Optimization on the Pareto set: a theory of multi-objective optimization

Geelon So

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