Global Convergence and Rate Analysis of the Steepest Descent Method for Uncertain Multiobjective Optimization via a Robust Optimization Approach
Shubham Kumar, Nihar Kumar Mahato, Debdas Ghosh
TL;DR
The paper addresses global convergence and convergence rate for the steepest descent method applied to uncertain multiobjective optimization via the robust counterpart OWRC, defined by $\Phi_j(x)=\max_{i\in\bar{\Lambda}} f_j(x,\xi_i)$. It constructs a descent-direction subproblem $\min_t\{\vartheta_x(t)+\tfrac{1}{2}\|t\|^2\}$ with $\vartheta_x(t)=\max_{j,i}\{ f_j(x,\xi_i)+\nabla f_j(x,\xi_i)^T t-\Phi_j(x)\}$, whose unique solution yields the descent direction $t(x)=-\sum_{j,i}\lambda_{ij}\nabla f_j(x,\xi_i)$, and uses an Armijo-type line search to determine step sizes. The authors prove that accumulation points of the generated sequence are robust Pareto critical points, and under convexity this leads to weak Pareto optimality; with strong convexity and smoothness and a fixed step, they establish a linear rate of convergence $\|x^{k+1}-x^*\|^2 \le (1-\alpha\beta)\|x^k-x^*\|^2$. These results strengthen the theoretical basis and broaden the applicability of steepest descent in robust multiobjective optimization with finite uncertainty.
Abstract
In this article, we extend our previous work (Applicable Analysis, 2024, pp. 1-25) on the steepest descent method for uncertain multiobjective optimization problems. While that study established local convergence, it did not address global convergence and the rate of convergence of the steepest descent algorithm. To bridge this gap, we provide rigorous proofs for both global convergence and the linear convergence rate of the steepest descent algorithm. Global convergence analysis strengthens the theoretical foundation of the steepest descent method for uncertain multiobjective optimization problems, offering deeper insights into its efficiency and robustness across a broader class of optimization problems. These findings enhance the method's practical applicability and contribute to the advancement of robust optimization techniques.