## A Convergent Infeasible Interior-Point Trust-Region Method For Constrained Minimization (1999)

Venue: | SIAM Journal on Optimization |

Citations: | 9 - 0 self |

### BibTeX

@ARTICLE{Tseng99aconvergent,

author = {Paul Tseng},

title = {A Convergent Infeasible Interior-Point Trust-Region Method For Constrained Minimization},

journal = {SIAM Journal on Optimization},

year = {1999},

volume = {13},

pages = {432--469}

}

### OpenURL

### Abstract

We study an infeasible interior-point trust-region method for constrained minimization. This method uses a logarithmic-barrier function for the slack variables and updates the slack variables using second-order correction. We show that if a certain set containing the iterates is bounded and the origin is not in the convex hull of the nearly active constraint gradients everywhere on this set, then any cluster point of the iterates is a 1st-order stationary point. If the cluster point satisfies an additional assumption (which holds when the constraints are linear or when the cluster point satisfies strict complementarity and a local error bound holds), then it is a 2nd-order stationary point. Key words. Nonlinear program, logarithmic-barrier function, interior-point method, trustregion strategy, 1st- and 2nd-order stationary points, semidefinite programming. 1 Introduction We consider the nonlinear program with inequality constraints: minimize f(x) subject to g(x) = [g 1 (x) g m (...

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Citation Context ... instead of (9) with # = 0, that the problem does not have "primal nondegenerate" 2nd-order stationary points. In the case of bound constraints (for which regularity holds everywhere), Colem=-=an and Li [7] proposed -=-a feasible a#ne-scaling trust-region method. They showed that if every cluster point of the generated iterates is "nondegenerate", then at least one cluster point is a 2ndorder stationary po... |

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Citation Context ...damped Newton steps on a reformulation of the 1st-order optimality condition for (4), and then decreasing , and so on. These methods and their global/local convergence were studied by El-Bakry et al. =-=[10]-=-, Yamashita [30], Yamashita and Yabe [31], and Akrotirianakis and Rustem [1] (also see [25] for a feasible method). Forsgren and Gill [15] and Gay et al. [16] studied implementation issues for these m... |

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Citation Context ...sary optimality condition for (1). 2 In fact, for quadratic f with rational coe#cients and g(x) = -x, it is NP-complete to decide whether the origin satisfies the strong 2nd-order necessary condition =-=[23]-=-. This suggests that weak 2nd-order necessary condition may be the best we can hope to achieve. In our notation, # n denotes the space of n-dimensional real column vectors, # m ++ denotes the positive... |

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Citation Context ...nvergence were studied by El-Bakry et al. [10], Yamashita [30], Yamashita and Yabe [31], and Akrotirianakis and Rustem [1] (also see [25] for a feasible method). Forsgren and Gill [15] and Gay et al. =-=[16]-=- studied implementation issues for these methods, including properties of the Newton direction, modified Newton directions (based on adding a suitable positive semidefinite matrix to # xx l(x, #)), te... |

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Citation Context ...heir global/local convergence were studied by El-Bakry et al. [10], Yamashita [30], Yamashita and Yabe [31], and Akrotirianakis and Rustem [1] (also see [25] for a feasible method). Forsgren and Gill =-=[15]-=- and Gay et al. [16] studied implementation issues for these methods, including properties of the Newton direction, modified Newton directions (based on adding a suitable positive semidefinite matrix ... |

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Citation Context ...s and global convergence to 1st-order stationary point was shown under reasonable assumptions. Numerical results on quadratic programs from the CUTE test set were also reported. In a series of papers =-=[4, 5, 6]-=-, Byrd, Nocedal, and coworkers proposed methods that combine interior-point approaches, trust-region strategies, and 2 successive quadratic programming (SQP) techniques. Global convergence to 1st-orde... |

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Citation Context ...ian-range assumptions, the method generates a 2nd-order stationary point [21, Thm. 4.13]. An extension of the Coleman-Li method to a case of implicit bound constraints was considered by Dennis et al. =-=[9]-=-. They showed that if the search directions have certain properties and the iterates are bounded, then at least one cluster point is a 2nd-order stationary point. Other trust-region methods, not using... |

30 |
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Citation Context ... the 1st-order optimality condition for (4), and then decreasing , and so on. These methods and their global/local convergence were studied by El-Bakry et al. [10], Yamashita [30], Yamashita and Yabe =-=[31]-=-, and Akrotirianakis and Rustem [1] (also see [25] for a feasible method). Forsgren and Gill [15] and Gay et al. [16] studied implementation issues for these methods, including properties of the Newto... |

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Citation Context ...hen decreasing , and so on. These methods and their global/local convergence were studied by El-Bakry et al. [10], Yamashita [30], Yamashita and Yabe [31], and Akrotirianakis and Rustem [1] (also see =-=[25]-=- for a feasible method). Forsgren and Gill [15] and Gay et al. [16] studied implementation issues for these methods, including properties of the Newton direction, modified Newton directions (based on ... |

26 |
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Citation Context ...eps on a reformulation of the 1st-order optimality condition for (4), and then decreasing , and so on. These methods and their global/local convergence were studied by El-Bakry et al. [10], Yamashita =-=[30]-=-, Yamashita and Yabe [31], and Akrotirianakis and Rustem [1] (also see [25] for a feasible method). Forsgren and Gill [15] and Gay et al. [16] studied implementation issues for these methods, includin... |

