A Quasi-Newton Method to Solve Uncertain Multiobjective Optimization Problems with Uncertainty Set of Finite Cardinality

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.