## On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming (2006)

Venue: | Mathematical Programming |

Citations: | 112 - 5 self |

### BibTeX

@ARTICLE{Wächter06onthe,

author = {Andreas Wächter and Lorenz T. Biegler},

title = {On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming},

journal = {Mathematical Programming},

year = {2006},

volume = {106},

pages = {25--57}

}

### OpenURL

### Abstract

We present a primal-dual interior-point algorithm with a filter line-search method for nonlinear programming. Local and global convergence properties of this method were analyzed in previous work. Here we provide a comprehensive description of the algorithm, including the feasibility restoration phase for the filter method, second-order corrections, and inertia correction of the KKT matrix. Heuristics are also considered that allow faster performance. This method has been implemented in the IPOPT code, which we demonstrate in a detailed numerical study based on 954 problems from the CUTEr test set. An evaluation is made of several line-search options, and a comparison is provided with two state-of-the-art interior-point codes for nonlinear programming.

### Citations

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Citation Context ...int Jacobian A T k , is positive definite 1 . These conditions are satisfied if the iteration matrix has the inertia (n, m, 0), i.e., if it has exactly n positive, m negative, and no zero eigenvalues =-=[20]-=-. Therefore, if the inertia of this matrix is not (n, m, 0), the linear system (13) is re-solved in our implementation with a modified iteration matrix for different trial values for the scalars δw, δ... |

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Citation Context ...tion min x∈sn f(x) (2a) s.t. c(x) = 0 (2b) x ≥ 0. (2c) The changes necessary to handle the general case (1) will be briefly outlined in Section 3.4. As a barrier method, like the methods discussed in =-=[2, 8, 11, 29]-=-, the proposed algorithm computes (approximate) solutions for a sequence of barrier problems min x∈sn ϕµ(x) := f(x) − µ n� ln(x (i) ) (3a) i=1 s.t. c(x) = 0 (3b) for a decreasing sequence of barrier p... |

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Citation Context ...case the evaluation of the constraints was repeatedly unsuccessful (producing the IEEE numbers Inf or NaN). For the comparisons in the next sections we make use of the Dolan-Moré performance profiles =-=[9]-=-. Given a test set P containing np problems, and ns runs (e.g., obtained with different solver options) for each problem, these profiles provide a way to graphically present the comparison of quantiti... |

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Citation Context ...n both trust region and line-search frameworks, have been developed that use exact penalty merit functions to enforce progress toward the solution [2, 21, 29]. On the other hand, Fletcher and Leyffer =-=[14]-=- recently proposed filter methods, offering an alternative to merit functions, as a tool to guarantee global convergence in algorithms for nonlinear programming. The underlying concept is that trial p... |

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Citation Context ... made for previously proposed line-search interior-point methods for nonlinear programming (e.g., [10, 29]). A number of interior-point methods have been implemented in robust software codes (such as =-=[3, 23]-=-), and numerical tests have shown them to be efficient and robust in practice. In this paper we describe the detailed development of a primal-dual interior-point algorithm with a filter linesearch, ba... |

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Citation Context ... from poor starting points, interior-point methods, in both trust region and line-search frameworks, have been developed that use exact penalty merit functions to enforce progress toward the solution =-=[2, 21, 29]-=-. On the other hand, Fletcher and Leyffer [14] recently proposed filter methods, offering an alternative to merit functions, as a tool to guarantee global convergence in algorithms for nonlinear progr... |

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Citation Context ...nd MC29) to the first derivative matrix � ∇xc(x0) J0 = T � ∇xf(x0) T to obtain scaling matrices Dx and Dcf = diag(Dc, df ) so that the nonzero elements in DcfJ0D −1 x are of order one (as proposed in =-=[6]-=-). Similarly, we computed scaling factors so that the matrix � �� D−1 x 0 ∇2 xxf(x0) ∇xc(x0) 0 Dc ∇xc(x0) T �� � D−1 x 0 0 0 Dc has non-zero entries close to one. While these strategies seem to work w... |

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Citation Context .... M. Ulbrich, S. Ulbrich, and Vicente [22] consider a trust region filter method that bases the acceptance of trial steps on the norm of the optimality conditions. Also, Benson, Shanno, and Vanderbei =-=[1]-=- proposed several heuristics based on the idea of filter methods, for which improved efficiency is reported compared to their previous merit function approach, although no convergence analysis ∗ IBM T... |

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Citation Context ...trategies in handling problems with large numbers of inequality constraints. Over the past 15 years, there has also been a better understanding of the convergence properties of interior-point methods =-=[16]-=- and efficient algorithms have been developed with desirable global and local convergence properties. To allow convergence from poor starting points, interior-point methods, in both trust region and l... |

