Optimization over the weakly Pareto set and multi-task learning
Lei Huang, Jiawang Nie, Jiajia Wang
TL;DR
This work develops a framework for optimization over the weakly Pareto set in convex polynomial multiobjective problems by deriving three polynomial representations of the weakly Pareto set via Lagrange multipliers and weight vectors. Each representation yields a reformulation of the optimization over weakly Pareto points (OWP) as a polynomial optimization problem, which is then tackled with the Moment-SOS hierarchy, including convergence guarantees under Archimedean conditions. The authors provide algorithms corresponding to each representation and demonstrate their effectiveness through numerical experiments, including applications to multi-task learning. The approach enables global optimization over Pareto trade-offs in polynomial settings and offers practical tools for MTLL tasks by exploiting structured representations to reduce computational burden. This work thus bridges multiobjective convex optimization, polynomial optimization, and SOS relaxations to yield tractable, globally convergent solvers for OWP.
Abstract
We study the optimization problem over the weakly Pareto set of a convex multiobjective optimization problem given by polynomial functions. Using Lagrange multiplier expressions and the weight vector, we give three types of representations for the weakly Pareto set. Using these representations, we reformulate the optimization problem over the weakly Pareto set as a polynomial optimization problem. We then apply the Moment--SOS hierarchy to solve it and analyze its convergence properties under certain conditions. Numerical experiments are provided to demonstrate the effectiveness of our methods. Applications in multi-task learning are also presented.