Execution time certification for gradient-based optimization in model predictive control

Giselsson, Pontus

doi:10.1109/cdc.2012.6427093

Cited by 17 publications

(16 citation statements)

References 12 publications

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“…Let Y * be the (bounded) optimal solution of (26). Then, the optimal solution of the primal QP (26) is…”

Section: A Pqp-based Solution Of Mpc Quadratic Programsmentioning

confidence: 99%

“…Solving the QP problem (24) via its dual (26) has the drawback that if in (24) there are more constraints than variables (n z < n y ), (26) has more variables than (24), and Q d ≥ 0 (while Q p > 0). On the other hand, solving the dual allows us to enforce termination conditions guaranteeing an ε-solution according to Definition 1 by using the duality gap.…”

Section: A Pqp-based Solution Of Mpc Quadratic Programsmentioning

confidence: 99%

“…A key property of the algorithms in [22], [23] is that, under mild assumptions, a bound on the number of iterations to converge to the solution can be computed, thus making the algorithm maximum number of operations predictable. An algorithm based on the fast gradient method combined with the Lagrange method of multipliers was proposed in [24] and accelerated gradient methods for problems with a structure, and in particular for distributed MPC, were proposed in [25], [26]. The iterative algorithms in [22]- [24] perform two steps at every iteration.…”

Section: Introductionmentioning

confidence: 99%

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Projection-free parallel quadratic programming for linear model predictive control

Cairano

Brand

Bortoff

2013

International Journal of Control

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Brand, M.; Bortoff, S. TR2013-059 July 2013Abstract A key component in enabling the application of model predictive control (MPC) in fields such as automotive, aerospace and factory automation is the availability of low-complexity fast optimization algorithms to solve the MPC finite horizon optimal control problem in architectures with reduced computational capabilities. In this paper we introduce a projection-free iterative optimization algorithm and discuss its application to linear MPC. The algorithm, originally developed by Brand for non-negative quadratic programs, is based on a multiplicative update rule and it is shown to converge to a fixed point which is the optimum. An acceleration technique based on a projection-free line search is also introduced, to speed-up the convergence to the optimum. The algorithm is applied to MPC through the dual of the quadratic program (QP) formulated from the MPC finite time optimal control problem. We discuss how termination conditions with guaranteed degree of suboptimality can be enforced, and how the algorithm performance can be optimized by pre-computing the matrices in a parametric form.We show computational results of the algorithm in three common case studies and we compare such results with the results obtained by other available free and commercial QP solvers. International Journal of ControlThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. AbstractA key component in enabling the application of model predictive control (MPC) in fields such as automotive, aerospace and factory automation is the availability of low-complexity fast optimization algorithms to solve the MPC finite horizon optimal control problem in architectures with reduced computational capabilities. In this paper we introduce a projection-free iterative optimization algorithm and discuss its application to linear MPC. The algorithm, originally developed by Brand for non-negative quadratic programs, is based on a multiplicative update rule and it is shown to converge to a fixed point which is the optimum. An acceleration technique based on a projectionfree line search is also introduced, to speed-up the convergence to the optimum. The algorithm is applied to MPC through the dual of the quadratic program (QP) formulated from the MPC finite time optimal control problem. We discuss how termination conditions with guaranteed degree of suboptimality can be enforced, and how the algo...

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“…Let Y * be the (bounded) optimal solution of (26). Then, the optimal solution of the primal QP (26) is…”

Section: A Pqp-based Solution Of Mpc Quadratic Programsmentioning

confidence: 99%

Section: A Pqp-based Solution Of Mpc Quadratic Programsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projection-free parallel quadratic programming for linear model predictive control

Cairano

Brand

Bortoff

2013

International Journal of Control

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“…In addition to updating the iterate x k+1 = x k + α k p k , the working set is updated by adding the first blocking constraint, i.e., the minimizing index of (17). Concretely, if m is the minimizing index in (17), the updated working set becomes (3), λ k can be obtained by solving…”

Section: Remarkmentioning

confidence: 99%

Exact Complexity Certification of a Standard Primal Active-Set Method for Quadratic Programming

Arnström

Axehill

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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In model predictive control (MPC) an optimization problem has to be solved at each time step, which in realtime applications makes it important to solve these optimization problems efficiently and to have good upper bounds on worst-case solution time. Often for linear MPC problems, the optimization problem in question is a quadratic program (QP) that depends on parameters such as system states and reference signals. A popular class of methods for solving such QPs is active-set methods, where a sequence of linear systems of equations are solved. We propose an algorithm for computing which sequence of subproblems an active-set algorithm will solve, for every parameter of interest. By knowing these sequences, a worst-case bound on how many iterations, and ultimately the maximum time, the active-set algorithm needs to converge can be determined. The usefulness of the proposed method is illustrated on a set of QPs, originating from MPC problems, by computing the exact worst-case number of iterations primal and dual active-set algorithms require to reach optimality.

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“…In [12], [13], fast-gradient algorithms have been introduced. In [14], algorithms based on the fast gradient method combined with the Lagrange method of multipliers have been developed , and accelerated gradient methods for distributed MPC have been developed in [15], [16]. For fast gradient-based algorithms is that a bound on the number of iterations to converge to the solution can be computed.…”

Section: Introductionmentioning

confidence: 99%

On a multiplicative update dual optimization algorithm for constrained linear MPC

Cairano

Brand

2013

52nd IEEE Conference on Decision and Control

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We discuss a multiplicative update quadratic programming algorithm with applications to model predictive control for constrained linear systems. The algorithm, named PQP, is very simple to implement and thus verify, does not require projection, offers a linear rate of convergence, and can be completely parallelized. The PQP algorithm is equipped with conditions that guarantee the desired bound on sub-optimality and with an acceleration step based on projection-free line search. We also show how PQP can take advantage of the parametric structure of the MPC problem, thus moving offline several calculations and avoiding large input/output dataflows. The algorithm is evaluated on two benchmark problems, where it is shown to compete with, and possibly outperform, other open source and commercial packages. IEEE Conference on Decision and Control (CDC)This work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Abstract-We discuss a multiplicative update quadratic programming algorithm with applications to model predictive control for constrained linear systems. The algorithm, named PQP, is very simple to implement and thus verify, does not require projection, offers a linear rate of convergence, and can be completely parallelized. The PQP algorithm is equipped with conditions that guarantee the desired bound on suboptimality and with an acceleration step based on projection-free line search. We also show how PQP can take advantage of the parametric structure of the MPC problem, thus moving offline several calculations and avoiding large input/output dataflows. The algorithm is evaluated on two benchmark problems, where it is shown to compete with, and possibly outperform, other open source and commercial packages.

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Execution time certification for gradient-based optimization in model predictive control

Cited by 17 publications

References 12 publications

Projection-free parallel quadratic programming for linear model predictive control

Projection-free parallel quadratic programming for linear model predictive control

Exact Complexity Certification of a Standard Primal Active-Set Method for Quadratic Programming

On a multiplicative update dual optimization algorithm for constrained linear MPC

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