Optimization on Pareto sets: On a theory of multi-objective optimization
This addresses the challenge of efficiently navigating large Pareto sets for decision-makers in multi-objective optimization, representing an incremental theoretical advancement.
The paper tackles the problem of optimizing a preference function constrained to the Pareto set in multi-objective optimization, where the Pareto set is implicitly defined and non-convex, by developing local methods with a last-iterate convergence rate of O(K^{-1/2}) for strongly convex and Lipschitz smooth objectives.
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.