A Cubic Regularization Method for Multiobjective Optimization

TL;DR

This paper presents a novel cubic regularization algorithm for nonconvex multiobjective optimization, achieving efficient convergence rates and accommodating finite difference derivative approximations.

Contribution

It introduces a cubic regularization method with adaptive parameters for multiobjective problems, analyzing its convergence and iteration complexity under various derivative approximation schemes.

Findings

Requires at most (Cpsilon^{-3/2}) iterations for -pproximate Pareto critical points

Finite difference derivative computation affects iteration complexity and evaluation count

Achieves superlinear and quadratic convergence under local convexity and exact derivatives

Abstract

This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective function is minimized. This model allows approximations for the first- and second-order derivatives, which must satisfy suitable error conditions. One interesting feature of the proposed algorithm is that the regularization parameter of the model and the accuracy of the derivative approximations are jointly adjusted using a nonmonotone line search criterion. Implementations of the method, where derivative information is computed using finite difference strategies, are discussed. It is shown that, under the assumption that the Hessians of the objectives are globally Lipschitz continuous, the method requires at most $\mathcal{O}(C\epsilon^{-3/2})$ iterations to generate an $\epsilon$-approximate Pareto critical. In particular, if the first- and second-order derivative information is computed using finite differences based solely on function values, the method requires at most $\mathcal{O}(n^{1-\beta}\epsilon^{-3/2})$ iterations, corresponding to $\mathcal{O}\left(mn^{3-\beta}\varepsilon^{-\frac{3}{2}}\right)$ function evaluations, where \(n\) is the dimension of the domain of the objective function, $m$ is the number of objectives, and $\beta \in [0,1]$ is a constant associated with the stepsize used in the finite-difference approximation. We further discuss the global convergence and local convergence rate of the method. Specifically, under the local convexity assumption, we show that the method achieves superlinear convergence when the first derivative is computed exactly, and quadratic convergence when both first- and second-order derivatives are exact.