## Adaptive Barrier Update Strategies for Nonlinear Interior Methods (2005)

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Citations: | 6 - 0 self |

### BibTeX

@TECHREPORT{Nocedal05adaptivebarrier,

author = {Jorge Nocedal and Richard A. Waltz},

title = {Adaptive Barrier Update Strategies for Nonlinear Interior Methods},

institution = {},

year = {2005}

}

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### Abstract

Abstract This paper considers strategies for selecting the barrier parameter at every iterationof an interior-point method for nonlinear programming. Numerical experiments suggest that adaptive choices, such as Mehrotra's probing procedure, outperform static strate-gies that hold the barrier parameter fixed until a barrier optimality test is satisfied. A new adaptive strategy is proposed based on the minimization of a quality function. Thepaper also proposes a globalization framework that ensures the convergence of adaptive interior methods. The barrier update strategies proposed in this paper are applica-ble to a wide class of interior methods and are tested in the two distinct algorithmic frameworks provided by the ipopt and knitro software packages.

### Citations

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Citation Context ...hat, in practice, this function is usually unimodal. Therefore, we implement a one-dimensional search scheme to compute an approximate minimizer of qL. It uses a golden bisection procedure (see, e.g. =-=[17]-=-), and ignores the fact that qL may not necessarily be unimodal. We first choose oemin and oemax, which define a minimum and maximum limit on the oe value, and define the two intervals [oemin, 1] and ... |

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Citation Context ...begin by considering the linear programming case. There are several ways of viewing the MPC method in this context. One is to consider the step computation as taking place in three stages (see, e.g., =-=[26]-=-). First, the algorithm computes the affine scaling step (3.2) and uses it to determine the target value of the barrier parameter u = oe x T z n ,where oe is given by (3.5). Next, the algorithm comput... |

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Citation Context ...on is eventually satisfied to some tolerance ffl. Second, by repeating this minimization process for decreasing values of u and ffl that converge to zero, one can establish global convergence results =-=[3, 9]-=- to stationary points of the nonlinear programming problem (2.1). We now propose two globalization schemes that monitor the performance of the itera8stions in reference to a mechanism that enforces gl... |

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Citation Context ...daptive barrier update strategies are well established in interior methods for linear and convex quadratic programming. The most popular approach of this type is Mehrotra's predictor-corrector method =-=[18]-=-. It computes, at every iteration, a probing (affine scaling) step that determines a target value of the barrier parameter, and then takes a primal-dual step using this target value. A corrector step ... |

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Citation Context ...at is effective in practice and is supported by a global convergence analysis. To show the generality of our technique, we implement it in the two different algorithmic contexts provided by the ipopt =-=[23]-=- and knitro [24] software packages. The global convergence properties of interior methods for nonlinear programming have recently received much attention [3, 8, 13, 14, 16, 19, 20, 22, 28]. Some of th... |

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Citation Context ... result in a robust method. As we discuss below, the main source of instability is the corrector step. Other adaptive barrier update strategies designed specifically for nonlinear programming include =-=[1, 8, 11, 19, 20, 21]-=-. In this paper we propose a new strategy for updating the barrier parameter that is effective in practice and is supported by a global convergence analysis. To show the generality of our technique, w... |

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Citation Context ...he so-called Fiacco-McCormick approach that fixes the barrier parameter until an approximate solution of the barrier problem is computed. It has been employed in various nonlinear interior algorithms =-=[2, 4, 10, 12, 23, 25, 27]-=- and has been implemented, for example, in the ipopt and knitro software packages. The FiaccoMcCormick strategy provides a framework for establishing global convergence [3, 22], but suffers from impor... |

65 |
A primal-dual interior method for nonconvex nonlinear programming
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Citation Context ... result in a robust method. As we discuss below, the main source of instability is the corrector step. Other adaptive barrier update strategies designed specifically for nonlinear programming include =-=[1, 8, 11, 19, 20, 21]-=-. In this paper we propose a new strategy for updating the barrier parameter that is effective in practice and is supported by a global convergence analysis. To show the generality of our technique, w... |

63 | Primal-dual interior methods for nonconvex nonlinear programming
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Citation Context ...he so-called Fiacco-McCormick approach that fixes the barrier parameter until an approximate solution of the barrier problem is computed. It has been employed in various nonlinear interior algorithms =-=[2, 4, 10, 12, 23, 25, 27]-=- and has been implemented, for example, in the ipopt and knitro software packages. The FiaccoMcCormick strategy provides a framework for establishing global convergence [3, 22], but suffers from impor... |

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Citation Context ...linear case) and propose a remedy (Section 7). To show the generality of our quality function approach, we implement it in the two different algorithmic contexts provided by the ipopt [27] and knitro =-=[6, 28]-=- software packages. Notation. For any vector z, we denote by Z the diagonal matrix whose diagonal entries are given by z. We let e denote the vector of ones, of appropriate dimension, that is, e = (1,... |

