## Solution Of Optimal Control Problems By A Pointwise Projected Newton Method (1995)

Venue: | SIAM J. Control Optim |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Kelley95solutionof,

author = {C. T. Kelley and E. W. Sachs},

title = {Solution Of Optimal Control Problems By A Pointwise Projected Newton Method},

journal = {SIAM J. Control Optim},

year = {1995},

volume = {33},

pages = {1731--1757}

}

### OpenURL

### Abstract

. In the context of optimal control of ordinary differential equations, we prove local superlinear convergence and constraint identification results for an extension of the projected Newton method of Bertsekas. The estimates are also valid for discretized versions of the method-problem pair. Key words. projected Newton iteration, optimal control AMS(MOS) subject classifications. 47H17, 49K15, 49M15, 65J15, 65K10 1. Introduction. In many areas of optimal control the problems are formulated with simple constraints on the control. For this type of problems, the gradient projection type algorithms have proven to be quite successful, because they are able to take into account the structure of the underlying optimization problem. Another interesting feature of these methods is that they often can be formulated in infinite dimensional spaces which is important for the application to optimal control problems. In general, let H denote a Hilbert space and for some closed convex subset U 2 H c...

### Citations

81 | On the Goldstein-Levitin-Polyak gradient projection method
- Bertsekas
- 1976
(Show Context)
Citation Context ...t #CRG 920067. POINTWISE PROJECTED NEWTON METHOD 2 This method utilizes again the simple projection but has the drawback that it does not always produce a descent in the objective function. Bertsekas =-=[1]-=- and [2] introduced for the finite dimensional case with simple constraints such as upper and lower bounds on the variables a projected Newton method which alleviated this problem. For H = R n let u+ ... |

59 |
Convex programming in Hilbert space
- Goldstein
- 1964
(Show Context)
Citation Context ...method iterates are given by u+ = P(u c \Gamma ff c rOE(u c )) (1.2) where ff ? 0 is determined by a step-size rule. In Hilbert space, this algorithm has been formulated and investigated by Goldstein =-=[10]-=- and Levitin and Polyak [14]. The books [4] and [3] discussed the convergence properties of gradient projection methods. In [7] a thorough convergence analysis of the gradient projection method was pr... |

29 |
The Approximate Minimization of Functionals
- Daniel
- 1971
(Show Context)
Citation Context ...rOE(u c )) (1.2) where ff ? 0 is determined by a step-size rule. In Hilbert space, this algorithm has been formulated and investigated by Goldstein [10] and Levitin and Polyak [14]. The books [4] and =-=[3]-=- discussed the convergence properties of gradient projection methods. In [7] a thorough convergence analysis of the gradient projection method was presented which yields various convergence rates of t... |

16 |
Quasi-Newton methods and unconstrained optimal control problems
- Kelley, Sachs
- 1987
(Show Context)
Citation Context ...abulate, for different values of �� p the progress of the iteration for the example above. The two point boundary value problem (6.4) was solved with the trapezoid rule extrapolation approach used=-= in [11]-=- and the integration in (4.3) was done with the trapezoid rule. A uniform mesh of 1400 points was used. For each iterate k we tabulate the norm of the nonlinear residual ae = k(sx \Gamma f;sp + H x ; ... |

9 | Rates of convergence for conditional gradient algorithms near singular and nonsingular extremals - Dunn - 1979 |

7 |
Newton methods for optimization problems with simple constraints
- Projected
- 1982
(Show Context)
Citation Context ...20067. POINTWISE PROJECTED NEWTON METHOD 2 This method utilizes again the simple projection but has the drawback that it does not always produce a descent in the objective function. Bertsekas [1] and =-=[2]-=- introduced for the finite dimensional case with simple constraints such as upper and lower bounds on the variables a projected Newton method which alleviated this problem. For H = R n let u+ = P(u c ... |

4 |
On the gradient projection method for optimal control problems with nonnegative L inputs
- Tian, Dunn
- 1994
(Show Context)
Citation Context ...rt space like L 2 to a L 1 -type norm in X poses in the analysis of the convergence various difficulties. In another context this aspect has been the focus of other research activities, see e.g. [9], =-=[8]-=-. The estimate in Theorem 3.5 enables us to show a result on the identification of the set of active indices. It estimates the measure of the set on which the active set at the current iterate differs... |

