## Interior-point methods for nonconvex nonlinear programming: Filter methods and merit functions (2002)

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Venue: | Computational Optimization and Applications |

Citations: | 85 - 8 self |

### BibTeX

@ARTICLE{Benson02interior-pointmethods,

author = {Hande Y. Benson and Arun Sen and David and F. Shanno},

title = {Interior-point methods for nonconvex nonlinear programming: Filter methods and merit functions},

journal = {Computational Optimization and Applications},

year = {2002}

}

### Years of Citing Articles

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### Abstract

Abstract. In this paper, we present global and local convergence results for an interior-point method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Q-quadratically for a large class of problems. The proposed approach is the first to simultaneously have all of the following properties while solving a general nonconvex nonlinear programming problem: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the Karush-Kuhn-Tucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. Numerical testing on a set of general nonlinear programming problems, including degenerate problems and infeasible problems, confirm the theoretical results. We also provide comparisons to a highly-efficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1.

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Citation Context ...t suite. In our testing, we also found that larger problems were more susceptible to failure due to the unboundedness of Lagrange multipliers. We modified the problem egout from the MIPLIB test suite =-=[10]-=- by replacing the binary requirement on each variable xj with the conditions xj ≥ 0, 1 − xj ≥ 0, −xj(1 − xj) ≥ 0, converting it into an MPEC. Starting with the default multiplier initialization of 1, ... |

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Citation Context ...e modification to the linear solver was necessary because the matrix defined by (12) needs to be kept positive definite, and loqo expects its corresponding matrix to be quasi-definite. As analyzed in =-=[31]-=-, a quasi-definite matrix is strongly factorizable, so the change made to the linear solver was simply to the signs of the matrix entries, which is trivial and does not impact the performance. The def... |

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Citation Context ...boundedness of Sd in the next section. In Section 4, we will introduce some additional assumptions to analyze the local convergence of our approach. Under these additional assumptions, it is shown in =-=[17]-=- that if MFCQ holds for (2) then (3) is exact, so exactness is a weaker assumption than MFCQ and thereby LICQ. Additionally, the ℓ1 penalty approach was shown in [1] to be exact for MPECs, even though... |

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Citation Context ...ing and overcoming these failures. Recent work includes the solution of mathematical programs with equilibrium constraints (MPECs), especially using interior-point solvers, as documented in [1], [3], =-=[21]-=-, and [28]. There is also work on general NLPs that do not satisfy certain constraint qualifications at the optimal solution, and work on such problems includes [24] and [34]. Identification of infeas... |

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Citation Context ...terizing and overcoming these failures. Recent work includes the solution of mathematical programs with equilibrium constraints (MPECs), especially using interior-point solvers, as documented in [1], =-=[3]-=-, [21], and [28]. There is also work on general NLPs that do not satisfy certain constraint qualifications at the optimal solution, and work on such problems includes [24] and [34]. Identification of ... |

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Citation Context ...onstraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method =-=[12]-=-. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Q-quadratically for a large class of problems. The proposed approach is the first... |

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Citation Context ...haracterizing and overcoming these failures. Recent work includes the solution of mathematical programs with equilibrium constraints (MPECs), especially using interior-point solvers, as documented in =-=[1]-=-, [3], [21], and [28]. There is also work on general NLPs that do not satisfy certain constraint qualifications at the optimal solution, and work on such problems includes [24] and [34]. Identificatio... |

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Citation Context ... to locally converge to optima that may not satisfy the Karush-Kuhn-Tucker conditions, and (4) the algorithm is applicable to MPECs. Therefore, we present a more complete approach than the results of =-=[23]-=-, [29], [2], and [16], all of which use constraint relaxation as well. In [23], Mayne and Polak present a penalty approach for handling problems with equality and inequality constraints, however, they... |

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Citation Context ... Therefore, we will not include any such constraint qualifications in this paper. Instead, we will make some assumptions about the neighborhood of a local minimizer and employ the theory developed in =-=[36]-=- to show fast local convergence of Algorithm 2. Coupled with the assumption of exactness, this will allow us to show fast local convergence of Algorithm 3. First, we need to make a modification to our... |

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Citation Context ...the corresponding problem with 0. Problems DEGEN13f, DEGEN13fb, and DEGEN14f are infeasible problems. Finally, in Table 5, we have presented results on some MPECs gathered from the MacMPEC test suite =-=[20]-=- along with a few created from MPEC representations of large integer programming problems. Results on the full suite [20] were reported in [3] for loqo. Results on the whole CUTEr [14] test set for lo... |

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Citation Context ...for this study uses 871 of them. These models vary greatly in size and difficulty and have proved useful in drawing meaningful conclusions. All of the ampl models used in our testing are available at =-=[9]-=-. The cute suite contains some problems that are excluded from our set. We have not yet converted to ampl any models requiring special functions as well as some of the more complex models. We will con... |

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Citation Context ...and [26] show that there are a number of good nonlinear programming (NLP) codes that work well and solve most problems in standard test sets such as CUTEr [14]. However, codes like ipopt [35], knitro =-=[27]-=-, loqo [33], minos [25], and snopt [15] do sometimes fail, and there is a great deal of computational work focused on characterizing and overcoming these failures. Recent work includes the solution of... |

