New merit functions and error bounds for non-convex multiobjective optimization.

Our aim is to propose merit functions and provide error bounds for non-convex multiobjective optimization problems. For such problems, the merit functions return zero at Pareto stationary points and strictly positive values otherwise. These functions are known to be important for establishing convergence rates in single-objective optimization, but the related studies for the multiobjective case are still recent. We then propose in this paper six merit functions for multiobjective optimization that differ, on whether they contain some regularization term and they linearize the objective functions. In particular, those with regularization terms necessarily have bounded values. We also compare the proposed merit functions and analyze the sufficient conditions for them to have error bounds. In addition, by considering the well-known Fenchel duality, we present efficient ways to compute these merit functions for particular problems, specifically for differentiable conic programming problems, differentiable constrained problems, and problems involving $\ell_1$ norms.