Optimization on Pareto sets: On a theory of multi-objective optimization
Abhishek Roy Geelon So Yi-An Ma
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.