Multi-dimensional Variational Control Problem with Data Uncertainty in Objective and Constraint Functionals

Abstract

AbstractInsufficient information, outdated sources, a big amount of data, sampling discrepancy, etc. arise the uncertainty in the data, which generates the barrier to find the exact solution to the problem. In this way, the robust approach is growing rapidly for finding the solution to the problem in the face of data uncertainty. The purpose of this method is to minimize the maximum uncertainty involved in the problem. This fact motivated the researchers to study various optimization problems in the face of data uncertainty and developed new results on this topic. Liu and Yuan [1] generated a robust algorithm that converges to a point that satisfies a certain first-order necessary efficiency condition when the considered problem is itself infeasible. Jeyakumar [2] characterized the robust approach when the uncertainty consists in the constraints function. Whereas, Wei et al. [3] examined the robust optimization problems under the strictly robust counterpart via the image space analysis. More precisely, a relationship between the uncertain optimization problem and its image problem was proved, which provides an approach to tackle minimax problems. Sun et al. [4] studied the robust convex optimization problem and developed the associated saddle-point theory. Besides this, Preeti et al. [5] extended the robust approach over multi-dimensional control problem in the face of data uncertainty and used the penalty function method to simplify the considered constrained problem.