constrained Nonlinear Least Squares](https://mlsanity.com/paper/2004.08259/) (2020)

## Built on by (most cited first, 7 of 7)

- [MUNBa: Machine Unlearning via Nash Bargaining](https://mlsanity.com/paper/2411.15537/) (2024)
- [PMGDA: A Preference-based Multiple Gradient Descent Algorithm](https://mlsanity.com/paper/2402.09492/) (2024)
- [Many-Objective Multi-Solution Transport](https://mlsanity.com/paper/2403.04099/) (2024)
- [Toward Finding Strong Pareto Optimal Policies in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2410.19372/) (2024)
- [UMOEA/D: A Multiobjective Evolutionary Algorithm for Uniform Pareto Objectives based on Decomposition](https://mlsanity.com/paper/2402.09486/) (2024)
- [Learning Pareto manifolds in high dimensions: How can regularization help?](https://mlsanity.com/paper/2503.08849/) (2025)
- [SURF: Steering the Scalarization Weight to Uniformly Traverse the Pareto Front](https://mlsanity.com/paper/2605.20619/) (2026)

## Abstract (arXiv)

In multi-objective optimization, a single decision vector must balance the trade-offs between many objectives. Solutions achieving an optimal trade-off are said to be Pareto optimal: these are decision vectors for which improving any one objective must come at a cost to another. But as the set of Pareto optimal vectors can be very large, we further consider a more practically significant Pareto-constrained optimization problem, where the goal is to optimize a preference function constrained to the Pareto set. We investigate local methods for solving this constrained optimization problem, which poses significant challenges because the constraint set is (i) implicitly defined, and (ii) generally non-convex and non-smooth, even when the objectives are. We define notions of optimality and stationarity, and provide an algorithm with a last-iterate convergence rate of O(K-1/2) to stationarity when the objectives are strongly convex and Lipschitz smooth.