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## Multiparametric Linear Complementarity Problems (2006)

Venue: | In IEEE Conference on Decision and Control |

Citations: | 4 - 2 self |

### Citations

7239 | Convex Optimization.
- Boyd, Vandenberghe
- 2004
(Show Context)
Citation Context ...f and only if the polytope (15) is full–dimensional. Remark 4: Note that a polytope can be tested for full– dimensionality through the use of a single linear program by computing the Chebyshev centre =-=[21]-=-. Remark 5: We consider only full–dimensional intersections in Theorem 4 because Theorem 3 demonstrates that this is sufficient to guarantee that all full–dimensional critical regions will be discover... |

322 |
The explicit linear quadratic regulator for constrained systems,”
- Bemporad, Morari
- 2002
(Show Context)
Citation Context ...constrained control allocation [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs =-=[2]-=-, [3], [7]–[11] and pLPs [10], [12], [13] have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enumerated with an algorithm based on the simple... |

90 | An algorithm for multi-parametric quadratic programming and explicit MPC solutions
- Tøndel, Johansen, et al.
- 2003
(Show Context)
Citation Context ... [7], [11], [13] can solve, the proposed approach will not execute a larger number of operations, since these problems will always fall into Case 1 in Section III-B.1. Furthermore, in every case that =-=[8]-=-, [9] can determine the adjacent critical region by inspection, (13) will contain exactly one element and therefore one pivot will be made to find the adjacent region. We therefore claim that the appr... |

84 |
Model predictive control based on linear programming — The explicit solution.
- Bemporad, Borelli, et al.
- 2002
(Show Context)
Citation Context ...ction off–line, the on–line calculation of the control input then becomes one of evaluating the PWA function at the current measured state, which allows for significant improvements in sampling speed =-=[1]-=-. If the system is linear, the constraints polyhedral and the cost linear or quadratic, then the optimisation problem to be solved is a linear or quadratic program. Pre–computation of the PWA control ... |

51 |
Geometric algorithm for multiparametric linear programming,”
- Borrelli, Bemporad, et al.
- 2003
(Show Context)
Citation Context ...on [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], [7]–[11] and pLPs =-=[10]-=-, [12], [13] have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enumerated with an algorithm based on the simplex approach. A geometric appro... |

48 | Post optimal Analyses, Parametric Programming, And Related Topics, - Gal - 1979 |

29 | Frequently asked questions in polyhedral computation,Zurich. http://www.ifor.math.ethz.ch/∼fukuda/ polyfaq/polyfaq.html
- Fukuda
- 2000
(Show Context)
Citation Context ...dundant inequalities of (11). Testing if an inequality is redundant requires a single linear program of dimension d. This is a standard redundancy elimination operation, and the reader is reffered to =-=[20]-=- for computational details. B. Adjacent Region Computation Given a full–dimensional critical region RB and a hyperplane f , { θ ∣∣ γT θ + b = 0}, such that f∩R̄B is a facet of the closure, the goal is... |

20 |
On multi-parametric nonlinear programming and explicit nonlinear model predictive control
- Johansen
- 2002
(Show Context)
Citation Context ...metric linear (pLP) or quadratic program (pQP) [1]–[3]. These algorithms have a wider use in control, such as for constrained control allocation [4] for solving sub–problems in nonlinear optimisation =-=[5]-=- and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], [7]–[11] and pLPs [10], [12], [13] have been published in the control literatur... |

19 | On the linear complementarity problem
- Murty
- 1978
(Show Context)
Citation Context ...tion in general. The paper [16] addressed this issue by combining the approach in [2] with that in [8], although some of the efficiency of [8] is lost. In this paper, we present a new method based on =-=[17]-=- for computing the solution to a multi–parametric linear complementarity problem (pLCP), which is defined by a positive semi–definite matrix. Linear complementarity problems are considered fundamental... |

17 |
An efficient algorithm for multi-parametric quadratic programming
- Baotić
- 2002
(Show Context)
Citation Context ...d control allocation [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], =-=[7]-=-–[11] and pLPs [10], [12], [13] have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enumerated with an algorithm based on the simplex approach... |

