An inexact projected gradient method with rounding and lifting by nonlinear programming for solving rank-one semidefinite relaxation of polynomial optimization

Yang, Heng; Liang, Ling; Carlone, Luca; Toh, Kim-Chuan

doi:10.1007/s10107-022-01912-6

Cited by 10 publications

(1 citation statement)

References 79 publications

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“…Such property is an improvement compared to other global optimization methods, such as combinatorial optimization with exponential complexity. To improve the scalability for realtime deployment, combining the global convergence property of SDP [76] and fast local search on Lie groups [77][78][79][80][81], should be considered in the future.…”

Section: Discussionmentioning

confidence: 99%

Lie Algebraic Cost Function Design for Control on Lie Groups

Teng

Clark

Bloch

et al. 2022

2022 IEEE 61st Conference on Decision and Control (CDC)

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This paper reports a novel result: with proper robot models on matrix Lie groups, one can formulate the kinodynamic motion planning problem for rigid body systems as exact polynomial optimization problems that can be relaxed as semidefinite programming (SDP). Due to the nonlinear rigid body dynamics, the motion planning problem for rigid body systems is nonconvex. Existing global optimization-based methods do not properly deal with the configuration space of the 3D rigid body; thus, they do not scale well to long-horizon planning problems. We use Lie groups as the configuration space in our formulation and apply the variational integrator to formulate the forced rigid body systems as quadratic polynomials. Then we leverage Lasserre's hierarchy to obtain the globally optimal solution via SDP. By constructing the motion planning problem in a sparse manner, the results show that the proposed algorithm has linear complexity with respect to the planning horizon. This paper demonstrates the proposed method can provide rank-one optimal solutions at relaxation order two for most of the testing cases of 1) 3D drone landing using the full dynamics model and 2) inverse kinematics for serial manipulators.

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