Adaptive generalized conditional gradient method for multiobjective optimization
Anteneh Getachew Gebrie, Ellen Hidemi Fukuda
TL;DR
This work extends the Frank-Wolfe/conditional gradient framework to multiobjective optimization with composite objectives $\mathcal{V}(x)=\mathcal{F}(x)+\mathcal{G}(x)$, where each $\mathcal{V}_i=f_i+g_i$. It introduces an adaptive $\varepsilon$-normalized descent direction combined with Armijo line search (A-GCG) to ensure descent and Pareto stationarity, without requiring exact Lipschitz constants. The authors establish convergence to Pareto stationary points, derive an $O(1/\mu^{2})$ iteration bound for obtaining a $\mu$-approximate Pareto critical point, and prove a sublinear merit-rate of $O(1/k)$; they also demonstrate competitive empirical performance against related methods. The results provide a robust, scalable framework for solving convex multiobjective optimization with composite objectives in constrained and unconstrained settings, with practical implications for large-scale problems and applications like portfolio optimization and machine learning contexts.
Abstract
In this paper, we propose a generalized conditional gradient method for multiobjective optimization, which can be viewed as an improved extension of the classical Frank-Wolfe (conditional gradient) method for single-objective optimization. The proposed method works for both constrained and unconstrained benchmark multiobjective optimization problems, where the objective function is the summation of a smooth function and a possibly nonsmooth convex function. The method combines the so-called normalized descent direction as an adaptive procedure and the line search technique. We prove the convergence of the algorithm with respect to Pareto optimality under mild assumptions. The iteration complexity for obtaining an approximate Pareto critical point and the convergence rate in terms of a merit function is also analyzed. Finally, we report some numerical results, which demonstrate the feasibility and competitiveness of the proposed method.