## KNITRO: An integrated package for nonlinear optimization (2006)

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Venue: | Large Scale Nonlinear Optimization, 35–59, 2006 |

Citations: | 39 - 3 self |

### BibTeX

@INPROCEEDINGS{Byrd06knitro:an,

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

title = {KNITRO: An integrated package for nonlinear optimization},

booktitle = {Large Scale Nonlinear Optimization, 35–59, 2006},

year = {2006},

pages = {35--59},

publisher = {Springer Verlag}

}

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### Abstract

This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an active-set method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1

### Citations

1915 |
Numerical optimization
- Nocedal, Wright
- 1999
(Show Context)
Citation Context ..., s + , y + , z + ) − ∇xL(x, s + , y + , z + ), ∆x = x + − x, and substitute the correction pairs (∆l, ∆x) in the standard definition of the BFGS, SR1 or limited memory BFGS update formulae (see e.g. =-=[31]-=-). To ensure positive definiteness of the BFGS and L-BFGS updates the vector ∆l is modified, if necessary, using Powell’s damping procedure. SR1 updating is safeguarded to avoid unboundedness, but is ... |

332 | SNOPT: An SQP algorithm for large-scale constrained optimization
- Gill, Murray, et al.
(Show Context)
Citation Context ...lasses. The quality and diversity of nonlinear optimization software has greatly improved during the last 10 years. Some of the established packages have matured, and new packages have emerged. Snopt =-=[18]-=- and FilterSQP [15] implement active-set sequential quadratic programming (SQP) methods. Snopt uses a line search approach, and in its default setting, employs quasi-Newton approximations to the Hessi... |

146 |
Pathways to the optimal set in linear programming
- Megiddo
- 1989
(Show Context)
Citation Context ...ne by switching from the interior to an active-set iteration, a process that is often called crossover. Although crossover techniques have received much attention in the context of linear programming =-=[27]-=-, to the best of our knowledge, none of the nonlinear interior codes provide an option for it. We regard it as essential to have this facility in our integrated system, both for computational efficien... |

145 | An interior point algorithm for nonconvex nonlinear programming
- Vanderbei, Shanno
- 1999
(Show Context)
Citation Context ...nt augmented Lagrangian methods. Another well established package is Conopt [14], which offers reduced Hessian and SQP methods. Most of the new packages are based on the interior-point approach. Loqo =-=[33]-=- implements a line search primal-dual algorithm that can be viewed as a direct extension of interior methods for linear and quadratic programming. The first release of Knitro [6] offered a trust regio... |

105 | A trust region method based on interior point techniques for nonlinear programming
- Byrd, Gilbert, et al.
- 2000
(Show Context)
Citation Context ... flexibility is a more complex iteration than in the line search approach. Interior/CG is an implementation of the algorithm described in [6], which is based on the approach described and analyzed in =-=[3]-=-. 6sAlgorithm 3.1: Knitro-Interior/Direct Choose x0, s0 > 0, and the parameters 0 < η, and 0 < αmin < 1. Compute initial values for the multipliers y0, z0 > 0, the trust-region radius ∆0 > 0, and the ... |

103 |
On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming
- Wächter, Biegler
- 2006
(Show Context)
Citation Context ...employing a conjugate gradient iteration in the step computation; the second release added a line search interior algorithm that is safeguarded by the trust region approach [38]. Barnlp [2] and Ipopt =-=[36]-=- implement line search interior-point approaches; Ipopt uses a filter globalization and includes a feasibility restoration phase. Mosek [1] is a primal-dual interior-point method for convex optimizati... |

96 |
LANCELOT: A Fortran Package for LargeScale Nonlinear Optimization (Release A
- Toint
- 1991
(Show Context)
Citation Context ..., employs quasi-Newton approximations to the Hessian. FilterSQP follows a trust region approach, with filter globalization, and makes use of second-derivative information. The Minos [29] and Lancelot =-=[12]-=- packages, which were the first widely available codes capable of solving problems with tens of thousands of variables and constraints, implement augmented Lagrangian methods. Another well established... |

76 |
Minos 5.4 user's guide
- Murtagh, Saunders
- 1993
(Show Context)
Citation Context ...ts default setting, employs quasi-Newton approximations to the Hessian. FilterSQP follows a trust region approach, with filter globalization, and makes use of second-derivative information. The Minos =-=[29]-=- and Lancelot [12] packages, which were the first widely available codes capable of solving problems with tens of thousands of variables and constraints, implement augmented Lagrangian methods. Anothe... |

