## An Algorithm for Nonlinear Optimization Using Linear Programming and Equality Constrained Subproblems (2003)

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Citations: | 38 - 12 self |

### BibTeX

@MISC{Byrd03analgorithm,

author = {Richard H. Byrd and Nicholas I. M. Gould and Jorge Nocedal and Richard A. Waltz},

title = {An Algorithm for Nonlinear Optimization Using Linear Programming and Equality Constrained Subproblems},

year = {2003}

}

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### Abstract

This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the ` 1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program.

### Citations

1544 |
Practical Optimization
- Gill, Murray, et al.
- 1981
(Show Context)
Citation Context ...is paper we describe a trust-region algorithm for nonlinear programming that does not require the solution of a general quadratic program at each iteration. It can be viewed as a so-called “EQP form” =-=[12]-=- of sequential quadratic programming, in which a guess of the active set is made (using linear programming techniques) and then an equality constrained quadratic program is solved to attempt to achiev... |

384 | SNOPT: An SQP algorithm for large-scale constrained optimization
- Gill, Murray, et al.
- 2005
(Show Context)
Citation Context ...st successful methods for large-scale nonlinear optimization. In recent years, active set SQP methods have proven to be quite effective at solving problems with thousands of variables and constraints =-=[9, 11]-=-, but are likely to become very expensive as the problems they are asked to solve become larger and larger. This is particularly so since the principal cost of SQP methods is often dominated by that o... |

269 | Benchmarking optimization software with performance profiles
- Dolan, Mor'e
- 2002
(Show Context)
Citation Context ... tests involving timing were carried out on a dedicated machine with no other jobs running. All the results in this section will be presented using the performance profiles proposed by Dolan and Moré =-=[7]-=-. In the plots πs(τ) denotes the logarithmic performance profile πs(τ) = no. of problems where log2(rp,s) ≤ τ , τ ≥ 0, (11.3) total no. of problems where rp,s is the ratio between the time to solve pr... |

181 | 2002), ‘Nonlinear programming without a penalty function
- Fletcher, Leyffer
(Show Context)
Citation Context ...st successful methods for large-scale nonlinear optimization. In recent years, active set SQP methods have proven to be quite effective at solving problems with thousands of variables and constraints =-=[9, 11]-=-, but are likely to become very expensive as the problems they are asked to solve become larger and larger. This is particularly so since the principal cost of SQP methods is often dominated by that o... |

132 |
CUTE: Constrained and unconstrained testing environment
- Toint
- 1995
(Show Context)
Citation Context ...s a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr =-=[1, 15]-=- test set. ∗ Department of Computer Science, University of Colorado, Boulder, CO 80309; richard@cs.colorado. edu. This author was supported by Air Force Office of Scientific Research grant F49620-00-1... |

106 |
LANCELOT: a Fortran package for Large-scale Nonlinear Optimization
- Toint
- 1992
(Show Context)
Citation Context ...) and 1(b) the performance of the three codes on 43 problems whose only constraints are simple bounds on the variables. Although there exist specialized approaches for solving these types of problems =-=[6, 18, 25]-=-, it is instructive to observe the performance of Slique when the feasible region has the geometry produced by simple bounds. Figures 1(a) and 1(b) indicate that Slique performs quite well on this cla... |

82 | Newton’s method for large bound-constrained optimization problems
- Lin, Moré
- 1999
(Show Context)
Citation Context ...) and 1(b) the performance of the three codes on 43 problems whose only constraints are simple bounds on the variables. Although there exist specialized approaches for solving these types of problems =-=[6, 18, 25]-=-, it is instructive to observe the performance of Slique when the feasible region has the geometry produced by simple bounds. Figures 1(a) and 1(b) indicate that Slique performs quite well on this cla... |

