## A Practical Algorithm For General Large Scale Nonlinear Optimization Problems (1994)

Venue: | SIAM Journal on Optimization |

Citations: | 22 - 10 self |

### BibTeX

@INPROCEEDINGS{Boggs94apractical,

author = {Paul T. Boggs and Anthony J. Kearsley and Jon and Jon W. Tolle},

title = {A Practical Algorithm For General Large Scale Nonlinear Optimization Problems},

booktitle = {SIAM Journal on Optimization},

year = {1994},

pages = {755--778}

}

### OpenURL

### Abstract

. We provide an effective and efficient implementation of a sequential quadratic programming (SQP) algorithm for the general large scale nonlinear programming problem. In this algorithm the quadratic programming subproblems are solved by an interior point method that can be prematurely halted by a trust region constraint. Numerous computational enhancements to improve the numerical performance are presented. These include a dynamic procedure for adjusting the merit function parameter and procedures for adjusting the trust region radius. Numerical results and comparisons are presented. Key words: nonlinear programming, interior point, SQP, merit function, trust region, large scale 1. Introduction. In a series of recent papers, [3], [6], and [8], the authors have developed a new algorithmic approach for solving large, nonlinear, constrained optimization problems. This proposed procedure is, in essence, a sequential quadratic programming (SQP) method that uses an interior point algorithm...

### Citations

1351 |
Practical Optimization
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- 1987
(Show Context)
Citation Context ...rsity. Note that if Q is positive semi-definite, then the matrix in (2.3) is positive definite for all interior points; otherwise, it may not be. In the latter case, a modification similar to that in =-=[20]-=- is used. In our application of this algorithm, using this procedure obviates the need for the matrix B k to be positive definite, which in turn allows us to use the Hessian of the Lagrangian or a fin... |

316 |
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall
- Schnabel
- 1983
(Show Context)
Citation Context ...nfeasibilities, but only when inside of C j . 4.2. Updatings. Our procedure for updatings, the trust region radius, in Step 6b is similar to the standard strategy used in trust region algorithms (see =-=[17]-=- or [31]) in that we base the decision on how to changeson a comparison of a predicted relative reduction, pred k ; and an actual relative reduction, ared k , in a function used to measure the progres... |

306 |
Optimal Control of Systems Governed by Partial Differential Equations
- Lions
- 1968
(Show Context)
Citation Context ...f the solution process. For some problems, these finite difference approximations are not convenient to use. This can be the case with control problems governed by partial differential equations (see =-=[Lio71]-=- or [Lio88]). If the partial differential equation is solved using a finite element method, with piecewise linear elements, then evaluating the derivative of the objective function with respect to the... |

234 |
Inexact Newton methods
- DEMBO, EISENSTAT, et al.
- 1982
(Show Context)
Citation Context ...is modified at each iteration according to how well the improvement in our merit function is predicted. Thus our algorithm can be said to be a "truncated Newton method" in the sense of [18] =-=(see also [15]). This pa-=-rticular merit function and a more useful "working version" are discussed in Section 3 and our strategy for updating the trust region radius is given in Section 4.2. The output of the (QP ) ... |

209 |
Computing a trust region step
- MORÉ, SORENSEN
- 1983
(Show Context)
Citation Context ...ies, but only when inside of C(j). Updating �� : Our procedure for updating �� , the trust region radius, in Step 6b is similar to the standard strategy used in trust region algorithms (see [D=-=S83] or [MS83]) in-=- that we base the decision on how to change �� on a comparison of a predicted relative reduction, pred k ; and an actual relative reduction, ared k , in a function used to measure the progress tow... |

201 |
Numerical Methods for Nonlinear Variational Problems
- Glowinski
- 1984
(Show Context)
Citation Context ...n our test suite, one can approximate the first derivatives of the objective function by solving an adjoint problem with a computational cost comparable to one function evaluation. (For examples, see =-=[22]-=-.) The objective function portion of the Hessian of the Lagrangian can then be approximated with forward finite differences. A set of eight problems was chosen as the first test suite. These problems ... |

152 | CUTE: Constrained and unconstrained testing environment
- Bongartz, Conn, et al.
- 1993
(Show Context)
Citation Context ...pping conditions as the problems above. Likewise, the same table format was used to present these numerical results. For detailed description of these problems, structure, motivation, and sources see =-=[9]-=-. While it appears that the CUTE test problem set is rich in both large and small scale unconstrained and equality constrained test problems, at present there are not many large scale problems that in... |

96 | LANCELOT: A Fortran Package for LargeScale Nonlinear Optimization (Release A - Toint - 1991 |

87 |
LANCELOT, A FORTRAN package for large-scale nonlinear optimization
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- 1992
(Show Context)
Citation Context ...ber of steps to be taken in O3D (our choice was 7) this problem disappeared. Recently a collection of test problems has become available for the testing and comparing of optimization algorithms, (see =-=[13]-=-). The Constrained and Unconstrained Testing Enviroment (CUTE), are quickly becoming standards with which researchers can establish the viability and effectiveness of their numerical algorithms. These... |