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Citation Context ...h of the third author is supported by NSF grant DMS-0107450. 12 HANDE Y. BENSON, ARUN SEN, AND DAVID F. SHANNO problems. As documented in our work [3] and [7], and those of others, including [1] and =-=[16]-=-, one technique that has proved particularly effective on all the issues raised above is constraint relaxation. Much of the work in this area has been to describe modifications to the basic algorithm ... |

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Citation Context ...on that can be helpful to reduce the number of linear subproblems that need to be solved. We similarly modified the mixed-integer nonlinear programming problem indextrack from the MacMINLP test suite =-=[19]-=-, and the convergent code solved this problem in 37 iterations, while loqo reached its iteration limit. It should be noted for both egout and indextrack that it is very hard for either algorithm to ap... |

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Citation Context ...olvers, as documented in [1], [3], [21], and [28]. There is also work on general NLPs that do not satisfy certain constraint qualifications at the optimal solution, and work on such problems includes =-=[24]-=- and [34]. Identification of infeasible problems has also been a persistent and difficult problem in NLP, as discussed in [3] and [15]. Many of these papers have dealt with modifications to a standard... |

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Citation Context .... If used, the penalty approach is generally employed to handle the equality constraints. Two recent works that propose and prove the convergence of interior-point methods with a penalty approach are =-=[22]-=- and [12]. In [22], Liu and Sun show that their interior-point algorithm converges globally under the standard assumptions of bounded primal and dual iterates. However, they also propose using a steep... |

3 |
Assessing the potential of interior-point methods for nonlinear optimization
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Citation Context ...ver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1. Introduction Computational studies including [7], [8], and =-=[26]-=- show that there are a number of good nonlinear programming (NLP) codes that work well and solve most problems in standard test sets such as CUTEr [14]. However, codes like ipopt [35], knitro [27], lo... |

3 |
Mathematical Programs with Equilibrium Constraints (MPECs
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Citation Context ...ercoming these failures. Recent work includes the solution of mathematical programs with equilibrium constraints (MPECs), especially using interior-point solvers, as documented in [1], [3], [21], and =-=[28]-=-. There is also work on general NLPs that do not satisfy certain constraint qualifications at the optimal solution, and work on such problems includes [24] and [34]. Identification of infeasible probl... |

2 |
Interior-point penalty methods for nonlinear programming. PhD. dissertation
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Citation Context ...tudies including [7] and [8] show, it is a highly efficient, state-of-the-art NLP code. Numerical results on an implementation of the algorithm of Chen and Goldfarb [12] are available and reported in =-=[11]-=-. These results were obtained by modifying ipopt to implement the key features of the solution algorithm, such as the penalty method and the barrier parameter updates. However, it retains many of the ... |

2 | On the fast local convergence of interior-point ℓ2 penalty methods for nonlinear programming
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(Show Context)
Citation Context ...tion of (2), if one exists. In this section, we examine the local convergence properties of the proposed algorithm, including its rate of convergence. Most local convergence proofs such as the one in =-=[13]-=- require the satisfaction of LICQ. However, the constraint Jacobian for (3) is given by [ ] A(x) I , 0 I and if LICQ does not hold for the original problem, then it will not hold for the penalty probl... |

2 |
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Citation Context ...al testing was conducted on a wide range of problems from literature. Tables 1-3 present the testing results on the Hock and Schittkowski [18] test suite, for which the ampl models were obtained from =-=[30]-=-. We also tested our algorithm on a set of degenerate problems described in [24] and present the results in Table 4. We made ampl models of the problems, and they can be obtained from [6]. Problems DE... |

2 |
More Test Samples for Nonlinear Programming Codes
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Citation Context ...except in the case of the ncvxqp family of problems and fminsurf, where the largest suggested sizes were beyond the capabilities of all solvers. In addition, we have expressed the entire Schittkowski =-=[7]-=- test suite in ampl. Together, this comprises a test suite of 889 ampl models. Some of these models are infeasible, unbounded, or nondifferentiable, so our final test set for this study uses 871 of th... |

1 |
Numerical testing results for the primal-dual penalty approach
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Citation Context ...rom MPEC representations of large integer programming problems. Results on the full suite [20] were reported in [3] for loqo. Results on the whole CUTEr [14] test set for loqo are also available from =-=[4]-=-. We have not included a discussion of the full test sets here as the resultsConvergence Analysis of an IPM for Nonconvex NLP 21 are long and Tables 1-5 proved adequate to make our points. The conver... |

1 |
Mixed-integer nonlinear programming using interior-point methods
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Citation Context ...onvergence Analysis of an IPM for Nonconvex NLP 27 dual variables. For many of the other problems in each test suite, both algorithms approach a local minimizer of infeasibility. However, as shown in =-=[5]-=-, the MPEC reformulation, combined with a penalty approach, can warmstart from the solution of the continuous relaxation and successfully locate an integer feasible solution for most problems. 6.6. Ba... |

1 |
AMPL models and numerical testing results for a globally convergent interior-point method. http://www.pages.drexel.edu/˜hvb22/Conlib
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(Show Context)
Citation Context ...tained from [30]. We also tested our algorithm on a set of degenerate problems described in [24] and present the results in Table 4. We made ampl models of the problems, and they can be obtained from =-=[6]-=-. Problems DEGEN1, DEGEN4, and DEGEN8 are identical to HS113, HS13, and HS32, respectively, but we included them again in Table 4 for completeness. Problems whose names end in “f” are feasibility prob... |