78 | An interior point algorithm for large scale nonlinear programming
- Byrd, Hribar, et al.
(Show Context)
Citation Context ...11 Numerical Tests In order to assess the potential of the SLP-EQP approach taken in Slique, we test it here on the CUTEr [1, 15] set of problems and compare it with the state-of-the-art codes Knitro =-=[3, 23]-=- and Snopt [11]. Slique 1.0 implements the algorithm outlined in the previous section. In all results reported in this section, Slique 1.0 uses the commercial LP software package ILOG CPLEX 8.0 [17] r... |

68 |
Trust-region methods
- Toint
- 2000
(Show Context)
Citation Context ...back that its constraints may be infeasible. This possible inconsistency of constraint linearizations and the trust region has received considerable attention in the context of SQP methods; see, e.g. =-=[5]-=- and the references therein. To deal with the possible inconsistency of the constraints we follow an ℓ1-penalty approach in which the constraints (3.1b)–(3.1c) are incorporated in the form of a penalt... |

52 |
A new algorithm for unconstrained optimization
- Powell
- 1970
(Show Context)
Citation Context ...search to compute α EQP , but rather use a backtracking line search that terminates as soon as the model q has decreased. The computation of the trial step d is similar to the dogleg method of Powell =-=[20, 21]-=- for approximately minimizing a quadratic objective subject to a trust-region constraint. As in the dogleg approach, the step is computed via a one dimensional line search along a piecewise path from ... |

50 | 2003), ‘On the global convergence of an SLP-filter algorithm that takes EQP steps
- Chin, Fletcher
(Show Context)
Citation Context ...n active set, followed by the solution of an equality constrained quadratic problem (EQP) was first proposed and analyzed by Fletcher and Sainz de la Maza [10], and more recently by Chin and Fletcher =-=[4]-=-, but has received little attention beyond this. This “sequential linear programming-EQP method”, or SLP-EQP in short, is motivated by the fact that solving quadratic subproblems with inequality const... |

48 |
CUTEr (and SifDec), a Constrained and Unconstrained Testing Environment, revisited
- Toint
(Show Context)
Citation Context ...s a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr =-=[1, 15]-=- test set. ∗ Department of Computer Science, University of Colorado, Boulder, CO 80309; richard@cs.colorado. edu. This author was supported by Air Force Office of Scientific Research grant F49620-00-1... |

34 |
Practical Methods of Optimization, volume 2. Constrained optimization
- Fletcher
- 1986
(Show Context)
Citation Context ...that make good progress toward the solution may be rejected by the penalty function φ, which may lead to slow convergence. We address this difficulty by computing a second order correction (SOC) step =-=[8]-=-, which incorporates second order curvature information on the constraints. If the trial point x T does not provide sufficient decrease of the merit function, we compute d SOC as the minimum norm solu... |

34 |
Solving the trust-region subproblem using the Lanczos method
- Toint
- 1999
(Show Context)
Citation Context ...tic problem (5.4), with its additional spherical trust-region constraint, will be solved using a projected Conjugate-Gradient/Lanczos iteration, as implemented in the GALAHAD code GLTR of Gould et al =-=[14]-=- (HSL routine VF05 [16]). Thissalgorithm has the feature of continuing for a few more iterations after the first negative curvature direction is encountered. The projected CG/Lanczos approach applies ... |

33 |
de la Maza. Nonlinear programming and non-smooth optimization by successive linear programming
- Fletcher, Sainz
- 1989
(Show Context)
Citation Context ...chnology Center Abstract This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza =-=[10]-=-. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approxima... |

30 |
Exact Penalty Function Algorithms for Finite Dimensional and Optimization Problems
- Maratos
- 1978
(Show Context)
Citation Context ...the choices of α LP and α EQP used to compute the trial step d), the acceptance rule (7.1) guarantees that we only accept steps which give a reduction in the merit function. As is well known (Maratos =-=[19]-=-) steps that make good progress toward the solution may be rejected by the penalty function φ, which may lead to slow convergence. We address this difficulty by computing a second order correction (SO... |