85 |
Globally convergent inexact Newton methods
- Eisenstat, Walker
- 1994
(Show Context)
Citation Context ...n radius" that is modified at each iteration according to how well the improvement in our merit function is predicted. Thus our algorithm can be said to be a "truncated Newton method" i=-=n the sense of [18] (see also-=- [15]). This particular merit function and a more useful "working version" are discussed in Section 3 and our strategy for updating the trust region radius is given in Section 4.2. The outpu... |

56 | Nonparametric Probability Density Estimation - Tapia, Thompson - 1978 |

54 | Molecular conformations from distance matrices - Glunt, Hayden, et al. - 1993 |

45 |
Exact controllability, stabilization and perturbations for distributed systems
- Lions
- 1988
(Show Context)
Citation Context ...ion process. For some problems, these finite difference approximations are not convenient to use. This can be the case with control problems governed by partial differential equations (see [Lio71] or =-=[Lio88]-=-). If the partial differential equation is solved using a finite element method, with piecewise linear elements, then evaluating the derivative of the objective function with respect to the control va... |

42 | A global convergence theory for general trust-region-based algorithms for equality constrained optimization
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- 1992
(Show Context)
Citation Context ...c(x; z) T k and a linear approximation to / k 2 (x; z) =sc(x; z) T (sA k ) \Gamma1sc(x; z): Note that / k d (x; z) = / k 1 (x; z) + (1=d)/ k 2 (x; z). Based on these considerations and the results of =-=[16]-=- we define the predicted relative reduction by pred k = ae \Gammaff k r/ k 1 (x k ; z k ) T (ffi k ; q k ) \Gamma (ff k ) 2 2 (ffi k ; q k ) T r 2 / k 1 (x k ; z k )(ffi k ; q k ) \Gamma ff k d r/ k 2... |

42 |
Test Examples for Nonlinear Programming
- Hock, Schittkowski
- 1981
(Show Context)
Citation Context ...rds with which researchers can establish the viability and effectiveness of their numerical algorithms. These problems are replacing the smaller and well scaled test problems of Hock and Schittkowski =-=[25]-=- and Schittkowski [32] which were not intended to be used to test large scale algorithms. Our results on the CUTE test problems are summarized in Tables 6, 7 and 8. These problems were solved to the s... |

24 | Nonparametric function estimation, modeling, and simulation - Thompson, Tapia - 1990 |

21 |
Robust trust-region algorithm with non-monotonic penalty parameter scheme for constrained optimization
- El-Alem
- 1992
(Show Context)
Citation Context ...lution. (Various formulas for the predicted relative reduction, pred k , have been suggested for different merit functions, especially for equality constrained programming problems; see, for example, =-=[19]-=-). What is distinctive about our procedure is that we use different functions for computing pred k and ared k depending on the current status of the algorithm. When the linearized constraints are sati... |

16 | Numerical Solution of a Nonlinear Parabolic Control Problem by a Reduced SQP Method - Kupfer, Sachs - 1992 |

13 |
A strategy for global convergence in a sequential quadratic programming algorithm
- Boggs, Tolle
- 1989
(Show Context)
Citation Context ...h the application of a type of trust region technique. The trust region strategy is based on a special merit function for measuring progress towards a solution that was developed in earlier research, =-=[BT89]-=- and [BTK91], on merit functions for equality- and inequality-constrained optimization problems. Over the past two years this approach has been extensively tested on a variety of large problems, inclu... |

11 |
An interior-point method for general large scale quadratic programming
- Boggs, Domich, et al.
- 1996
(Show Context)
Citation Context ... There are two significant points to be made concerning this phase of our algorithm. First, we apply an interior point quadratic program solver to (QP ); more specifically, we use the method found in =-=[1]-=- where solutions are calculated by solving a sequence of low dimensional quadratic programs. Pertinent details of this solver and its properties relative to its use in our SQP method can be found in S... |

11 |
Constrained nonlinear programming
- Gill, Murray, et al.
- 1989
(Show Context)
Citation Context ... possible avenues of research to improve its efficiency. For a discussion of the theoretical and practical questions related to large scale nonlinear programming see the recent surveys [12], [14] and =-=[21]-=-. 2. An Interior Point QP Solver. Interior point methods for linear programming have been demonstrated to be very successful, especially on large problems, and recent research has lead to their extens... |

10 |
On the role of slack variables in quasi-Newton methods for constrained optimization
- TAPIA
- 1980
(Show Context)
Citation Context ...1 i ! 0 then set z k+1 i = ( ffl Mach g i (x k+1 )s0 \Gammag i (x k+1 ) g i (x k+1 ) ! 0 where ffl Mach is machine epsilon. This is sometimes referred to as `closing' the constraints (see for example =-=[Tap80]-=-). The Hessian Matrix: While the Hessian of the Lagrangian at the solution is known to be positive definite on the appropriate subspace, it may be indefinite in general. We experimented with two diffe... |

