## Interior Point Methods For Optimal Control Of Discrete-Time Systems (1993)

Venue: | Journal of Optimization Theory and Applications |

Citations: | 34 - 6 self |

### BibTeX

@ARTICLE{Wright93interiorpoint,

author = {Stephen J. Wright},

title = {Interior Point Methods For Optimal Control Of Discrete-Time Systems},

journal = {Journal of Optimization Theory and Applications},

year = {1993},

volume = {77},

pages = {161--187}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discrete-time optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete time linear-quadratic regulator problem with mixed state/control constraints, and show how it can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interior-point method is the narrow-banded structure of the coefficient matrix which is factorized at each iteration. Key words. interior point algorithms, optimal control, banded linear systems. 1. Introduction. The problem of optimal control of an initial value ordinary differential equation, with Bolza objectives and mixed constraints, is min x;u Z T 0 L(x(t); u(t); t) dt + OE f (x(T )); x(t) = f(x(t); u(t); t); x(0) = x init ; (1.1) g(x(t); u(...

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Citation Context ...atic convergence of the duality gap to zero. The second algorithm we consider is of the "predictor-corrector" type. For linear programming problems, this algorithm is described by Mizuno, To=-=dd and Ye [22] and Ye, T-=-apia and Zhang [34]. The analysis is extended to linear complementarity problems by Ji, Potra and Huang [15]. The algorithm makes use of the idea of an "ff-neighborhood" of the central path ... |

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Citation Context ... both control and state constraints are present) the problem can be artificially augmented by one extrasand y component, in such a way that a feasible choice can be made trivially. Monteiro and Adler =-=[23]-=- and Kojima, Mizuno and Yoshise [18] show how to construct feasible initial points that are on or near the central path by introducing artificial variables into convex quadratic programs and linear co... |

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Citation Context ...solution of the mixed nonlinear complementarity problem (NLCP) dL dz + d h d z T w + d g d z T y = 0; h(z) = 0; (4.5) g(z)s0; ys0; y T g(z) = 0: ("Regularity" is referred to as "strong =-=regularity" in [29]-=-.) A common variant of the sequential quadratic programming algorithm for (4.1) obtains a new iterate (z j+1 ; w j+1 ; y j+1 ) from the current iterate (z j ; w j ; y j ) by solving the quadratic prog... |

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Citation Context ...r various special classes of these problems. In the "unconstrained" case (that is, when g, g i and g f are absent), Newton-like methods and conjugate gradient methods for (1.1) are described=-= by Polak [27]-=-; for (1.2), Newton's method, and its efficient implementation, is discussed in Dunn and Bertsekas [8]. A variety of quasi-Newton approaches have also been applied to the unconstrained version of (1.1... |

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Citation Context ...fl (M k + `ffiM k )(Y k + `ffiY k )e \Gamma (y k + `ffiy k ) T ( k + `ffi k ) m e fl fl fl fl fl 2 \Gamma 1 4 (y k + `ffiy k ) T ( k + `ffi k ) 2 m 2 = 0 by using a safeguarded search in the interval =-=[0; 1]-=- with an algorithm that has local third-order convergence. Typically about three iterations are required. The cost of this step is much less than the cost of solving the system (2.12). ffl The choice ... |

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Citation Context ... /(y k ;sk ), where / is defined in (2.8). The following result can be used to demonstrate polynomial complexity of the basic potential reduction algorithm. It is proved by Kojima, Mizuno and Yoshise =-=[17] for the l-=-inear complementarity problem in standard form, but can be easily extended to the "mixed" linear complementarity problem (2.3)--(2.7). Theorem 2.2. Suppose that (i) The set C is nonempty (ii... |

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Citation Context ...gle change to the active set, which causes poor performance when there are many constraints. More recently, Newtonian scaling has been added to gradient projection algorithms (see Gafni and Bertsekas =-=[12]-=-, Dunn [7]) to improve their asymptotic rate of convergence, and the resulting methods have proven to be useful for the control-constrained version of (1.2), as we see in Section 6. Pantoja and Mayne ... |

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Citation Context ...teps were usually observed to produce the largest reductions in the duality gap. The heuristic above was found, after some experimentation, to be quite successful. As noted by Zhang, Tapia and Dennis =-=[35], the-=- choice ae k = 1=\Delta k (which takes effect during the last few iterations) ensures quadratic convergence of the duality gap to zero. The second algorithm we consider is of the "predictor-corre... |