23 |
2001), ‘On the solution of equality constrained quadratic problems arising in optimization
- Gould, Hribar, et al.
(Show Context)
Citation Context ...ep ¯ d in the null-space of AW. The projection of a vector v, say w = P v, is computed by solving the system � I A T W (x) AW(x) 0 � � w u � = � v 0 � 9 (5.9) where u is an auxiliary vector; see also =-=[13]-=-. We use the routine MA27 from the HSL library [16] to factor this system. The CG iteration can be preconditioned to speed up convergence by replacing the identity matrix in the (1,1) block of the coe... |

20 |
On the superlinear convergence of a trust region algorithm for non-smooth optimization
- Yuan
- 1985
(Show Context)
Citation Context ...n 11 indicate, in our opinion, that the algorithm holds much promise. In addition, the algorithm is supported by the global convergence theory presented in [2], which builds upon the analysis of Yuan =-=[24]-=-. Our approach differs significantly from the SLP-EQP algorithm described by Fletcher and Chin [4]. These authors use a filter for step acceptance. In the event that the constraints in the LP subprobl... |

15 |
KNITRO user's manual
- Waltz, Nocedal
- 2003
(Show Context)
Citation Context ...11 Numerical Tests In order to assess the potential of the SLP-EQP approach taken in Slique, we test it here on the CUTEr [1, 15] set of problems and compare it with the state-of-the-art codes Knitro =-=[3, 23]-=- and Snopt [11]. Slique 1.0 implements the algorithm outlined in the previous section. In all results reported in this section, Slique 1.0 uses the commercial LP software package ILOG CPLEX 8.0 [17] r... |

9 |
A catalogue of subroutines (HSL
- Library, Technology, et al.
- 2000
(Show Context)
Citation Context ... its additional spherical trust-region constraint, will be solved using a projected Conjugate-Gradient/Lanczos iteration, as implemented in the GALAHAD code GLTR of Gould et al [14] (HSL routine VF05 =-=[16]-=-). Thissalgorithm has the feature of continuing for a few more iterations after the first negative curvature direction is encountered. The projected CG/Lanczos approach applies orthogonal projections ... |

7 |
Algorithms for large-scale nonlinear optimization
- Waltz
- 2002
(Show Context)
Citation Context ...ed in this paper is quite robust and efficient at solving simpler classes of 15s16 problems (e.g., LPs, unconstrained problems, equality constrained problems and feasibility problems) as evidenced in =-=[22]-=-. We should note that there are a few problems in CUTEr for which a solution does not exist (for example the problem may be infeasible or unbounded). Although, it is important for a code to recognize ... |

7 |
Algorithm 78: L-BFGS-B: Fortran subroutines for large-scale bound constrained optimization
- Zhu, Byrd, et al.
- 1997
(Show Context)
Citation Context ...) and 1(b) the performance of the three codes on 43 problems whose only constraints are simple bounds on the variables. Although there exist specialized approaches for solving these types of problems =-=[6, 18, 25]-=-, it is instructive to observe the performance of Slique when the feasible region has the geometry produced by simple bounds. Figures 1(a) and 1(b) indicate that Slique performs quite well on this cla... |

4 | On the convergence of successive linear programming algorithms
- Byrd, Gould, et al.
- 2002
(Show Context)
Citation Context ...The experimental results presented in Section 11 indicate, in our opinion, that the algorithm holds much promise. In addition, the algorithm is supported by the global convergence theory presented in =-=[2]-=-, which builds upon the analysis of Yuan [24]. Our approach differs significantly from the SLP-EQP algorithm described by Fletcher and Chin [4]. These authors use a filter for step acceptance. In the ... |

4 |
A Fortran subroutine for unconstrained minimization requiring first derivatives of the objective function
- Powell
- 1970
(Show Context)
Citation Context ...search to compute α EQP , but rather use a backtracking line search that terminates as soon as the model q has decreased. The computation of the trial step d is similar to the dogleg method of Powell =-=[20, 21]-=- for approximately minimizing a quadratic objective subject to a trust-region constraint. As in the dogleg approach, the step is computed via a one dimensional line search along a piecewise path from ... |