A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality
K. Gupta, D. Ghosh, C. Tammer, X. Zhao, J. C. Yao
TL;DR
The article tackles robust weakly efficient solutions for uncertain multiobjective optimization problems with a finite set of scenarios. It reformulates the problem as a deterministic set-valued optimization using the upper set less relation and leverages Gerstewitz scalarization to define a partition-based sequence of vector optimization subproblems. A quasi-Newton scheme with BFGS Hessian updates computes descent directions for these subproblems, coupled with Armijo line searches to guarantee progress. Theoretical results guarantee global convergence to a robust stationary point under regularity, and local superlinear convergence when the Hessian approximations are uniformly continuous; numerical experiments demonstrate effectiveness relative to a classical Newton method in convex settings. The approach provides a practical framework for obtaining robust weakly efficient points in UMOPs and suggests directions for extending to other set relations and broader uncertainty structures.
Abstract
In this article, we derive an iterative scheme through a quasi-Newton technique to capture robust weakly efficient points of uncertain multiobjective optimization problems under the upper set less relation. It is assumed that the set of uncertainty scenarios of the problems being analyzed is of finite cardinality. We also assume that corresponding to each given uncertain scenario from the uncertainty set, the objective function of the problem is twice continuously differentiable. In the proposed iterative scheme, at any iterate, by applying the \emph{partition set} concept from set-valued optimization, we formulate an iterate-wise class of vector optimization problems to determine a descent direction. To evaluate this descent direction at the current iterate, we employ one iteration of the quasi-Newton scheme for vector optimization on the formulated class of vector optimization problems. As this class of vector optimization problems differs iterate-wise, the proposed quasi-Newton scheme is not a straight extension of the quasi-Newton method for vector optimization problems. Under commonly used assumptions, any limit point of a sequence generated by the proposed quasi-Newton technique is found to be a robust weakly efficient point of the problem. We analyze the well-definedness and global convergence of the proposed iterative scheme based on a regularity assumption on stationary points. Under the uniform continuity of the Hessian approximation function, we demonstrate a local superlinear convergence of the method. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed method.