Related to work in mixed-integer optimal control, we are interested in an efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. The classical condensing algorithm preprocesses the large but structured quadratic programs to obtain small but dense ones. It can be observed that this approach is not optimal when applied in conjunction with outer convexification.
We developed a new complementary condensing algorithm for quadratic programs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block structure of the quadratic programs that is due to direct multiple shooting. An assessment of the theoretical run time complexity reveals significant advantages of the proposed algorithm.
Selected publications
@article{Weber2019,
author = {Weber, T. and Sager, S. and Gleixner, A.},
title = {Solving Quadratic Programs to High Precision using Scaled Iterative Refinement},
journal = {Mathematical Programming Computation},
year = {2019},
volume = {11},
number = {3},
pages = {421--455},
doi = {10.1007/s12532-019-00154-6}
}
@article{Janka2016,
author = {Janka, D. and Kirches, C. and Sager, S. and W\"achter, A.},
title = {An {SR1/BFGS SQP} algorithm for nonconvex nonlinear programs with block-diagonal {H}essian matrix},
journal = {Mathematical {P}rogamming {C}omputation},
year = {2016},
volume = {8},
number = {4},
pages = {435--459},
doi = {10.1007/s12532-016-0101-2}
}
@article{Frasch2015,
author = {Frasch, J. V. and Sager, S. and Diehl, M.},
title = {A parallel quadratic programming method for dynamic optimization problems},
journal = {Mathematical {P}rogramming {C}omputation},
year = {2015},
volume = {7},
number = {3},
pages = {289--329}
}
@phdthesis{Frasch2014,
author = {Frasch, J.},
title = {{P}arallel {A}lgorithms for {O}ptimization of {D}ynamic {S}ystems in {R}eal-{T}ime},
school = {Otto von Guericke University Magdeburg},
year = {2014},
url = {https://mathopt.de/publications/Frasch2014.pdf}
}
@article{Frasch2013c,
author = {Frasch, J. V. and Vukov, M. and Ferreau, H.J. and Diehl, M.},
title = {{A} dual {N}ewton strategy for the efficient solution of sparse quadratic programs arising in {SQP}-based nonlinear {MPC}},
journal = {{O}ptimization {O}nline},
year = {2013},
note = {Optimization Online 3972},
url = {http://www.optimization-online.org/DB_HTML/2013/07/3972.html}
}
@inproceedings{Kozma2013a,
author = {Kozma, A. and Frasch, J. V. and Diehl, M.},
title = {{A} {D}istributed {M}ethod for {C}onvex {Q}uadratic {P}rogramming {P}roblems {A}rising in {O}ptimal {C}ontrol of {D}istributed {S}ystems},
booktitle = {{P}roceedings of the 52nd {C}onference on {D}ecision and {C}ontrol ({CDC})},
year = {2013}
}
@mastersthesis{Burgdoerfer2011,
author = {Burgd\"orfer, H.},
title = {{S}trukturausnutzende {A}lgorithmen der {Q}uadratischen {P}rogrammierung f\"ur gemischt-ganzzahlige {O}ptimalsteuerung},
school = {Universit\"at Heidelberg},
year = {2011}
}
@article{Kirches2010b,
author = {Kirches, C. and Bock, H.G. and Schl\"{o}der, J.P. and Sager, S.},
title = {{B}lock {S}tructured {Q}uadratic {P}rogramming for the {D}irect {M}ultiple {S}hooting {M}ethod for {O}ptimal {C}ontrol},
journal = {{O}ptimization {M}ethods and {S}oftware},
year = {2011},
volume = {26},
number = {2},
pages = {239--257},
url = {http://www.informaworld.com/smpp/content~db=all~content=a919763304~frm=titlelink},
doi = {10.1080/10556781003623891}
}