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

TL;DR

This paper develops a theoretical framework for multi-objective optimization focusing on Pareto sets, introducing local methods for Pareto-constrained problems with convergence guarantees under certain conditions.

Contribution

It defines new notions of optimality and stationarity for Pareto-constrained problems and proposes an algorithm with provable convergence rates.

Findings

The algorithm achieves an $O(K^{-1/2})$ convergence rate to stationarity.

The framework handles implicit, non-convex, and non-smooth Pareto constraints.

Provides theoretical insights into local methods for multi-objective optimization.

Abstract

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.