## On Interior-Point Newton Algorithms For Discretized Optimal Control Problems With State Constraints (1998)

Venue: | OPTIM. METHODS SOFTW |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Vicente98oninterior-point,

author = {Luís N. Vicente},

title = {On Interior-Point Newton Algorithms For Discretized Optimal Control Problems With State Constraints},

journal = {OPTIM. METHODS SOFTW},

year = {1998},

volume = {8},

pages = {249--275}

}

### OpenURL

### Abstract

In this paper we consider a class of nonlinear programming problems that arise from the discretization of optimal control problems with bounds on both the state and the control variables. For this class of problems, we analyze constraint qualifications and optimality conditions in detail. We derive an affine-scaling and two primal-dual interior-point Newton algorithms by applying, in an interior-point way, Newton's method to equivalent forms of the first-order optimality conditions. Under appropriate assumptions, the interior-point Newton algorithms are shown to be locally well-defined with a q-quadratic rate of local convergence. By using the structure of the problem, the linear algebra of these algorithms can be reduced to the null space of the Jacobian of the equality constraints. The similarities between the three algorithms are pointed out, and their corresponding versions for the general nonlinear programming problem are discussed.

### Citations

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Citation Context ...nt methods for minimization problems with simple bounds (see [3], [6], [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in [12], [30], [31] (primal dual), =-=[4]-=-, [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], [19], [20] (affine scaling). For discretized optimal control problems see also [1... |

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Citation Context ...ed affine--scaling interior--point methods for minimization problems with simple bounds (see [3], [6], [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in =-=[12]-=-, [30], [31] (primal dual), [4], [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], [19], [20] (affine scaling). For discretized optim... |

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Citation Context ...is to present an affine--scaling and two primal--dual interior--point Newton algorithms to solve problems of the form (1.1). The affine--scaling framework extends the approach followed by [10], [17], =-=[28]-=- for problems of the type (1.1) but with no bounds on the state variables y. First, it is shown that the Karush--Kuhn--Tucker constraint qualifications hold for most feasible points of (1.1). We also ... |

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Citation Context ...of x and z respectively, and e is a vector of ones with n components. The equation XZe = e is a relaxation of the complementarity condition x ? z = 0 that includes a perturbation term e (see [12] and =-=[32]-=- for more details). This algorithm is also of interior--point type, meaning that x and z are required always to be strictly feasible with respect to the bound constraints, i.e., x and z have to satisf... |

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Citation Context ...m Primal x DualsDual z Affine Scaling p p Primal Dual p p p Reduced Primal Dual p p Coleman and Li introduced affine--scaling interior--point methods for minimization problems with simple bounds (see =-=[3]-=-, [6], [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in [12], [30], [31] (primal dual), [4], [24] (primal using trust regions and approximation to the m... |

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Citation Context ...are easily extended for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], [5], =-=[10]-=-, [15], [16], [17], [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears frequent... |

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Citation Context ...4] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], [19], [20] (affine scaling). For discretized optimal control problems see also [18], =-=[29]-=-. Some of the ideas presented in this paper were discovered independently by Das [9] for the general nonlinear programming problem, namely the formulation of the optimality conditions and the applicat... |

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Citation Context ...caling interior--point methods for minimization problems with simple bounds (see [3], [6], [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in [12], [30], =-=[31]-=- (primal dual), [4], [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], [19], [20] (affine scaling). For discretized optimal control p... |

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Citation Context ... algorithms in the absence of regularity or strict complementarity for particular classes of problems like linear programming, monotone variational inequalities, and linear complementarity (see [13], =-=[25]-=-, [27] and the references therein). In problems of the ON INTERIOR--POINT NEWTON ALGORITHMS 19 form (1.1), conditions like regularity and strict complementarity (or nondegeneracy) are likely to be abs... |

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Citation Context ...mal x DualsDual z Affine Scaling p p Primal Dual p p p Reduced Primal Dual p p Coleman and Li introduced affine--scaling interior--point methods for minimization problems with simple bounds (see [3], =-=[6]-=-, [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in [12], [30], [31] (primal dual), [4], [24] (primal using trust regions and approximation to the multip... |

