Modified Quasi-Newton Method for Nonconvex Multiobjective Optimization Problems with Barzilai-Borwein diagonal matrix

TL;DR

This paper introduces a novel Barzilai-Borwein diagonal Quasi-Newton method for large-scale nonconvex multiobjective optimization, which uses a shared Hessian approximation to improve efficiency and convergence without convexity assumptions.

Contribution

It proposes a new BB-DQN method that uses a single shared Hessian approximation for all objectives, reducing computational cost and enabling convergence analysis in nonconvex settings.

Findings

Outperforms existing methods like M-BFGSMO in computational efficiency.

Proves global convergence and R-linear convergence under mild conditions.

Effective for large-scale nonconvex multiobjective problems.

Abstract

This paper addresses the challenge of developing efficient algorithms for large-scale nonconvex multiobjective optimization problems (MOPs). While quasi-Newton methods are effective, their traditional application to MOPs is computationally expensive as they require maintaining and inverting separate Hessian approximations for each objective function. To overcome this limitation, we propose a novel Barzilai-Borwein diagonal-type Quasi-Newton method (BB-DQN). Our key innovation is the use of a single, shared, and modified BB-type matrix, updated iteratively using function and gradient information, to approximate the Hessians of all objectives simultaneously. We theoretically demonstrate that this approximation matrix remains positive definite throughout the iterative process. Furthermore, we establish the global convergence of the BB-DQN method without convexity assumptions and prove its R-linear convergence under mild conditions. Numerical experiments on a diverse set of test problems confirm that BB-DQN outperforms existing methods like M-BFGSMO, achieving superior performance in terms of computational time, iteration count, and reliability, especially for large-scale instances.