Approximation properties and tight bounds for constrained mixed-integer optimal control

• Verfasst von: Kirches, Christian [VerfasserIn]
• Verfasst von: Lenders, Felix [VerfasserIn]
• Verfasst von: Manns, Paul [VerfasserIn]
• Titel: Approximation properties and tight bounds for constrained mixed-integer optimal control
• Verf.angabe: C. Kirches, F. Lenders, and P. Manns
• E-Jahr: 2020
• Jahr: May 20,2020
• Umfang: 32 S.
• Fussnoten: Gesehen am 05.08.2020
• Titel Quelle: Enthalten in: Society for Industrial and Applied MathematicsSIAM journal on control and optimization
• Ort Quelle: Philadelphia, Pa. : Soc., 1966
• Jahr Quelle: 2020
• Band/Heft Quelle: 58(2020), 3, Seite 1371-1402
• ISSN Quelle: 1095-7138
• DOI: doi:10.1137/18M1182917
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: approximation theory
• Sach-SW: mathematical programs
• Sach-SW: mixed-integer optimization
• Sach-SW: optimal control
• Sach-SW: optimization
• Sach-SW: ordinary differential equations
• Sach-SW: switched dynamic systems
• Sach-SW: vanishing constraints
• K10plus-PPN: 1726253813

Abstract

We extend recent work on solving mixed-integer nonlinear optimal control problems (MIOCPs) to the case of integer control functions subject to constraints that involve a pointwise coupling of the state with the integer controls. We extend a theorem due to [S. Sager, H. Bock, and M. Diehl, Math. Program. Ser. A, 133 (2012), pp. 1-23] to the case of MIOCPs with constraints on the integer control and show that the integrality gap vanishes in function space when the coarseness of the rounding grid is driven to zero even after adding constraints of this type. For the time-discretized problem, we extend a sum-up rounding (SUR) scheme due to [S. Sager, C. Reinelt, and H. Bock, Math. Program. Ser. A, 118 (2009), pp. 109-149] to the new problem class. Our scheme permits one to constructively obtain an epsilon-feasible and epsilon-optimal binary feasible control. We derive new, tight upper bounds on the integer control approximation error made by SUR. For unconstrained binary controls on equidistant grids, we reduce the approximation error bound from O(vertical bar Omega vertical bar) to O(log vertical bar Omega vertical bar) asymptotically for vertical bar Omega vertical bar -> infinity and a fixed coarseness of the rounding grid, where vertical bar Omega vertical bar is the number of binary controls. For constrained binary controls, we show that the approximation problem is more difficult, and we give a proof of an approximation error bound of complexity O(vertical bar Omega vertical bar). A numerical example compares our approach to a state-of-the-art mixed-integer nonlinear programming solver and illustrates the applicability of our results when solving MIOCPs using the direct and simultaneous approach.