Optimization over the weakly Pareto set and multi-task learning

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.