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Sequential Hierarchical Least-Squares Programming for Prioritized Non-Linear Optimal Control (2302.11891v2)

Published 23 Feb 2023 in math.OC, cs.NA, cs.RO, and math.NA

Abstract: We present a sequential hierarchical least-squares programming solver with trust-region and hierarchical step-filter with application to prioritized discrete non-linear optimal control. It is based on a hierarchical step-filter which resolves each priority level of a non-linear hierarchical least-squares programming via a globally convergent sequential quadratic programming step-filter. Leveraging a condition on the trust-region or the filter initialization, our hierarchical step-filter maintains this global convergence property. The hierarchical least-squares programming sub-problems are solved via a sparse reduced Hessian based interior point method. It leverages an efficient implementation of the turnback algorithm for the computation of nullspace bases for banded matrices. We propose a nullspace trust region adaptation method embedded within the sub-problem solver towards a comprehensive hierarchical step-filter. We demonstrate the computational efficiency of the hierarchical solver on typical test functions like the Rosenbrock and Himmelblau's functions, inverse kinematics problems and prioritized discrete non-linear optimal control.

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References (49)
  1. Ansary, Md Abu Talhamainuddin. 2023. “A Newton-type proximal gradient method for nonlinear multi-objective optimization problems.” Optimization Methods and Software 0 (0): 1–21.
  2. ApS, MOSEK. 2019. MOSEK Fusion API for C++ 10.1.12. https://docs.mosek.com/latest/cxxfusion/index.html.
  3. Berry, M., Michael Heath, Ikuyo Kaneko, Michael Lawo, Robert Plemmons, and Robert Ward. 1985. “An algorithm to compute a sparse basis of the null space.” Numerische Mathematik 47: 483–504.
  4. Bongartz, I., A. R. Conn, Nick Gould, and Ph. L. Toint. 1995. “CUTE: Constrained and Unconstrained Testing Environment.” ACM Trans. Math. Softw. 21 (1): 123–160.
  5. Broyden, C G. 1970. “The Convergence of a Class of Double-rank Minization Algorithms.” Journal of the Mathematics and its Applications 6: 76–90.
  6. Candès, Emmanuel, Michael Wakin, and Stephen Boyd. 2007. “Enhancing Sparsity by Reweighted L1 Minimization.” Journal of Fourier Analysis and Applications 14: 877–905.
  7. Carpentier, Justin, and Nicolas Mansard. 2018. “Analytical Derivatives of Rigid Body Dynamics Algorithms.” Robotics: Science and Systems XIV https://api.semanticscholar.org/CorpusID:44070783.
  8. Chin, Choong Ming, Abdul Halim Abdul Rashid, and Khalid Mohamed Nor. 2007. “Global and local convergence of a filter line search method for nonlinear programming.” Optimization Methods and Software 22 (3): 365–390. https://doi.org/10.1080/10556780600565489.
  9. Cococcioni, Marco, Massimo Pappalardo, and Yaroslav D. Sergeyev. 2018. “Lexicographic multi-objective linear programming using grossone methodology: Theory and algorithm.” Applied Mathematics and Computation 318: 298–311. Recent Trends in Numerical Computations: Theory and Algorithms.
  10. Courant, R. 1943. “Variational methods for the solution of problems of equilibrium and vibrations.” Bulletin of the American Mathematical Society 49 (1): 1 – 23.
  11. Dang, Thuy V, Keck Voon Ling, and Jan Maciejowski. 2017. “Banded Null Basis and ADMM for Embedded MPC.” IFAC-PapersOnLine 50 (1): 13170–13175. 20th IFAC World Congress.
  12. Escande, Adrien, Nicolas Mansard, and Pierre-Brice Wieber. 2014. “Hierarchical quadratic programming: Fast online humanoid-robot motion generation.” The International Journal of Robotics Research 33 (7): 1006–1028.
  13. Evtushenko, Yu.G., and M.A. Posypkin. 2014. “A deterministic algorithm for global multi-objective optimization.” Optimization Methods and Software 29 (5): 1005–1019.