45 |
CUTEr (and SifDec), a Constrained and Unconstrained Testing Environment, revisited
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Citation Context ...un using the latest development versions of ipopt and knitro as of October 2005. The first results are for the linear programming problems in the NETLIB collection, as specified in the CUTEr test set =-=[15]-=-. No preprocessing was performed, and no initial point strategy was employed (i.e., the default starting point x0 = (0, . . . , 0) was used). Figure 2 compares the performance of the quality function ... |

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Line search filter methods for nonlinear programming: motivation and global convergence
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Citation Context ...lgorithmic contexts provided by the ipopt [23] and knitro [24] software packages. The global convergence properties of interior methods for nonlinear programming have recently received much attention =-=[3, 8, 13, 14, 16, 19, 20, 22, 28]-=-. Some of these studies focus on the effects of merit functions or filters, and on regularization techniques. With the exception of [8, 19, 20], however, these papers do not consider the numerical or ... |

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Citation Context ...he so-called Fiacco-McCormick approach that fixes the barrier parameter until an approximate solution of the barrier problem is computed. It has been employed in various nonlinear interior algorithms =-=[2, 4, 10, 12, 23, 25, 27]-=- and has been implemented, for example, in the ipopt and knitro software packages. The FiaccoMcCormick strategy provides a framework for establishing global convergence [3, 22], but suffers from impor... |

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Citation Context ...ificantly increase the complementarity value xT z. This behaviour can be sustained and lead to very slow convergence or failure, as shown in Table 1. The results in this table were obtained using PCx =-=[6]-=-, an interior-point code for linear programming that implements the MPC method, applied to problem Forplan from the NETLIB collection. Practical implementations of the MPC use a procedure for choosing... |

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Citation Context ...lgorithmic contexts provided by the ipopt [23] and knitro [24] software packages. The global convergence properties of interior methods for nonlinear programming have recently received much attention =-=[3, 8, 13, 14, 16, 19, 20, 22, 28]-=-. Some of these studies focus on the effects of merit functions or filters, and on regularization techniques. With the exception of [8, 19, 20], however, these papers do not consider the numerical or ... |

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Citation Context ... designed specifically for nonlinear programming include [2, 11, 15, 23, 24, 25]. The global convergence properties of interior methods for nonlinear programming have recently received much attention =-=[4, 8, 11, 17, 20, 23, 24, 26, 32]-=-. Some of these studies focus on the effects of merit functions or filters, and on regularization techniques. With the exception of [11, 23, 24], however, these papers do not consider the numerical or... |

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Knitro 4.0 User's Manual
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Citation Context ...linear case) and propose a remedy (Section 7). To show the generality of our quality function approach, we implement it in the two different algorithmic contexts provided by the ipopt [27] and knitro =-=[6, 28]-=- software packages. Notation. For any vector z, we denote by Z the diagonal matrix whose diagonal entries are given by z. We let e denote the vector of ones, of appropriate dimension, that is, e = (1,... |

5 |
Numerical methods for large-scale nonlinear optimization
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- 2005
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Citation Context ...d features that guarantee superlinear convergence. This could be done by using various techniques proposed in the literature for controlling the asymptotic behavior of the barrier parameter; see e.g. =-=[19]-=- and the references therein. In particular, we could implement the strategies recently proposed by Armand et al. [1, 2] in conjunction with the quality function approach. We have not done so because t... |

4 |
Some disadvantages of a Mehrotra-type primal-dual corrector interior-point algorithm for linear programming
- Cartis
- 2004
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Citation Context ...observation that harmful effects of the corrector steps manifest themselves in a significant increase in complementarity. The failure of convergence of the MPC method has also been analyzed by Cartis =-=[5]-=-. Finally we compare the monotone and quality function approaches described in Section 3 with the conditional MPC approach on the nonlinear programming problems used in that section. The conditional M... |

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Citation Context ...erature for controlling the asymptotic behavior of the barrier parameter; see e.g. [19] and the references therein. In particular, we could implement the strategies recently proposed by Armand et al. =-=[1, 2]-=- in conjunction with the quality function approach. We have not done so because the asymptotic behavior of the method proposed in this paper has proved to be acceptable in practice. In fact, the quali... |

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Citation Context ...lgorithmic contexts provided by the ipopt [23] and knitro [24] software packages. The global convergence properties of interior methods for nonlinear programming have recently received much attention =-=[3, 8, 13, 14, 16, 19, 20, 22, 28]-=-. Some of these studies focus on the eects of merit functions orslters, and on regularization techniques. With the exception of [8, 19, 20], however, these papers do not consider the numerical or the... |

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1 |
Interpretation of nonlinear interior methods as damped Newton methods, methods for nonlinear programming
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Citation Context ...esult in a robust method. As we discuss below, the main source of instability is the corrector step. Further adaptive barrier update strategies designed specifically for nonlinear programming include =-=[2, 11, 15, 23, 24, 25]-=-. The global convergence properties of interior methods for nonlinear programming have recently received much attention [4, 8, 11, 17, 20, 23, 24, 26, 32]. Some of these studies focus on the effects o... |