25 | On the local behavior of an interior point method for nonlinear programming - Byrd, Liu, et al. - 1997 |

19 |
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Citation Context ...d 2 successive quadratic programming (SQP) techniques. Global convergence to 1st-order stationary points and local superlinear convergence were studied in, respectively, [4] and [6]. Yamashita et al. =-=[33]-=- proposed a primal-dual interior-point trust-region method and, under certain assumptions, show global convergence and local superlinear convergence to a 1st-order stationary point of (1). However, th... |

16 | A primal-dual algorithm for minimizing a nonconvex function subject to bounds and nonlinear constraints
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Citation Context ...tionary point of (1) has been shown. In [30], only convergence to 1st-order stationary point of (4) is shown (also see [32] for related results using an l 2 logarithmic barrier function). Conn et al. =-=[8]-=- studied an infeasible primal-dual method for the case of linear constraints. Their method uses modified Newton directions and global convergence to 1st-order stationary point was shown under reasonab... |

9 |
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Citation Context ...), i # I(x), globally [1, 10, 25]. In addition, only convergence to 1st-order stationary point of (1) has been shown. In [30], only convergence to 1st-order stationary point of (4) is shown (also see =-=[32]-=- for related results using an l 2 logarithmic barrier function). Conn et al. [8] studied an infeasible primal-dual method for the case of linear constraints. Their method uses modified Newton directio... |

8 | Convergence to second order stationary points in inequality constrained optimization
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Citation Context ...): g(x) # 0, # # 0, g(x) T # = 0, # x l(x, #) = 0, (2) and x is a 2nd-order stationary point if in addition it satifies, together with #, the weak 2nd-order necessary optimality condition (see, e.g., =-=[3, 11, 17]-=-): d T # xx l(x, #)d # 0 #d with #g i (x) T d = 0 #i # I(x), (3) where I(x) := {i # {1, ..., m} : g i (x) = 0} and we define the Lagrangian l(x, #) := f(x) + g(x) T #. 1 This research is supported by ... |

7 | Newton methods for large-scale linear inequality-constrained minimization
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Citation Context ...r is weaker than the common regularity assumption of linear independence of #g i (x), i # I(x). The feasible active-set Newton method of Forsgren and Murray for linear inequality-constrained problems =-=[14] also gene-=-rates 2nd-order stationary points, but assumes, instead of (9) with # = 0, that the problem does not have "primal nondegenerate" 2nd-order stationary points. In the case of bound constraints... |

5 |
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Citation Context ...lts with these methods were reported in [16, 27, 30]. Numerical comparison of a primal-dual method, a primal-dual trust-region method and a primal method on sparse problems was given by Lasdon et al. =-=[20]-=-. For some methods, local superlinar convergence can also be shown under suitable assumptions. However, as Newton directions may not be defined everywhere, global convergence of methods using Newton d... |

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Citation Context ...vergence to a 1st-order stationary point of (1). However, the assumptions seem to be di#cult to verify. Implementation issues were studied in [5, 33], with promising numerical results reported. Jarre =-=[19]-=- considered a trust-region strategy for computing a 1st-order stationary point of (5), which involves a line search on the trust-region multiplier at each iteration and requires a strictly feasible st... |

3 | A Globally Convergent Interior-Point Algorithm for Nonlinear Programming Problems
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(Show Context)
Citation Context ...for (4), and then decreasing , and so on. These methods and their global/local convergence were studied by El-Bakry et al. [10], Yamashita [30], Yamashita and Yabe [31], and Akrotirianakis and Rustem =-=[1]-=- (also see [25] for a feasible method). Forsgren and Gill [15] and Gay et al. [16] studied implementation issues for these methods, including properties of the Newton direction, modified Newton direct... |

3 |
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(Show Context)
Citation Context ...): g(x) # 0, # # 0, g(x) T # = 0, # x l(x, #) = 0, (2) and x is a 2nd-order stationary point if in addition it satifies, together with #, the weak 2nd-order necessary optimality condition (see, e.g., =-=[3, 11, 17]-=-): d T # xx l(x, #)d # 0 #d with #g i (x) T d = 0 #i # I(x), (3) where I(x) := {i # {1, ..., m} : g i (x) = 0} and we define the Lagrangian l(x, #) := f(x) + g(x) T #. 1 This research is supported by ... |

1 | A note on the second-order convergence of optimization algorithms using barrier functions - unknown authors - 1998 |

1 |
Trust region a#ne scaling algorithms for linearly constrained convex and concave programs
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Citation Context ... every cluster point of the generated iterates is "nondegenerate", then at least one cluster point is a 2ndorder stationary point [7, Thm. 3.10(ii)]. A related method was studied by Monteiro=-= and Wang [21]-=- for the case of linear constraints and f being either convex or concave. Under certain nondegeneracy and constant Hessian-range assumptions, the method generates a 2nd-order stationary point [21, Thm... |

1 |
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Citation Context ...the introduction of [11]). In the case of nona#ne g, the only other methods we are aware of that can generate 2nd-order stationary point of (1) are the line search methods proposed by Mukai and Polak =-=[22]-=- and by Facchinei and Lucidi [11]. The method in [22] reformulates the inequalities as equalities (by expressing the slacks as the square of artificial variables), which in turn are handled by an exac... |