  14. Fletcher, Roger, Sven Leyffer, and Philippe L. Toint. 2002. “On the Global Convergence of a Filter–SQP Algorithm.” SIAM Journal on Optimization 13 (1): 44–59.
  15. Frison, Gianluca, and Moritz Diehl. 2020. “HPIPM: a high-performance quadratic programming framework for model predictive control.” IFAC-PapersOnLine 53 (2): 6563–6569. 21st IFAC World Congress, https://www.sciencedirect.com/science/article/pii/S2405896320303293.
  16. Geffken, Sören, and Christof Büskens. 2017. “Feasibility refinement in sequential quadratic programming using parametric sensitivity analysis.” Optimization Methods and Software 32 (4): 754–769. https://doi.org/10.1080/10556788.2016.1200045.
  17. Gilbert, John R., and Michael T. Heath. 1987. “Computing a Sparse Basis for the Null Space.” SIAM Journal on Algebraic Discrete Methods 8 (3): 446–459.
  18. Gill, Philip E., Walter Murray, and Michael A. Saunders. 2005. “SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization.” SIAM Rev. 47 (1): 99–131.
  19. Gill, Philip E., Walter Murray, Michael A. Saunders, and Matgaret H. Wright. 1987. “Maintaining LU factors of a general sparse matrix.” Linear Algebra and its Applications 88-89: 239–270.
  20. Gould, Nicholas Ian Mark. 1989. “On the Convergence of a Sequential Penalty Function Method for Constrained Minimization.” SIAM Journal on Numerical Analysis 26 (1): 107–128.
  21. Grimminger, Felix, Thomas Flayols, Jonathan Fiene, Alexander Badri-Spröwitz, Ludovic Righetti, Avadesh Meduri, Majid Khadiv, et al. 2020. “An Open Torque-Controlled Modular Robot Architecture for Legged Locomotion Research.” IEEE Robotics and Automation Letters PP: 1–1.
  22. Guennebaud, Gaël, Benoît Jacob, et al. 2010. “Eigen v3.” http://eigen.tuxfamily.org.
  23. Gurobi Optimization, LLC. 2023. “Gurobi Optimizer Reference Manual.” https://www.gurobi.com.
  24. Han, Shih-Ping. 1975. “A globally convergent method for nonlinear programming.” Journal of Optimization Theory and Applications 22: 297–309.
  25. Hespanhol, Pedro, and Rien Quirynen. 2021. “Adjoint-based SQP method with block-wise quasi-Newton Jacobian updates for nonlinear optimal control.” Optimization Methods and Software 36 (5): 1030–1058. https://doi.org/10.1080/10556788.2019.1653869.
  26. Hestenes, Magnus R. 1969. “Multiplier and gradient methods.” Journal of Optimization Theory and Applications 4: 303–320.
  27. Higham, N. 1986. “Computing the Polar Decomposition with Applications.” SIAM Journal on Scientific and Statistical Computing 7 (4): 1160–1174.
  28. Jäger, H., and E.W. Sachs. 1997. “Global convergence of inexact reduced sqp methods.” Optimization Methods and Software 7 (2): 83–110.
  29. Kaneko, Ikuyo, Michael Lawo, and Georg Thierauf. 1982. “On computational procedures for the force method.” International Journal for Numerical Methods in Engineering 18: 1469–1495.
  30. Kanoun, Oussama, Florent Lamiraux, Pierre-Brice Wieber, Fumio Kanehiro, Eiichi Yoshida, and Jean-Paul Laumond. 2009. “Prioritizing linear equality and inequality systems: Application to local motion planning for redundant robots.” 2009 IEEE International Conference on Robotics and Automation (May): 2939–2944.
  31. Kao, Chiang, and Shih Pen Chen. 1994. “A sequential quadratic programming algorithm utilizing QR matrix factorization.” Engineering Optimization 22 (4): 283–296.
  32. Lai, Leonardo, Lorenzo Fiaschi, Marco Cococcioni, and Kalyanmoy Deb. 2021. “Handling Priority Levels in Mixed Pareto-Lexicographic Many-Objective Optimization Problems.” In Evolutionary Multi-Criterion Optimization, edited by Hisao Ishibuchi, Qingfu Zhang, Ran Cheng, Ke Li, Hui Li, Handing Wang, and Aimin Zhou, Cham, 362–374. Springer International Publishing.
