Robust Optimization with Continuous Decision-Dependent Uncertainty

We consider a robust optimization problem with continuous decision-dependent uncertainty (RO-CDDU). RO-CDDU has two main features that have not been addressed in the literature: an uncertainty set with linear dependence on continuous decision variables and a convex piecewise-linear objective function. We prove that RO-CDDU is NP-hard in general. To address the computational challenges, we reformulate RO-CDDU into an equivalent mixed-integer nonlinear program (MINLP) with a decomposable structure. We show that such an MINLP model can be further transformed into a mixed-integer linear program (MILP) given the extreme points of the uncertainty set. We propose an alternating direction algorithm (ADA) and a column generation algorithm (CGA) for the RO-CDDU problem. We further apply RO-CDDU to an important decision problem in electricity markets. In particular, we propose a novel robust demand response (DR) management model as RO-CDDU, where electricity users can reduce or shift their consumption, and the actual realization of demand reduction is uncertain and depends on the DR planning decision. Extensive computational results demonstrate the promising performance of the proposed algorithms in both computational speed and the quality of solutions. The results also shed light on the usefulness of modeling decision-dependent uncertainty in the demand response management problem.

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