Sensitivity Generation in an Adaptive BDF-Method

SUMMARYWe present a numerical method and results for a recently published benchmark problem (Optim. Contr. Appl. Met. 2005; 26:1-18; Optim. Contr. Appl. Met. 2006; 27(3):169-182) in mixed-integer optimal control. The problem has its origin in automobile test-driving and involves discrete controls for the choice of gears. Our approach is based on a convexification and relaxation of the integer controls constraint. Using the direct multiple shooting method we solve the reformulated benchmark problem for two cases: we were able to reduce the overall computing time considerably, applying our algorithm. We give theoretical evidence on why our convex reformulation is highly beneficial in the case of time-optimal mixed-integer control problems as the chosen benchmark problem basically is (neglecting a small regularization term).

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“…[28][29][30]. Obviously we obtain from the above IVPs N shoot trajectories, which in general will not combine to a single continuous trajectory.…”

Section: The Direct Multiple Shooting Methodsmentioning

confidence: 99%

“…For the solution of the ODE system, an adaptive fourth/fifth-order Runge-Kutta-Fehlberg method equipped with internal numerical differentiation (IND), as described in [28,29], was used. An integration tolerance of 10 −9 was used.…”

Section: Solutionsmentioning

confidence: 99%

Time‐optimal control of automobile test drives with gear shifts

Kirches

Säger

Bock

et al. 2009

Optim Control Appl Methods

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“…Consistency of the derivatives is ensured by applying the principle of internal numerical differentiation (IND) [1].…”

Section: Block Sparse Quadratic Subproblemmentioning

confidence: 99%

Complementary Condensing for the Direct Multiple Shooting Method

Kirches

Bock

Schlöder

et al. 2012

Modeling, Simulation and Optimization of Complex Processes

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Summary. In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems typically arise from the convexification of integer control decisions. We treat this problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using an SQP method. Concerning the solution of the quadratic subproblems we present a factorization of the QP's KKT system, based on a combined null-space range-space approach exploiting the problem's block sparse structure. We demonstrate the merit of this approach for a vehicle control problem in which the integer gear decision is convexified.

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“…To ensure consistency of the derivatives, this should be done according to the principle of internal numerical differentiation (IND) [1,6], i.e. by computing nominal solution and its derivatives using the same discretization scheme.…”

Section: Constraint Discretizationmentioning

confidence: 99%

Block-structured quadratic programming for the direct multiple shooting method for optimal control

Kirches

Bock

Schlöder

et al. 2011

Optimization Methods and Software

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Abstract. In this contribution we address the 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. We review the classical condensing algorithm that preprocesses the large but sparse quadratic programs to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present 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 sparse 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. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally we demonstrate the merit of the new complementary condensing approach by comparing the behavior of both methods for a vehicle control problem in which the integer gear decision is convexified.

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Sensitivity Generation in an Adaptive BDF-Method

Cited by 20 publications

References 7 publications

Time‐optimal control of automobile test drives with gear shifts

Time‐optimal control of automobile test drives with gear shifts

Complementary Condensing for the Direct Multiple Shooting Method

Block-structured quadratic programming for the direct multiple shooting method for optimal control

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