  33. Markowitz, Harry M. 1957. “The Elimination Form of the Inverse and Its Application to Linear Programming.” Management Science 3 (3): 255–269.
  34. Meduri, Avadesh, Paarth Shah, Julian Viereck, Majid Khadiv, Ioannis Havoutis, and Ludovic Righetti. 2023. “BiConMP: A Nonlinear Model Predictive Control Framework for Whole Body Motion Planning.” IEEE Transactions on Robotics 1–18.
  35. Pfeiffer, K., A. Escande, and A. Kheddar. 2018. “Singularity Resolution in Equality and Inequality Constrained Hierarchical Task-Space Control by Adaptive Nonlinear Least Squares.” IEEE Robotics and Automation Letters 3 (4): 3630–3637.
  36. Pfeiffer, Kai, Adrien Escande, Pierre Gergondet, and Abderrahmane Kheddar. 2023. “The Hierarchical Newton’s Method for Numerically Stable Prioritized Dynamic Control.” IEEE Transactions on Control Systems Technology 1–14.
  37. Pfeiffer, Kai, Adrien Escande, and Ludovic Righetti. 2023. “𝒩𝒩\mathcal{N}caligraphic_NIPM-HLSP: an efficient interior-point method for hierarchical least-squares programs.” Optimization and Engineering 1573–2924.
  38. Pfeiffer, Kai, and Ludovic Righetti. 2021. “𝒩𝒩\mathcal{N}caligraphic_NIPM-MPC: An Efficient Null-Space Method Based Interior-Point Method for Model Predictive Control.” https://arxiv.org/abs/2109.03338.
  39. Schwan, Roland, Yuning Jiang, Daniel Kuhn, and Colin N. Jones. 2023. “PIQP: A Proximal Interior-Point Quadratic Programming Solver.” .
  40. Sherali, Hanif D., and Allen L. Soyster. 1983. “Preemptive and nonpreemptive multi-objective programming: Relationship and counterexamples.” Journal of Optimization Theory and Applications 39: 173–186.
  41. Stellato, B., G. Banjac, P. Goulart, A. Bemporad, and S. Boyd. 2020. “OSQP: an operator splitting solver for quadratic programs.” Mathematical Programming Computation 12 (4): 637–672.
  42. Sun, Z.B., Y.Y. Sun, Y. Li, and K.P. Liu. 2019. “A new trust region–sequential quadratic programming approach for nonlinear systems based on nonlinear model predictive control.” Engineering Optimization 51 (6): 1071–1096.
  43. Topcu, A. 1979. “A contribution to the systematic analysis of finite element structures using the force method.” Ph.D. thesis, University of Essen, Germany .
  44. Ulbrich, Michael, Stefan Ulbrich, and Luís Vicente. 2004. “A globally convergent primal-dual interior-point filter method for nonlinear programming.” Math. Program. 100: 379–410.
  45. Vanderbei, R.J. 2013. Linear Programming: Foundations and Extensions. International Series in Operations Research & Management Science. Springer US.
  46. Wächter, Andreas, and Lorenz T. Biegler. 2006. “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming.” Mathematical Programming 106: 25–57.
  47. Wang, Y., and S. Boyd. 2010. “Fast Model Predictive Control Using Online Optimization.” IEEE Transactions on Control Systems Technology 18 (2): 267–278.
  48. Yang, Jiaheng, T.J. Meijer, V.S. Dolk, Bram de Jager, and W.P.M.H. Heemels. 2019. “A System-Theoretic Approach to Construct a Banded Null Basis to Efficiently Solve MPC-Based QP Problems.” In 2019 IEEE Conference on Decision and Control, 1410–1415.
  49. Zhong, Shaopeng, Yu Jiang, and Otto Anker Nielsen. 2022. “Lexicographic multi-objective road pricing optimization considering land use and transportation effects.” European Journal of Operational Research 298 (2): 496–509.
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