On the convergence analysis of a proximal gradient method for multiobjective optimization

We propose a proximal gradient method for unconstrained nondifferentiable multiobjective optimization problems with the objective function being the sum of a proper lower semicontinuous convex function and a continuously differentiable function. We have shown under appropriate assumptions that each accumulation point of the sequence generated by the algorithm is Pareto stationary. Further, when imposing convexity on the smooth component of the objective function, the convergence of the generated iterative sequence to a weak Pareto optimal point of the problem is established. Meanwhile, the convergence rate of the proposed method is analyzed when the smooth component function in the objective function is non-convex ( \(\mathcal {O}(\sqrt{1/k})\) $\mathcal{O}(\sqrt{1/k})$ ), convex ( \(\mathcal {O}(1/k)\) $\mathcal{O}(1/k)$ ), and strongly convex ( \(\mathcal {O}(r^k)\) $\mathcal{O}(r^k)$ for some \(r\in (0,1)\) $r\in (0,1)$ ), respectively, here k is the number of iterations. The performance of the proposed method on a few test problems with an \(\ell _1\) ${\ell }_1$ -norm function and with the indicator function is provided.