37 |
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Citation Context ...es subprogram of the Office of Energy Research, U. S. Department of Energy, under Contract W-31-109-Eng-38. projection class are easily implementable (see, for example, Demyanov and Rubinov [3], Dunn =-=[5, 6]). In the -=-finite-dimensional problem, these methods have the advantage that the set of currently-active constraints can change extensively at each iteration, whereas "active set" methods only allow a ... |

37 |
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Citation Context ...ty gap to zero. The second algorithm we consider is of the "predictor-corrector" type. For linear programming problems, this algorithm is described by Mizuno, Todd and Ye [22] and Ye, Tapia =-=and Zhang [34]. The anal-=-ysis is extended to linear complementarity problems by Ji, Potra and Huang [15]. The algorithm makes use of the idea of an "ff-neighborhood" of the central path C, defined as follows: C(ff) ... |

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Citation Context ...y almost independent of N though, as we show in the next section, formal analyses suggest that it should be O(N 1=2 ). Interesting algorithms have recently been proposed by Rockafellar and co-workers =-=[30, 36]-=- for extended linear-quadratic programming, a class of problems that includes discrete-time linear-quadratic optimal control problems. They aim to find the saddle point of a Lagrangian which, for mult... |

32 |
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Citation Context ...r linear programming problems, this algorithm is described by Mizuno, Todd and Ye [22] and Ye, Tapia and Zhang [34]. The analysis is extended to linear complementarity problems by Ji, Potra and Huang =-=[15]. The algo-=-rithm makes use of the idea of an "ff-neighborhood" of the central path C, defined as follows: C(ff) = ( (y; ) 2 F+ fi fi fi fi fi fl fl fl fl fl MY e (y T =m) \Gamma e fl fl fl fl flsff ) ;... |

22 |
Simultaneous optimization and solution methods for batch reactor control profiles
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Citation Context ...ration. In the numerical results of Section 6, we restrict attention to simple discretizations of the continuous problems. Higher-order discretizations are possible: for example, Cuthrell and Biegler =-=[2]-=- use collocation at Gauss points to convert (1.1) to a nonlinear programming problem. Many issues arise in the discretization process, particularly when the solution of (1.1) contains singular arcs (a... |

18 |
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(Show Context)
Citation Context ...are absent), Newton-like methods and conjugate gradient methods for (1.1) are described by Polak [27]; for (1.2), Newton's method, and its efficient implementation, is discussed in Dunn and Bertsekas =-=[8]-=-. A variety of quasi-Newton approaches have also been applied to the unconstrained version of (1.1); see, for example, Edge and Powers [9] and Kelley and Sachs [16], and the references therein. In the... |

17 | Primal-dual projected gradient algorithms for extended linearquadratic programming
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(Show Context)
Citation Context ...y almost independent of N though, as we show in the next section, formal analyses suggest that it should be O(N 1=2 ). Interesting algorithms have recently been proposed by Rockafellar and co-workers =-=[30, 36]-=- for extended linear-quadratic programming, a class of problems that includes discrete-time linear-quadratic optimal control problems. They aim to find the saddle point of a Lagrangian which, for mult... |

16 |
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(Show Context)
Citation Context ..., is discussed in Dunn and Bertsekas [8]. A variety of quasi-Newton approaches have also been applied to the unconstrained version of (1.1); see, for example, Edge and Powers [9] and Kelley and Sachs =-=[16]-=-, and the references therein. In the control-constrained case (in which g f is absent, and the states x and x i do not appear in g and g i ), the problem is traditionally treated as a constrained opti... |

15 |
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Citation Context ...ffi zsg(z j ); ys0; y T (C j ffi z + g(z j )) = 0: When Q j is positive semidefinite, (4.6) has the form (2.1) and the coefficient matrix on the left hand side of (4.7) is positive semidefinite. Pang =-=[25]-=- suggests the following algorithm for solving (4.5) and hence (4.1): initially: choose (z 0 ; w 0 ; y 0 ), set j = 0 each iteration: find (ffi z ; w; y) such that kH j (ffi z ; w; y)ksj j fl fl fl fl ... |