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Citation Context ...0 together with “x, z ≥ 0” are the Karush-Kuhn-Tucker (KKT) conditions for the original problem (2). Those are the first order optimality conditions for (2) if constraint qualifications are satisfied =-=[7]-=-. The presented method computes an approximate solution to the barrier problem (3) for a fixed value of µ, then decreases the barrier parameter, and continues the solution of the next barrier problem ... |

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An interior point algorithm for large-scale nonlinear programming
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Citation Context ... made for previously proposed line-search interior-point methods for nonlinear programming (e.g., [10, 29]). A number of interior-point methods have been implemented in robust software codes (such as =-=[3, 23]-=-), and numerical tests have shown them to be efficient and robust in practice. In this paper we describe the detailed development of a primal-dual interior-point algorithm with a filter linesearch, ba... |

44 |
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Citation Context ...ility becomes arbitrarily small. • Case II: Otherwise. Here, we hope to overcome possible inefficiencies by tentatively ignoring the filter criteria for one iteration, similar to a watchdog procedure =-=[5]-=- (with one relaxed step). In the next iteration, k + 1, we choose αk+1 = αmax k+1 without any backtracking line search. The filter is not augmented in Step A-7 for iteration k + 1, and the search dire... |

41 |
An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications
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Citation Context ...similar to the method proposed here, and since its practical behavior seems promising (in particular, it performs considerable better than the penalty function approach used in our earlier comparison =-=[24]-=-). In the following we only briefly state the algorithm; its motivation can be found in [27]. We should point out, however, that the algorithm proposed in [27] is more complex and, in particular, reve... |

39 |
CUTEr (and SifDec), a Constrained and Unconstrained Testing Environment, revisited
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- 2003
(Show Context)
Citation Context ... the filter procedure, as well as several heuristics to improve the performance of the overall method. Section 4 presents numerical results of our implementation, called IPOPT, for the CUTEr test set =-=[18]-=-, including a comparison of the filter method with a penalty function approach, and a comparison with two state-of-the-art nonlinear optimization codes, KNITRO [3, 28] and LOQO [23]. 1.1 Notation The ... |

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Citation Context ...ticular, reverts, under certain circumstances, to a trust region approach, ensuring global convergence (in contrast to the penalty function option using only a backtracking line-search procedure, see =-=[25]-=-). For the penalty function based option, the search direction is computed from (13) in an iteration k, and the maximum step sizes are obtained from (15). After this, the penalty parameter is updated ... |

33 | A globally convergent primal-dual interior point filter method for nonconvex nonlinear programming
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(Show Context)
Citation Context ...ad of a combination of those two measures defined by a merit function. More recently, this filter approach has been adapted to barrier methods in a number of ways. M. Ulbrich, S. Ulbrich, and Vicente =-=[22]-=- consider a trust region filter method that bases the acceptance of trial steps on the norm of the optimality conditions. Also, Benson, Shanno, and Vanderbei [1] proposed several heuristics based on t... |

30 | A primal-dual interior-point method for nonlinear programming with strong global and local convergence properties
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Citation Context ... from poor starting points, interior-point methods, in both trust region and line-search frameworks, have been developed that use exact penalty merit functions to enforce progress toward the solution =-=[2, 21, 29]-=-. On the other hand, Fletcher and Leyffer [14] recently proposed filter methods, offering an alternative to merit functions, as a tool to guarantee global convergence in algorithms for nonlinear progr... |

28 |
Line search filter methods for nonlinear programming: motivation and global convergence
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Citation Context ...atson.ibm.com † Carnegie Mellon University, Pittsburgh, PA; E-mail: lb01@andrew.cmu.edu 1sis given. Finally, global convergence of an interior-point algorithm with a filter line search is analyzed in =-=[26]-=-. The assumptions made for the analysis of the interior-point method in [26] are less restrictive than those made for previously proposed line-search interior-point methods for nonlinear programming (... |

26 |
A globally convergent primaldual interior point method for constrained optimization
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Citation Context ... from poor starting points, interior-point methods, in both trust region and line-search frameworks, have been developed that use exact penalty merit functions to enforce progress toward the solution =-=[2, 21, 29]-=-. On the other hand, Fletcher and Leyffer [14] recently proposed filter methods, offering an alternative to merit functions, as a tool to guarantee global convergence in algorithms for nonlinear progr... |

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Citation Context ... interpreted as applying a homotopy method to the primal-dual equations, ∇f(x) + ∇c(x)λ − z = 0 (4a) c(x) = 0 (4b) XZe − µe = 0, (4c) with the homotopy parameter µ, which is driven to zero (see e.g., =-=[4, 17]-=-). Here, λ ∈ Rm and z ∈ Rn correspond to the Lagrangian multipliers for the equality constraints (2b) and the bound constraints (2c), respectively. Note, that the equations (4) for µ = 0 together with... |

