Decision-Focused Learning under Joint Uncertainty: Local Geometry and Complementarity Structure

Decision-focused learning trains predictors for decisions in downstream optimisation problems. While objective uncertainty leaves the feasible set unchanged, constraint uncertainty can reshape the feasible set, render the represented decision infeasible for the true problem, and alter the active set governing local sensitivity. To study these effects, we focus on strongly convex quadratic programs and introduce $\text{SIGNAL}$ by formalising a principled training loss function integrating objective-induced decision divergence, true-constraint violation, and active-set disagreement. We establish finite-sample bounds linking the $\text{SIGNAL}$ loss to operational regret, true-violation frequency and magnitude, and dormant-bound prediction error. We then characterise the local learning information available through the represented optimiser, i.e., identify decision-sensitivity components alongside decision-null bound directions that remain observable through the primal–dual complementarity structure. We further show that $\text{SIGNAL}$ is regionally convex quadratic in the prediction errors, with an affine output-space gradient and explicit active-set mismatch offsets. We finally demonstrate the effectiveness of $\text{SIGNAL}$ on IEEE power-grid benchmarks, with particular emphasis on constraint shifts and their consequences for decision quality and true feasibility.