3 |
Another Jacobi Sufficiency Criterion for Optimal Control with Smooth Constraints
- ORRELL, ZEIDAN
- 1988
(Show Context)
Citation Context ...) we have omitted the arguments for the derivatives of the functions. The regularity assumptions in Assumption 2.4 are related to second order sufficiency conditions in optimal control. In the papers =-=[16]-=- and [15] second order sufficiency conditions of the following type are used. A strengthened Legendre-Clebsch condition is posed with the existence of a solution to a Riccati equation, both appropriat... |

2 | Convergence of algorithms for perturbed optimization problems - Sachs - 1990 |

1 |
Rubinov, Approximate Methods in Optimization Theory
- Demyanov, M
- 1970
(Show Context)
Citation Context ...ma ff c rOE(u c )) (1.2) where ff ? 0 is determined by a step-size rule. In Hilbert space, this algorithm has been formulated and investigated by Goldstein [10] and Levitin and Polyak [14]. The books =-=[4]-=- and [3] discussed the convergence properties of gradient projection methods. In [7] a thorough convergence analysis of the gradient projection method was presented which yields various convergence ra... |

1 |
method and the Goldstein step-length rule for constrained minimization problems
- Newton's
- 1980
(Show Context)
Citation Context ...m of the type (1.1) one would have to solve Minimize (rOE(u c ); u \Gamma u c ) + 1 2 (u \Gamma u c ; r 2 OE(u c )(u \Gamma u c )) subject to u 2 U: (1.3) This algorithm has been analyzed in [14] and =-=[6]-=-. The disadvantage of the method (1.3) is that at each step a quadratic problem with constraints needs to be solved. The simplicity of the constraints cannot be used in a direct way through the projec... |

1 |
and asymptotic convergence rate estimates for a class of projected gradient processes
- Global
- 1981
(Show Context)
Citation Context ...space, this algorithm has been formulated and investigated by Goldstein [10] and Levitin and Polyak [14]. The books [4] and [3] discussed the convergence properties of gradient projection methods. In =-=[7]-=- a thorough convergence analysis of the gradient projection method was presented which yields various convergence rates of the algorithm under various assumptions. Since the gradient projection method... |

1 |
of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions
- Variants
(Show Context)
Citation Context ...Hilbert space like L 2 to a L 1 -type norm in X poses in the analysis of the convergence various difficulties. In another context this aspect has been the focus of other research activities, see e.g. =-=[9]-=-, [8]. The estimate in Theorem 3.5 enables us to show a result on the identification of the set of active indices. It estimates the measure of the set on which the active set at the current iterate di... |

1 |
independence of the gradient projection method for optimal control problems
- Mesh
- 1992
(Show Context)
Citation Context ...imal control problems are problems formulated in function space, an analysis of the projected Newton method in this framework would give some insight for the case of fine discretizations. As shown in =-=[12]-=- this issue is important because the identification of finite indices is only mesh independent if proper measures are taken. The goal of this paper is to extend the algorithm to the infinite dimension... |

1 |
Constrained optimization methods
- Levitin, Polyak
- 1966
(Show Context)
Citation Context ... u+ = P(u c \Gamma ff c rOE(u c )) (1.2) where ff ? 0 is determined by a step-size rule. In Hilbert space, this algorithm has been formulated and investigated by Goldstein [10] and Levitin and Polyak =-=[14]-=-. The books [4] and [3] discussed the convergence properties of gradient projection methods. In [7] a thorough convergence analysis of the gradient projection method was presented which yields various... |

1 |
The two-norm approach for second order sufficiency conditions in mathematical programming and optimal control
- Maurer
- 1992
(Show Context)
Citation Context ... omitted the arguments for the derivatives of the functions. The regularity assumptions in Assumption 2.4 are related to second order sufficiency conditions in optimal control. In the papers [16] and =-=[15]-=- second order sufficiency conditions of the following type are used. A strengthened Legendre-Clebsch condition is posed with the existence of a solution to a Riccati equation, both appropriately alter... |