72 | An interior point algorithm for large scale nonlinear programming
- Byrd, Hribar, et al.
- 1999
(Show Context)
Citation Context ...point approach. Loqo [33] implements a line search primal-dual algorithm that can be viewed as a direct extension of interior methods for linear and quadratic programming. The first release of Knitro =-=[6]-=- offered a trust region interior-point algorithm employing a conjugate gradient iteration in the step computation; the second release added a line search interior algorithm that is safeguarded by the ... |

72 | A.J.: Constraint preconditioning for indefinite linear systems
- Keller, Gould, et al.
- 2000
(Show Context)
Citation Context ...lgorithm and the EQP phase (4.6) of the Active algorithm both require the solution of an equality constrained quadratic program. We solve these problems using a projected conjugate gradient iteration =-=[10, 20, 24, 26, 32]-=-, which is well suited for large problems and can handle the negative curvature case without the need for Hessian modifications. We now outline this iteration and refer the reader to [20] for a more d... |

66 |
The MOSEK interior point optimization for linear programming: an implementation of the homogeneous algorithm
- ANDERSEN, ANDERSEN
- 1999
(Show Context)
Citation Context ...ded by the trust region approach [38]. Barnlp [2] and Ipopt [36] implement line search interior-point approaches; Ipopt uses a filter globalization and includes a feasibility restoration phase. Mosek =-=[1]-=- is a primal-dual interior-point method for convex optimization, and Pennon [25] follows an augmented Lagrangian approach. 2sCG INTERIOR−POINT DIRECT KNITRO crossover ACTIVE−SET Figure 1: The main alg... |

61 |
Trust-region methods
- Toint
- 2000
(Show Context)
Citation Context ... that λi ≥ 0 for i ∈ I. Penalty Parameter Update Strategy. A novel feature of our SLQP algorithm is the procedure for updating the penalty parameter. Unlike most strategies proposed in the literature =-=[11]-=-, which hold the penalty parameter ν fixed for a series of iterations and only update it if insufficient progress toward feasibility is made, our algorithm chooses an appropriate value of ν at each it... |

47 | On the global convergence of an SLP-filter algorithm that takes EQP steps
- Chin, Fletcher
(Show Context)
Citation Context ...VE−SET Figure 1: The main algorithmic options in the Knitro 5.0 package. New active-set methods based on Sequential Linear-Quadratic Programming (SLQP) have recently been studied by Chin and Fletcher =-=[9]-=- and Byrd et al. [5]. Unlike SQP methods, which combine the active-set identification and the step computation in one quadratic subproblem, SLQP methods decouple these tasks into two subproblems. The ... |

44 |
A GRG Code for Large Sparse Dynamic Nonlinear Optimization
- Drud
- 1985
(Show Context)
Citation Context ...re the first widely available codes capable of solving problems with tens of thousands of variables and constraints, implement augmented Lagrangian methods. Another well established package is Conopt =-=[14]-=-, which offers reduced Hessian and SQP methods. Most of the new packages are based on the interior-point approach. Loqo [33] implements a line search primal-dual algorithm that can be viewed as a dire... |

42 |
Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numerical Linear Algebra with Applications 5(3
- Luk˘san, Vl˘cek
- 1998
(Show Context)
Citation Context ...lgorithm and the EQP phase (4.6) of the Active algorithm both require the solution of an equality constrained quadratic program. We solve these problems using a projected conjugate gradient iteration =-=[10, 20, 24, 26, 32]-=-, which is well suited for large problems and can handle the negative curvature case without the need for Hessian modifications. We now outline this iteration and refer the reader to [20] for a more d... |

41 |
An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications
- Wächter
- 2002
(Show Context)
Citation Context ...m� φν(x, s) = f(x) − µ log si + ν�cE(x)�2 + ν�cI(x) − s�2, (3.3) i=1 where ν > 0. A step is acceptable only if it provides a sufficient decrease in φν. Although it has been reported in the literature =-=[22, 34]-=- that merit functions of this type can interfere with rapid progress of the iteration, our experience indicates that the implementation described in Section 3.3 overcomes these difficulties. These obs... |

39 | An algorithm for nonlinear optimization using linear programming and equality constrained subproblems
- Byrd, Gould, et al.
(Show Context)
Citation Context ... main algorithmic options in the Knitro 5.0 package. New active-set methods based on Sequential Linear-Quadratic Programming (SLQP) have recently been studied by Chin and Fletcher [9] and Byrd et al. =-=[5]-=-. Unlike SQP methods, which combine the active-set identification and the step computation in one quadratic subproblem, SLQP methods decouple these tasks into two subproblems. The active-set algorithm... |

