## Decomposition of mixed-integer optimal control problems using branch and bound and sparse direct collocation (2000)

Venue: | IN PROCEEDINGS OF ADPM 2000 – AUTOMATION OF MIXED PROCESSES: HYBRID DYNAMIC SYSTEMS |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Stryk00decompositionof,

author = {Oskar von Stryk and Markus Glocker},

title = {Decomposition of mixed-integer optimal control problems using branch and bound and sparse direct collocation},

booktitle = {IN PROCEEDINGS OF ADPM 2000 – AUTOMATION OF MIXED PROCESSES: HYBRID DYNAMIC SYSTEMS},

year = {2000},

pages = {18--19},

publisher = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

A large class of optimal control problems for hybrid dynamic systems can be formulated as mixed-integer optimal control problems (MIOCPs). It is the intrinsic combinatorial complexity, in addition to the nonlinearity of the continuous, multi-phase optimal control problems that is largely responsible for the challenges in the theoretical and numerical solution of MIOCPs. We present a new decomposition approach to numerically solving fairly general MICOPs with binary control variables. A Branch and Bound (B&B) technique is applied to efficiently search the entire discrete solution space performing a truncated binary tree search for the discrete variables maintaining upper and lower bounds on the performance index. The partially relaxed binary variables at an inner node define an optimal control problem with dynamic equations defined in multiple phases. Its global solution provides a lower bound on the performance index for all nodes of the subtree. If the lower bound for a given subtree is greater than the current global upper bound then that entire subtree need no longer be searched. The many optimal control problems with nonlinear, continuous state dynamics defined in multiple phases subject to nonlinear constraints are solved most efficiently by a sparse direct collocation transcription. Hereby, the multi-phase optimal control problem is transcribed to a sparse, large-scale nonlinear programming problem being solved efficiently by a tailored SQP method. Despite the high efficiency of the sparse direct collocation method, the efficiency of the decomposition technique for MIOCPs strongly depends on

### Citations

335 | SNOPT: An SQP algorithm for large-scale constrained optimization
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Citation Context ...ion a current guess of the solution y is improved by the solution of a quadratic subproblem derived from a quadratic approximation of the Lagrangian of the NLP subject to the linearized constraints [=-=3, 8]-=-. The NLPs resulting from a direct collocation discretization have several special properties [15]: The NLPs are of large-scale with very many variables and very many constraints. Most of the NLP cons... |

314 |
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- 1985
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Citation Context ...re he started. In what order should he visit them to minimize the overall travel time? The Traveling Salesman Problem (TSP) is one of the most prominent members of combinatorial optimization problems =-=[7, 11]-=-. Here, we introduce a hybrid dynamical extension of the TSP to demonstrate the strong interaction of continuous and discrete dynamics in hybrid optimal control [15] being first presented to the scien... |

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222 | Combinatorial Optimization
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131 |
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56 |
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21 |
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Citation Context ...bles of MIOCPs, the resulting mixed-integer nonlinearly constrained optimization problems (MINLPs) are in generally nonconvex. Consequently, the recently developed numerical methods for convex MINLPs =-=[1, 9]-=- cannot be applied, since the bounding properties of the relaxed problem cannot be achieved [2]. However, there is another reason why MINLP techniques are not suited at all: The actual value of the di... |

20 |
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Citation Context ...tion although the EL-DEQs have not been solved explicitly. Local optimality error estimates can be derived that enable efficient strategies for successively refining a first solution on a coarse grid =-=[12, 14]-=-. Thus, a sequence of related NLPs must be solved whose dimensions increase with the number of grid points. NLPs can be solved most efficiently numerically by SQP methods. In each SQP iteration a curr... |

19 | SQP methods and their application to numerical optimal control
- Barclay, Gill, et al.
- 1998
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Citation Context ...ion a current guess of the solution y is improved by the solution of a quadratic subproblem derived from a quadratic approximation of the Lagrangian of the NLP subject to the linearized constraints [=-=3, 8]-=-. The NLPs resulting from a direct collocation discretization have several special properties [15]: The NLPs are of large-scale with very many variables and very many constraints. Most of the NLP cons... |

16 |
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Citation Context ...d indirect methods [16]. Indirect methods approximate a solution by explicitly solving first and second order optimality conditions resulting from EL-DEQs and the MP. For reasons already discussed in =-=[6, 12, 16]-=- they are not flexible enough for the purpose needed here. Direct methods are based on a transcription of optimal control problems into (finite dimensional) nonlinearly constrained optimization proble... |

7 | Mixed-integer nonlinear optimization in process synthesis
- Adjiman, Schweiger, et al.
- 1998
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Citation Context ...bles of MIOCPs, the resulting mixed-integer nonlinearly constrained optimization problems (MINLPs) are in generally nonconvex. Consequently, the recently developed numerical methods for convex MINLPs =-=[1, 9]-=- cannot be applied, since the bounding properties of the relaxed problem cannot be achieved [2]. However, there is another reason why MINLP techniques are not suited at all: The actual value of the di... |

4 | Towards hybrid optimal control
- Buss, von, et al.
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Citation Context ...d from a quadratic approximation of the Lagrangian of the NLP subject to the linearized constraints [3, 8]. The NLPs resulting from a direct collocation discretization have several special properties =-=[15]-=-: The NLPs are of large-scale with very many variables and very many constraints. Most of the NLP constraints are active at the solution, e.g., the equality constraints from collocation. Thus, the num... |

3 |
Mixed integer dynamic optimization
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Citation Context ... are in generally nonconvex. Consequently, the recently developed numerical methods for convex MINLPs [1, 9] cannot be applied, since the bounding properties of the relaxed problem cannot be achieved =-=[2]-=-. However, there is another reason why MINLP techniques are not suited at all: The actual value of the discrete variable determines the sequence, type and number of phase dynamics. Thus, the actual dy... |

2 |
Modellierung und Numerik gemischt-ganzzahliger Optimalsteuerungsprobleme, Diploma thesis
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- 2000
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Citation Context ...mize the overall travel time? We consider a hybrid, cooperative dynamic game extension of the MTSP for two salesmen representing the class of rather autonomous dynamic agents that cooperate optimally =-=[10, 13, 15]-=-. The solution for 5 cities is displayed in Fig. 6. There VI Oskar von Stryk and Markus Glocker 0 100 200 300 400 0 200 400 600 Figure 6: The tours of two cooperating motorized salesmen to 5 cities. e... |

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Discrete-continuous optimal control of dynamic systems
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Citation Context ...duce a hybrid dynamical extension of the TSP to demonstrate the strong interaction of continuous and discrete dynamics in hybrid optimal control [15] being first presented to the scientific public in =-=[13]-=-. The salesman is supposed Decomposition of Mixed-Integer Optimal Control Problems V to drive a car (Fig. 4). The task is to determine the steering angle velocity g and the accelerating or braking for... |

1 |
Towards hybrid optimal control. at --- Automatisierungstechnik (to appear
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- 2000
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Citation Context ...t index R t 0 L(x;u;q;v)dt is minimized subject to the system dynamicssx = f(x;u;q;v; t) and further constraints, where x denotes the continuous state and q : [0; t f ] ! Q Z n q the discrete state [=-=6]-=-. Assuming a finite, either given or bounded, number of switchings for both q and v, then the hybrid optimal control problem can be transformed into a MIOCP with integer variables which may be represe... |