Robust Optimization Under Objective Functional Uncertainty

TL;DR

This paper introduces a novel robust optimization framework under objective functional uncertainty, with a convergence-guaranteed solution algorithm and practical application to battery charging scheduling.

Contribution

It formulates the ObRO model, develops a convergent algorithm using operator theory, and proposes a PWL approximation for practical numerical solutions.

Findings

Algorithm converges to a semi-global saddle point.

PWL approximation is numerically consistent with the original problem.

Application to battery scheduling demonstrates practical effectiveness.

Abstract

This paper proposes a new robust optimization (RO) formulation namely the RO under objective functional uncertainty (ObRO). The ObRO adopts a min-max structure where the inner problem finds the worst-case objective function in a continuous function space to maximize the cost, and the outer problem finds the optimal decision in a Euclidean space to minimize the cost. A solution algorithm is designed to alternately generate the worst-case objective function at the current decision and the optimal decision for the current collection of objective functions. Using operator theory, we prove that this algorithm converges to the defined ``semi-global'' saddle point of the ObRO problem. In addition, we propose a numerical solver based on the piece-wise linearization (PWL) approximation of objective functions. The PWL approximate problem is proved to be numerically consistent with the original ObRO problem. The obtained results are applied to the degradation-aware battery charging scheduling in distribution networks.