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.
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© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 11 January 2021
Accepted: 12 December 2021
Published online: 31 May 2022
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Grainger Foundation https://doi.org/10.13039/100008074
Funding Information
National Science Foundation https://doi.org/10.13039/100000001 : CNS-1545046
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U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-AC02-06CH11347
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