Second-order Methods for Multiobjective Composite Optimization: Preconditioning Strategies, Subspace Variants and Inexact Solutions

Multiobjective composite optimization problems arise in sparse regularization, constrained multiobjective models, and multi-task learning, but their numerical solution remains challenging when the smooth components are ill-conditioned.

Proximal gradient methods are inexpensive per iteration but may converge slowly, while proximal Newton and quasi-Newton methods exploit curvature information at the cost of evaluating expensive metric proximal mappings.

To address these issues, we propose a preconditioned proximal Barzilai--Borwein method for multiobjective composite optimization.

The method combines objective-wise Barzilai--Borwein scaling, which reduces imbalance among objectives, with a common preconditioner that captures shared curvature information.

To avoid non-diagonal metric proximal mappings, we develop a subspace variant in which the search direction is computed in a two-dimensional subspace generated by a proximal-gradient-type direction and a projected historical direction.

By constructing a conjugate basis with respect to the preconditioning metric, the subspace model decomposes into tractable one-dimensional subproblems.

The framework is further extended to nonsmooth terms of the form $g_i(Ax)$ through a linear-operator-aware preconditioner, yielding explicit proximal evaluations via dual subproblems.

We also analyze an inexact version based on relaxed descent conditions.

We establish the global convergence of the inexact algorithm in the nonconvex setting and prove a linear convergence rate under an error-bound condition.

Numerical experiments on ill-conditioned $\ell_1$-regularized, structured $\ell_1$-regularized, and linearly constrained problems demonstrate the effectiveness of the proposed method.