Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

Abstract

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.

Cited by in corpus (6)

• Near-Optimal Distributed Linear-Quadratic Regulator for Networked Systems
• Separable approximations of optimal value functions under a decaying sensitivity assumption
• A Unifying Modeling Abstraction for Infinite-Dimensional Optimization
• Event Constrained Programming
• Optimal state estimation: Turnpike analysis and performance results
• Constrained Policy Optimization for Stochastic Optimal Control under Nonstationary Uncertainties