# Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

arXiv: https://arxiv.org/abs/2101.03067 · HTML: https://mlsanity.com/paper/2101.03067/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2021.

## Built on by (most cited first, 4 of 4)

- [Sub-linear Regret in Adaptive Model Predictive Control](https://mlsanity.com/paper/2310.04842/) (2023)
- [Optimization Solution Functions as Deterministic Policies for Offline Reinforcement Learning](https://mlsanity.com/paper/2408.15368/) (2024)
- [Horizon-Uniform Sensitivity and Decay of Terminal Reward Perturbations in Discrete-Time Pontryagin Systems](https://mlsanity.com/paper/2606.17762/) (2026)
- [A Scalable Learned Iterative Solver for Nonlinear MPC via Temporal Decomposition](https://mlsanity.com/paper/2610.13480/) (2026)

## Abstract (arXiv)

We study solution sensitivity for nonlinear programs (NLPs) whose structures are induced by graphs. These NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that for a given pair of nodes, the sensitivity of the primal-dual solution at one node against a data perturbation at the other node decays exponentially with respect to the distance between these two nodes on the graph. In other words, the solution sensitivity decays as one moves away from the perturbation point. This result, which we call exponential decay of sensitivity, holds under the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the decay rate remains uniformly bounded; this allows us to characterize the sensitivity behavior of NLPs defined over subgraphs of infinite graphs. The theoretical developments are illustrated with numerical examples.