9 | Optimal signal sets for nonGaussian detectors
- Gockenbach, Kearsley
- 1999
(Show Context)
Citation Context ...ption of our perturbation procedure. We do note, however, that we have encountered important application problems where this procedure was crucial to the performance of our algorithm (see for example =-=[24]-=-). 4.8. Updating slack variables. One difficulty in our algorithm is the updating of slacks in the event that the SQP step does not satisfy the linearized constraints well enough, i.e., ` k is not sma... |

8 |
An interior-point method for linear and quadratic programming problems
- Boggs, Domich, et al.
- 1994
(Show Context)
Citation Context ...tic programs. A particular method, the method of optimizing over low-dimensional subspaces, has performed well on linear programs and has been extended to the quadratic programming case (see [1], and =-=[2]-=- and the references contained therein). This method, for which good numerical results for quadratic programs have been reported, has properties that make it particularly compatible with the SQP algori... |

8 | Functional and numerical solution of a control problem originating from heat transfer - Burger, Pogu - 1991 |

8 |
Optimizing over three-dimensional subspaces in an interior-point method for linear programming", Linear Algebra and Its
- Domich, Boggs, et al.
- 1991
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Citation Context ...A particular method, the method of optimizing over low-dimensional subspaces, has performed well on linear programs and has been extended to the quadratic programming case (see [BDRW94], [BDRW91] and =-=[DBRW91]-=- and the references contained therein). This method, for which good numerical results for quadratic programs have been reported, has properties that make it particularly compatible with the SQP algori... |

7 |
A family of descent functions for constrained optimization
- Boggs, Tolle
- 1984
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Citation Context ...olution to (NLP ). Because a reduction in this function implies that progress is being made towards the solution, it can be used to determine an appropriate steplength in a given search direction. In =-=[5]-=- and [6] a merit function for equality-constrained problems was derived that has important properties vis-a-vis the steps generated by the SQP algorithm. Using a slack-variable formulation of (NLP ) a... |

7 |
Large-scale nonlinear constrained optimization
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Citation Context ...ossible avenues of research that will improve its efficiency. For a discussion of the theoretical and practical questions related to large scale nonlinear programming see the recent surveys [Col] and =-=[CGT92b]-=-. 2. An Interior Point QP Solver Interior point methods for linear programming have been demonstrated to be very successful, especially on large problems, and recent research has lead to their extensi... |

6 | On the exact and approximate boundary controllabilities for the heat equation: a numerical approach - Carthel, Glowinski, et al. - 1994 |

5 | A merit function for inequality constrained nonlinear programming problems
- Boggs, Tolle, et al.
- 1991
(Show Context)
Citation Context ...adius. Numerical results and comparisons are presented. Key words: nonlinear programming, interior point, SQP, merit function, trust region, large scale 1. Introduction. In a series of recent papers, =-=[3]-=-, [6], and [8], the authors have developed a new algorithmic approach for solving large, nonlinear, constrained optimization problems. This proposed procedure is, in essence, a sequential quadratic pr... |

5 | A truncated SQP algorithm for large scale nonlinear programming problems
- Boggs, Tolle, et al.
- 1994
(Show Context)
Citation Context ...al results and comparisons are presented. Key words: nonlinear programming, interior point, SQP, merit function, trust region, large scale 1. Introduction. In a series of recent papers, [3], [6], and =-=[8]-=-, the authors have developed a new algorithmic approach for solving large, nonlinear, constrained optimization problems. This proposed procedure is, in essence, a sequential quadratic programming (SQP... |

4 |
Large Scale Numerical Optimization: Introduction and overview
- Coleman
- 1992
(Show Context)
Citation Context ...thm and suggest possible avenues of research to improve its efficiency. For a discussion of the theoretical and practical questions related to large scale nonlinear programming see the recent surveys =-=[12]-=-, [14] and [21]. 2. An Interior Point QP Solver. Interior point methods for linear programming have been demonstrated to be very successful, especially on large problems, and recent research has lead ... |

2 |
quadratic programming, Acta Numerica
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(Show Context)
Citation Context ...be inferred from [6]. Nonlinear equality constraints are included in our code and in some of the problems we tested. The sequential quadratic programming method is the backbone of our algorithm. (See =-=[7]-=- for a review of these techniques.) At the kth step we have an iterate, x k , denoting the current approximation to the solution of (NLP ). In addition to the x-iterate we also maintain a non-negative... |

1 |
nonlinear constrained optimization
- Large-scale
- 1992
(Show Context)
Citation Context ...d suggest possible avenues of research to improve its efficiency. For a discussion of the theoretical and practical questions related to large scale nonlinear programming see the recent surveys [12], =-=[14]-=- and [21]. 2. An Interior Point QP Solver. Interior point methods for linear programming have been demonstrated to be very successful, especially on large problems, and recent research has lead to the... |