38 |
The conjugate gradient method in extremal problems
- Polyak
- 1969
(Show Context)
Citation Context ...lgorithm and the EQP phase (4.6) of the Active algorithm both require the solution of an equality constrained quadratic program. We solve these problems using a projected conjugate gradient iteration =-=[10, 20, 24, 26, 32]-=-, which is well suited for large problems and can handle the negative curvature case without the need for Hessian modifications. We now outline this iteration and refer the reader to [20] for a more d... |

34 | Failure of Global Convergence for a Class of Interior Point Methods for Nonlinear Programming, Mathematical Programming 88(3), p. 565574
- Wächter, Biegler
- 2000
(Show Context)
Citation Context ...ng a null-space approach, without modifying the Hessian ∇ 2 xxL. An additional benefit of invoking the trust region step is that it guarantees progress in cases when the line search approach can fail =-=[7, 35]-=-. Since it is known that, when line search iterations converge to non-stationary points, the steplengths αs or αz in (3.9) converge to zero, we monitor these steplengths. If one of them is smaller tha... |

31 | An interior algorithm for nonlinear optimization that combines line search and trust region steps
- Waltz, Morales, et al.
(Show Context)
Citation Context ...n interior-point algorithm employing a conjugate gradient iteration in the step computation; the second release added a line search interior algorithm that is safeguarded by the trust region approach =-=[38]-=-. Barnlp [2] and Ipopt [36] implement line search interior-point approaches; Ipopt uses a filter globalization and includes a feasibility restoration phase. Mosek [1] is a primal-dual interior-point m... |

30 |
de la Maza. Nonlinear programming and nonsmooth optimization by successive linear programming
- Fletcher, Sainz
(Show Context)
Citation Context ... solved in practice. In addition, the incorporation of second derivative information in SQP methods has proved to be difficult. We use, instead a sequential linear-quadratic programming (SLQP) method =-=[5, 9, 16]-=- that computes a step in two stages, each of which scales up well with the number of variables. First, a linear program (LP) is solved to identify a working set. This is followed by an equality constr... |

20 |
On the Convergence of Successive Linear-Quadratic Programming Algorithms
- Byrd, Gould, et al.
(Show Context)
Citation Context ...ncreasing the current trial value of ν by 10 and re-solving the linear program. The penalty update algorithm above guarantees that ν is chosen large enough to ensure convergence to a stationary point =-=[4]-=-. Although the procedure does require the solution of some additional linear programs, our experience is that it results in an overall savings in iterations (and total LP solves) by achieving a better... |

20 |
On the solution of equality constrained quadratic problems arising in optimization
- Gould, Hribar, et al.
(Show Context)
Citation Context |

17 | User manual for filterSQP
- Fletcher, Leyffer
- 1998
(Show Context)
Citation Context ... and diversity of nonlinear optimization software has greatly improved during the last 10 years. Some of the established packages have matured, and new packages have emerged. Snopt [18] and FilterSQP =-=[15]-=- implement active-set sequential quadratic programming (SQP) methods. Snopt uses a line search approach, and in its default setting, employs quasi-Newton approximations to the Hessian. FilterSQP follo... |

15 | Feasible interior methods using slacks for nonlinear optimization
- Byrd, Nocedal, et al.
- 2003
(Show Context)
Citation Context ...egion because such steps increase the barrier term −µ � m i=1 log(si) in the merit function (3.3). Apart from the reset (6.21), 18sin the Interior/CG algorithm we must introduce a slight modification =-=[8]-=- in the normal step computation to ensure that this step makes sufficient progress toward feasibility. Initial Point Strategy. As is well known, interior methods can perform poorly if the initial poin... |

15 | Optimality measures for performance profiles
- Dolan, Moré, et al.
(Show Context)
Citation Context ...ribed in [5]. Interior-point and active-set methods offer competing state-of-the-art approaches for solving nonlinear optimization problems — each with its own set of advantages. Benchmarking studies =-=[13, 28]-=- have tried to identify the classes of problems for which each approach is best suited, but the rapid pace of software development makes it difficult to arrive at concrete conclusions at this time. We... |

10 |
Linearly constrained optimization and projected preconditioned conjugate gradients
- Coleman
- 1994
(Show Context)
Citation Context |

10 |
Applications of Optimization with Xpress-MP. Dash Optimization. Translated and revised by Susanne Heipcke. http://www.dashoptimization. com/home/downloads/book/booka4.pdf
- Guéret, Prins, et al.
- 2002
(Show Context)
Citation Context ...imization: the interior-point approach and the active-set approach. The impressive success of an integrated approach of this sort for linear and integer programming, particularly over the past decade =-=[21, 23]-=-, argues for a similar approach to be taken in nonlinear optimization. Knitro is capable of applying features of an interior-point method or an active-set method — or possibly both — depending on prob... |

