Robust Optimization Approach for Solving Uncertain Multiobjective Optimization Problems Using the Projected Gradient Method
Shubham Kumar, Nihar Kumar Mahatoa, Debdas Ghosh
TL;DR
This work addresses robust, uncertain multiobjective optimization by extending the projected gradient method to the objective-wise worst-case counterparty $OWC_{P(U)}$ over a finite uncertainty set $U$. By reformulating the robust problem into the deterministic, non-smooth $H(x)=(H_1(x),\dots,H_m(x))^T$ with $H_j(x)=\max_{\xi_i\in U} h_j(x,\xi_i)$, the authors develop a projected gradient descent framework that solves a subproblem to obtain descent directions and employs an Armijo-type line search to guarantee progress. They establish that accumulation points of the generated sequences are Pareto critical (and under convexity assumptions, globally convergent) to a robust Pareto optimum of $P(U)$, and validate the method numerically against scalarization-based approaches. The results demonstrate that the proposed method produces high-quality, well-spread Pareto fronts in both convex and nonconvex settings without the need for prespecified weights, highlighting its practical appeal for uncertain multiobjective decision problems. Future work includes extending step-size rules (Wolfe, Zoutendijk) and handling infinite uncertainty sets.
Abstract
Numerous real-world applications of uncertain multiobjective optimization problems (UMOPs) can be found in science, engineering, business, and management. To handle the solution of uncertain optimization problems, robust optimization is a relatively new field. An extended version of the projected gradient method (PGM) for a deterministic smooth multiobjective optimization problem (MOP) is presented in the current study as a PGM for UMOP. An objective-wise worst-case cost (OWWC) type robust counterpart is considered, and the PGM is used to solve a UMOP by using OWWC. A projected gradient descent algorithm is created using theoretical findings. It is demonstrated that the projected gradient descent algorithm's generated sequence converges to the robust counterpart's weak Pareto optimal solution, which will be the robust weak Pareto optimal solution for UMOP. Under a few reasonable presumptions, the projected gradient descent algorithm's full convergent behavior is also justified. Finally, numerical tests are presented to validate the proposed method.