20 |
A primal-dual trust-region algorithm for nonconvex nonlinear programming
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(Show Context)
Citation Context ...tion min x∈sn f(x) (2a) s.t. c(x) = 0 (2b) x ≥ 0. (2c) The changes necessary to handle the general case (1) will be briefly outlined in Section 3.4. As a barrier method, like the methods discussed in =-=[2, 8, 11, 29]-=-, the proposed algorithm computes (approximate) solutions for a sequence of barrier problems min x∈sn ϕµ(x) := f(x) − µ n� ln(x (i) ) (3a) i=1 s.t. c(x) = 0 (3b) for a decreasing sequence of barrier p... |

20 |
Global convergence of a trust-region SQP-filter algorithm for general nonlinear programming
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(Show Context)
Citation Context ...al point is approached. Enforcing decrease in the objective function by (20) then prevents the method from converging to such a point. In accordance with previous publications on filter methods (e.g. =-=[13, 15]-=-) we may call a trial step size αk,l for which (19) holds, a “ϕ-step size.” The algorithm also maintains a “filter”, a set Fk ⊆ {(θ, ϕ) ∈ R2 : θ ≥ 0} for each iteration k. The filter Fk contains those... |

20 |
A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization,MathematicalProgramming,SeriesA,102(2005),pp.111–151
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Citation Context ...σ (i) k+1 of Σk+1 is in the interval σ (i) 1 k+1 ∈ [ µj/(x κΣ (i) k )2 , κΣµj/(x (i) k )2 ]. (17) Such safeguards are common for the global convergence proof of primal-dual methods for NLP (see e.g., =-=[8, 30]-=-), and do not interfere with the primal-dual spirit of the method in terms of local convergence, when the parameter κΣ is chosen sufficiently large. In our implementation, κΣ = 10 10 . 2.3 A Line-Sear... |

16 |
Superlinear Convergence of Primal-Dual Interior Point Algorithms for Nonlinear Programming
- Toint
(Show Context)
Citation Context ... interpreted as applying a homotopy method to the primal-dual equations, ∇f(x) + ∇c(x)λ − z = 0 (4a) c(x) = 0 (4b) XZe − µe = 0, (4c) with the homotopy parameter µ, which is driven to zero (see e.g., =-=[4, 17]-=-). Here, λ ∈ Rm and z ∈ Rn correspond to the Lagrangian multipliers for the equality constraints (2b) and the bound constraints (2c), respectively. Note, that the equations (4) for µ = 0 together with... |

13 |
On the global convergence of a filter-SQP algorithm
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(Show Context)
Citation Context ...al point is approached. Enforcing decrease in the objective function by (20) then prevents the method from converging to such a point. In accordance with previous publications on filter methods (e.g. =-=[13, 15]-=-) we may call a trial step size αk,l for which (19) holds, a “ϕ-step size.” The algorithm also maintains a “filter”, a set Fk ⊆ {(θ, ϕ) ∈ R2 : θ ≥ 0} for each iteration k. The filter Fk contains those... |

13 |
and J.Nocedal, “Knitro user’s manual
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(Show Context)
Citation Context ...alled IPOPT, for the CUTEr test set [18], including a comparison of the filter method with a penalty function approach, and a comparison with two state-of-the-art nonlinear optimization codes, KNITRO =-=[3, 28]-=- and LOQO [23]. 1.1 Notation The i-th component of a vector v ∈ R n is written as v (i) . Norms �·� denote a fixed vector norm and its compatible matrix norm unless explicitly noted. We further introd... |

8 |
A catalogue of subroutines (HSL
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(Show Context)
Citation Context ...tia is as desired. The inertia of the iteration matrix is readily available from several symmetric indefinite linear solvers such as the factorization routine MA27 from the Harwell subroutine library =-=[19]-=- used in our implementation. Note that the desired inertia is obtained if δw is sufficiently large and the constraint Jacobian ∇c(xk) T has full rank. If ∇c(xk) T is rank-deficient, the matrix is sing... |

4 |
KNITRO-Direct: A hybrid interior algorithm for nonlinear optimization
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(Show Context)
Citation Context ...based on the exact penalty function φν(x) = ϕµj (x) + ν�c(x)�. (39) The update rule and step acceptance criteria chosen for the comparison in this paper has been proposed recently by Waltz et. al. in =-=[27]-=- as part of a hybrid trust region and line-search interiorpoint method. We chose this option since the algorithm in [27] is in many aspects similar to the method proposed here, and since its practical... |