10 |
Sequential Quadratic Programming Methods Based on Approximating a Projected Hessian
- Gurwitz
- 2001
(Show Context)
Citation Context ...m� φν(x, s) = f(x) − µ log si + ν�cE(x)�2 + ν�cI(x) − s�2, (3.3) i=1 where ν > 0. A step is acceptable only if it provides a sufficient decrease in φν. Although it has been reported in the literature =-=[22, 34]-=- that merit functions of this type can interfere with rapid progress of the iteration, our experience indicates that the implementation described in Section 3.3 overcomes these difficulties. These obs... |

9 | On the Convergence of Newton Iterations to Non-Stationary
- Byrd, Marazzi, et al.
- 2001
(Show Context)
Citation Context ...ng a null-space approach, without modifying the Hessian ∇ 2 xxL. An additional benefit of invoking the trust region step is that it guarantees progress in cases when the line search approach can fail =-=[7, 35]-=-. Since it is known that, when line search iterations converge to non-stationary points, the steplengths αs or αz in (3.9) converge to zero, we monitor these steplengths. If one of them is smaller tha... |

9 |
Adaptive barrier strategies for nonlinear interior methods
- Nocedal, Wächter, et al.
- 2005
(Show Context)
Citation Context ... µ; (iii) a so-called quality-function approach; (iv) variants of option (ii) which possibly utilize safeguarded corrector steps. These rules are described and tested in Nocedal, Wäechter 4sand Waltz =-=[30]-=-. Since it is not known at present which one is the most effective in practice, Knitro allows the user to experiment with the barrier update strategies just mentioned. To control the quality of the st... |

7 | Assessing the potential of interior methods for nonlinear optimization
- Morales, Nocedal, et al.
- 2001
(Show Context)
Citation Context ...ribed in [5]. Interior-point and active-set methods offer competing state-of-the-art approaches for solving nonlinear optimization problems — each with its own set of advantages. Benchmarking studies =-=[13, 28]-=- have tried to identify the classes of problems for which each approach is best suited, but the rapid pace of software development makes it difficult to arrive at concrete conclusions at this time. We... |

5 |
An interior-point nonlinear programming algorithm for large scale optimization
- Betts, Eldersveld, et al.
- 2003
(Show Context)
Citation Context ...int algorithm employing a conjugate gradient iteration in the step computation; the second release added a line search interior algorithm that is safeguarded by the trust region approach [38]. Barnlp =-=[2]-=- and Ipopt [36] implement line search interior-point approaches; Ipopt uses a filter globalization and includes a feasibility restoration phase. Mosek [1] is a primal-dual interior-point method for co... |

5 |
Knitro 4.0 User’s Manual
- Waltz
- 2004
(Show Context)
Citation Context ... norm weighted by D. 6 Special Algorithmic Features The Knitro package provides many algorithmic options and features that are listed comprehensibly in the documentation that accompanies the software =-=[37]-=-. Here we highlight x0 17ssome of these options and discuss their relationship to the algorithms presented in the previous sections. Hessian Options The user can supply first and second derivatives, w... |

5 |
Applications of optimization with Xpress-MP
- Guret, Prins, et al.
- 2000
(Show Context)
Citation Context ...imization: the interior-point approach and the active-set approach. The impressive success of an integrated approach of this sort for linear and integer programming, particularly over the past decade =-=[21, 23]-=-, argues for a similar approach to be taken in nonlinear optimization. Knitro is capable of applying features of an interior-point method or an active-set method — or possibly both — depending on prob... |

2 |
A Starting-Point Strategy for Nonlinear Interior Methods
- Gertz, Nocedal, et al.
(Show Context)
Citation Context ...ngian using z0 and not z1, i.e., ∇ 2 xxL(x0, s0, y0, z0) (this Hessian is independent of s, so the choice of that variable is irrelevant). More details about the initial point strategies are given in =-=[17]-=-. Special Problem Classes When the nonlinear program (1.1) has a special form, the algorithms in Knitro often reduce to well-known special-purpose methods. For unconstrained optimization problems, the... |

2 |
Available at http://www2.am.unierlangen.de/˜kocvara/pennon/problems.html
- Kočvara, Stingl
(Show Context)
Citation Context ... search interior-point approaches; Ipopt uses a filter globalization and includes a feasibility restoration phase. Mosek [1] is a primal-dual interior-point method for convex optimization, and Pennon =-=[25]-=- follows an augmented Lagrangian approach. 2sCG INTERIOR−POINT DIRECT KNITRO crossover ACTIVE−SET Figure 1: The main algorithmic options in the Knitro 5.0 package. New active-set methods based on Sequ... |