Pose graph optimization is recast as damped Riemannian dynamics on Lie groups, enabling a fully distributed algorithm with a semi-implicit integrator that converges under both synchronous and asynchronous communication.
Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs
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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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Uniform controllability and observability imply exponential decay of sensitivity under uniform Hessian boundedness, uSOSC, and uLICQ in dynamic optimization.
A Julia framework combines Plasmo.jl and MadNLP.jl to model and solve graph-structured nonlinear optimization problems, demonstrated on a large stochastic gas network instance with over 1.7 million variables.
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Distributed Pose Graph Optimization via Continuous Riemannian Dynamics
Pose graph optimization is recast as damped Riemannian dynamics on Lie groups, enabling a fully distributed algorithm with a semi-implicit integrator that converges under both synchronous and asynchronous communication.
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Controllability and Observability Imply Exponential Decay of Sensitivity in Dynamic Optimization
Uniform controllability and observability imply exponential decay of sensitivity under uniform Hessian boundedness, uSOSC, and uLICQ in dynamic optimization.
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A Julia framework combines Plasmo.jl and MadNLP.jl to model and solve graph-structured nonlinear optimization problems, demonstrated on a large stochastic gas network instance with over 1.7 million variables.