9 |
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Citation Context ...of ae k and ` k that yield the most efficient practical algorithms lie outside the scope of this analysis. We use the following heuristics (which are similar to those utilized by Han, Pardalos and Ye =-=[13]): i-=-nitially: ae min / m 1:5 the k-th iteration: ae k / max (ae min ; 1=\Delta k ); calculate the step (ffiz k ; ffi k ; ffiw k ; ffiy k ); set �� ` k = maxf` j `s1; y k i + `ffiy k is0;sk i + `ffi k ... |

6 |
Sequential Quadratic Programming Algorithm for Discrete Optimal Control Problems with Control Inequality Constraints
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(Show Context)
Citation Context ..., Dunn [7]) to improve their asymptotic rate of convergence, and the resulting methods have proven to be useful for the control-constrained version of (1.2), as we see in Section 6. Pantoja and Mayne =-=[26]-=- have described a stagewise algorithm for the control-constrained case that, in a neighborhood of a solution of (1.2), produces iterates that are identical to sequential quadratic programming iterates... |

5 |
A Class of Structured Quasi-Newton Algorithms for Optimal Control Problems
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(Show Context)
Citation Context ...y constraints. Evtushenko [10, Chapter 6] describes a variety of augmented Lagrangian penalty function methods, in which (1.2) is reduced to an unconstrained problem. Di Pillo, Grippo and Lampariello =-=[4]-=- describe a structured quasi-Newton method for a particular augmented Lagrangian, and take advantage of the same feature which we exploit in this paper: bandedness of the coefficient matrix which is f... |

5 |
Partitioned dynamic programming for optimal control
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(Show Context)
Citation Context ...re identical to sequential quadratic programming iterates. Instead of making use of the inherent structure in (1.2) at the level of the linear algebra computations, as we do in this paper and also in =-=[32, 31]-=-, Pantoja and Mayne exploit the structure at a somewhat higher level. The most general cases of (1.1),(1.2), in which the functions g, g i , g f are nontrivial in both states and controls, are known t... |

4 |
Recent Advances in Gradient Algorithms for Optimal Control Problems
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Citation Context ...he most promising. In these algorithms, both states and controls are treated as unknowns and the state equation and "auxiliary" constraints as equality and inequality constraints, respective=-=ly. Miele [21]-=- deals mainly with the case in which the auxiliary constraintssg in (1.1) are equalities (rather than inequalities) and proposes algorithms of the reduced gradient type, with features added to ensure ... |

4 |
A new approach of differential dynamic programming for discrete time systems
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Citation Context ...o those reported by Zhu and Rockafellar [36], who performed computations with randomly generated examples on similar workstation equipment. Our approach bears some similarity to one described by Ohno =-=[24]-=-. Ohno treats the firstorder stationarity conditions for (1.2), and the complementarity conditions y T i g i (x i ; u i ) = 0 as a system of nonlinear equations, which are then solved by a differentia... |

3 |
A Method of Centers Based on Barrier Functions for Solving Optimal Control Problems with Continuum State and Control Constraints
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(Show Context)
Citation Context ...raintssg in (1.1) are equalities (rather than inequalities) and proposes algorithms of the reduced gradient type, with features added to ensure near-feasibility of all iterates. Polak, Yang and Mayne =-=[28]-=- describe a first-order algorithm which makes use of barrier functions for the inequality constraints. Evtushenko [10, Chapter 6] describes a variety of augmented Lagrangian penalty function methods, ... |

1 |
Function space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs
- Edge, Powers
- 1976
(Show Context)
Citation Context ... efficient implementation, is discussed in Dunn and Bertsekas [8]. A variety of quasi-Newton approaches have also been applied to the unconstrained version of (1.1); see, for example, Edge and Powers =-=[9]-=- and Kelley and Sachs [16], and the references therein. In the control-constrained case (in which g f is absent, and the states x and x i do not appear in g and g i ), the problem is traditionally tre... |

1 |
Efficient determination of optimal control profiles for differential algebraic systems
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(Show Context)
Citation Context ...:2)s1; i = 1; : : : ; 4: Versions of this problem have been discussed by a number of authors, including Jacobson and Mayne [14, page 85] (who exclude the terminal inequality constraints) and Longsdon =-=[19]-=-. The problem has a bang-bang solution with eight switching times. It is solved in [19] by using a discrete nonlinear programming formulation, and in [14] by using a second-order method Table 6.3 Resu... |