17 |
Computation of the constrained infinite time linear quadratic regulator,”
- Grieder, Borelli, et al.
- 2004
(Show Context)
Citation Context ...ntrol allocation [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], [7]–=-=[11]-=- and pLPs [10], [12], [13] have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enumerated with an algorithm based on the simplex approach. A g... |

14 | Polyhedral functions and multiparametric linear programming - Schechter - 1987 |

11 |
Efficient optimal constrained control allocation via multi-parametric programming
- Johansen, Fossen, et al.
- 2005
(Show Context)
Citation Context ...ontrol law then requires the solution of a (multi) parametric linear (pLP) or quadratic program (pQP) [1]–[3]. These algorithms have a wider use in control, such as for constrained control allocation =-=[4]-=- for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], [7]–[11] and pLPs [10], ... |

11 |
The existence of a short sequence of admissible pivots to an optimal basis
- Fukuda, Lüthi, et al.
- 1997
(Show Context)
Citation Context ... if and only if the matrix A−1B,∗ [ q I ] is lexico-positive; such a basis is called lexico-feasible. We now state the main result of lexicographic perturbation: Theorem 1 (Lexicographic Perturbation =-=[19]-=-): If an LCP (q,M) is feasible, then there exists an ǫ1 > 0 such that for all 0 < ǫ0 < ǫ1 the lexicographically perturbed LCP (q+ǫ,M) has a unique copmlementary feasible basis, where ǫ , ( ǫ0, ǫ 2 0, ... |

10 |
Doná, “Characterisation of receding horizon control for constrained linear systems
- Seron, Goodwin, et al.
- 2003
(Show Context)
Citation Context ...timisation problem to be solved is a linear or quadratic program. Pre–computation of the PWA control law then requires the solution of a (multi) parametric linear (pLP) or quadratic program (pQP) [1]–=-=[3]-=-. These algorithms have a wider use in control, such as for constrained control allocation [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constr... |

10 | Soft constraints and exact penalty functions in model predictive control, Control
- Kerrigan, Maciejowski
- 2000
(Show Context)
Citation Context ...s have a wider use in control, such as for constrained control allocation [4] for solving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC =-=[6]-=-. A large number of algorithms for pQPs [2], [3], [7]–[11] and pLPs [10], [12], [13] have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enume... |

10 | Lexicographic perturbation for multiparametric linear programming with applications to control
- Jones, Maciejowski, et al.
- 2007
(Show Context)
Citation Context ...olving sub–problems in nonlinear optimisation [5] and for calculating penalty weights in soft–constrained linear MPC [6]. A large number of algorithms for pQPs [2], [3], [7]–[11] and pLPs [10], [12], =-=[13]-=- have been published in the control literature in the past few years. In [3], [12], [14] all optimal bases are enumerated with an algorithm based on the simplex approach. A geometric approach has been... |

2 |
results on multi-parametric quadratic programming
- “Further
- 2003
(Show Context)
Citation Context ... addresses this problem by enumerating the regions in a non-recursive manner by stepping a sufficiently small distance over the facets of each region to find a point in a neighbouring region. In [8], =-=[9]-=- the algorithm for pQPs was again improved and it was shown that the adjacent critical region can be determined by inspection if it satisfies fairly strict non– degeneracy assumptions. The algorithms ... |

1 |
On-line Tuning of Controllers for Systems with Constraints
- Baric, Baotic, et al.
- 2005
(Show Context)
Citation Context ... [2], [7]–[11] can be applied only to strictly convex pQPs and they implicitly make the assumption that the intersection of two polyhedral critical regions is a face of each. However, it was shown in =-=[15]-=-, [16] that this property does not hold either for strictly, or non-strictly convex pQPs or pLPs/pQPs with parameterised costs and constraints, although it does hold for non–degenerate pLPs [13]. As a... |

1 |
On the facet-to-facet property of convex parametric quadratic programs
- Spjøtvold, Kerrigan, et al.
(Show Context)
Citation Context ...[7]–[11] can be applied only to strictly convex pQPs and they implicitly make the assumption that the intersection of two polyhedral critical regions is a face of each. However, it was shown in [15], =-=[16]-=- that this property does not hold either for strictly, or non-strictly convex pQPs or pLPs/pQPs with parameterised costs and constraints, although it does hold for non–degenerate pLPs [13]. As a resul... |