Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

TL;DR

This paper demonstrates that in graph-structured nonlinear programs, the sensitivity of solutions to data perturbations diminishes exponentially with distance on the graph, under certain conditions.

Contribution

It establishes the exponential decay of solution sensitivity in graph-structured NLPs and characterizes conditions for uniform decay rates, extending understanding of local versus global effects.

Findings

Sensitivity decays exponentially with graph distance.

Decay rate remains bounded under specific conditions.

Numerical examples illustrate theoretical results.

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.