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Citation Context ...ine--scaling interior--point methods for minimization problems with simple bounds (see [3], [6], [7], [11]). Interior--point methods for nonlinear programming have been proposed and analyzed in [12], =-=[30]-=-, [31] (primal dual), [4], [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], [19], [20] (affine scaling). For discretized optimal con... |

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Citation Context ... affine--scaling interior--point Newton algorithms have already been implemented and tested with problems from [17]. The preliminary numerical results are encouraging. The preconditioners proposed in =-=[1]-=- will certainly play an important role in the implementation of efficient and robust algorithms for this class of problems. Table 1.1 Interior--point Newton algorithms corresponding to the application... |

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Citation Context ...aper are easily extended for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], =-=[5]-=-, [10], [15], [16], [17], [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears fr... |

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Citation Context ...ear programming have been proposed and analyzed in [12], [30], [31] (primal dual), [4], [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), =-=[8]-=-, [19], [20] (affine scaling). For discretized optimal control problems see also [18], [29]. Some of the ideas presented in this paper were discovered independently by Das [9] for the general nonlinea... |

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Citation Context ...d for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], [5], [10], [15], [16], =-=[17]-=-, [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears frequently in engineering ... |

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Citation Context ...in the place of the adjoint multiplier update (5.10), we use the least squares multiplier update (x; z) = argmin fkr x `(x; ) \Gamma zkg. Then, the analysis follows by using the result established in =-=[14]-=-. Now let us consider the reduced--space versions of Algorithms 4.1, 5.1, and 5.2, where the step s is computed along the null space of the Jacobian of the equality constraints. The extension to probl... |

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Citation Context ...bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], [5], [10], [15], [16], [17], =-=[18]-=-). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears frequently in engineering design... |

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Citation Context ...his paper are easily extended for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references =-=[2]-=-, [5], [10], [15], [16], [17], [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appea... |

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Citation Context ...inequality constraints), [8], [19], [20] (affine scaling). For discretized optimal control problems see also [18], [29]. Some of the ideas presented in this paper were discovered independently by Das =-=[9]-=- for the general nonlinear programming problem, namely the formulation of the optimality conditions and the application of Newton's method using affine--scaling matrices. The full--space version of th... |

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Citation Context ...xtended for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], [5], [10], [15], =-=[16]-=-, [17], [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears frequently in engine... |

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Citation Context ...rogramming have been proposed and analyzed in [12], [30], [31] (primal dual), [4], [24] (primal using trust regions and approximation to the multipliers corresponding to inequality constraints), [8], =-=[19]-=-, [20] (affine scaling). For discretized optimal control problems see also [18], [29]. Some of the ideas presented in this paper were discovered independently by Das [9] for the general nonlinear prog... |

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Citation Context ...;s)D(xs;s) + E(xs;s) is positive definite on the null--space of the matrix J(xs)D(xs;s) and its transpose, D(xs;s)J(xs) ? , has full column rank, we conclude that the matrix (3.2) is nonsingular (see =-=[26]-=-). If xsis a regular point then the optimality conditions can be stated by using a null--space basis representation of J(xs)D(xs;s). Proposition 3.4. Let xsbe a regular point and W (xs;s) a matrix who... |

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Citation Context ...ithms in the absence of regularity or strict complementarity for particular classes of problems like linear programming, monotone variational inequalities, and linear complementarity (see [13], [25], =-=[27]-=- and the references therein). In problems of the ON INTERIOR--POINT NEWTON ALGORITHMS 19 form (1.1), conditions like regularity and strict complementarity (or nondegeneracy) are likely to be absent an... |

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Citation Context ...sily extended for bounds of the form a ysysb y and a ususb u . The nonlinear programming problem (1.1) often arises from the discretization of optimal control problems (see references [2], [5], [10], =-=[15]-=-, [16], [17], [18]). In this case y is the vector of state variables, u is the vector of control variables, and C(y; u) = 0 is the discretized state equation. Problem (1.1) also appears frequently in ... |