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## Feasible Interior Methods Using Slacks for Nonlinear Optimization (2002)

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Venue: | Computational Optimization and Applications |

Citations: | 20 - 2 self |

### Citations

437 |
Nonlinear Programming, Sequential Unconstrained Minimization Techniques
- Fiacco, McCormick
- 1968
(Show Context)
Citation Context ...hods that satisfy some or all of the inequality constraints by resetting slack variables. The relationship between these slack-based feasible methods and the classical methods of Fiacco and McCormick =-=[11]-=- is discussed. In section 4 we describe potential di culties with our strategy when equality constraints are present, and provide guidelines to deal with them. A concrete procedure for modifying the s... |

193 | An interior-point algorithm for nonconvex nonlinear programming.
- Vanderbei, Shanno
- 1999
(Show Context)
Citation Context ...phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms =-=[5, 10, 20, 23]-=- whichmayenter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack-based infeasible methods into feasible methods. In this... |

153 | A trust region method based on interior point techniques for nonlinear programming
- Byrd, Gilbert, et al.
(Show Context)
Citation Context ... case in methods that handle constraints by a trust region method. To illustrate this, we now present an example that occurs with a feasible version of the algorithm implemented in the KNITRO package =-=[4, 5]-=-. Example 1. Consider the problem GAUSSELM from the CUTE collection [2]. One version of this problem has 385 nonlinear equality constraints along with 750 linear inequality constraints and some bounds... |

130 |
CUTEr, a Constrained and Unconstrained Test Environment, revisited. http://cuter.rl.ac.uk/cuter-www
- Toint
(Show Context)
Citation Context ...strate this, we now present an example that occurs with a feasible version of the algorithm implemented in the KNITRO package [4, 5]. Example 1. Consider the problem GAUSSELM from the CUTE collection =-=[2]-=-. One version of this problem has 385 nonlinear equality constraints along with 750 linear inequality constraints and some bounds on the variables. Using an initial point whichwas feasible with respec... |

105 | On the formulation and theory of the Newton interior–point method for nonlinear programming
- EL-BAKRY, TAPIA, et al.
- 1996
(Show Context)
Citation Context ...phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms =-=[5, 10, 20, 23]-=- whichmayenter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack-based infeasible methods into feasible methods. In this... |

91 |
Trust-region methods.
- Toint
- 2000
(Show Context)
Citation Context ...ain g = G ;1 (x)e� rf(x) ; Ah h ; Ag g = 0 h(x) = 0 (3.8) ; e + G(x) g = 0: Applying Newton's method (in the variables, x� h� g) to this system gives a primal-dual interior method for (3:7)� s=-=ee e.g. [8]-=-. 5sTo study the relationship between this method and Algorithm Feas-Reset, let us consider a slack-based, feasible method for solving (2:3) that computes steps by applying Newton's method in the vari... |

88 | An interior point algorithm for large scale nonlinear programming
- Byrd, Hribar, et al.
- 1999
(Show Context)
Citation Context ...phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programming are based, however, on infeasible algorithms =-=[5, 10, 20, 23]-=- whichmayenter and leave the feasible region during the course of the minimization. In this paper we describe a framework for transforming slack-based infeasible methods into feasible methods. In this... |

88 | A trust region strategy for nonlinear equality constrained optimization - Celis, Dennis, et al. - 1985 |

75 | Primal-dual interior methods for nonconvex nonlinear programming
- FORSGREN, GILL
- 1998
(Show Context)
Citation Context ...recent approaches [16] aim at decreasing the cost per iteration. Interior point approaches provide a natural framework for deriving feasible methods for nonlinear programming. The methods proposed in =-=[1, 7, 12, 13]-=- either start with a feasible point or apply a phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programmin... |

71 | A trust region algorithm for equality constrained optimization
- Powell, Yuan
- 1991
(Show Context)
Citation Context ...g that the constraints (4:19b)-(4:19d) are compatible, and that the step makes su cient progress towards achieving feasibility. The methods of Vardi [21], Celis, Dennis and Tapia [6], Powell and Yuan =-=[19]-=-, Byrd and Omojokun [3, 17], and Yamashita and Yabe [22] can be described in this framework. If there is a step d satisfying (4:19b)-(4:19d) for rh = rg = 0, then these trust region methods will usual... |

69 |
A primal-dual interior method for nonconvex nonlinear programming
- GAY, OVERTON, et al.
- 1998
(Show Context)
Citation Context ...recent approaches [16] aim at decreasing the cost per iteration. Interior point approaches provide a natural framework for deriving feasible methods for nonlinear programming. The methods proposed in =-=[1, 7, 12, 13]-=- either start with a feasible point or apply a phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programmin... |

67 |
A Hybrid Method for Nonlinear Equations,"
- Powell
- 1970
(Show Context)
Citation Context ...h = h(x)+Ah(x) T x� rg = g(x) ; s + Ag(x) T x ; s: (5.22) To determine whether rg is set to zero when g(x) ; s = 0, we note that in KNITRO problem (5:21) is solved approximately using a dogleg metho=-=d [18]-=-: is a linear combination of a Newton step N and a Cauchy direction. Speci cally, it is the minimizer of (5:21a) along the piecewise linear path from 0 to C to N subject to the trust region constraint... |

55 | A Computationally Efficient Feasible Sequential Quadratic Programming Algorithm." - Lawrence, Tits - 2001 |

50 |
Trust region algorithms for optimization with nonlinear equality and inequality constraints
- Omojokun
- 1989
(Show Context)
Citation Context ...4:19b)-(4:19d) are compatible, and that the step makes su cient progress towards achieving feasibility. The methods of Vardi [21], Celis, Dennis and Tapia [6], Powell and Yuan [19], Byrd and Omojokun =-=[3, 17]-=-, and Yamashita and Yabe [22] can be described in this framework. If there is a step d satisfying (4:19b)-(4:19d) for rh = rg = 0, then these trust region methods will usually de ne rh and rg to be ze... |

44 |
A trust region algorithm for equality constrained minimization: convergence properties and implementation
- Vardi
- 1985
(Show Context)
Citation Context ...nd rg are chosen with the dual objective of ensuring that the constraints (4:19b)-(4:19d) are compatible, and that the step makes su cient progress towards achieving feasibility. The methods of Vardi =-=[21]-=-, Celis, Dennis and Tapia [6], Powell and Yuan [19], Byrd and Omojokun [3, 17], and Yamashita and Yabe [22] can be described in this framework. If there is a step d satisfying (4:19b)-(4:19d) for rh =... |

37 |
Superlinear and quadratic convergence of some primal-dual interior point methods for constrained optimization
- YAMASHITA, YABE
- 1996
(Show Context)
Citation Context ...nd that the step makes su cient progress towards achieving feasibility. The methods of Vardi [21], Celis, Dennis and Tapia [6], Powell and Yuan [19], Byrd and Omojokun [3, 17], and Yamashita and Yabe =-=[22]-=- can be described in this framework. If there is a step d satisfying (4:19b)-(4:19d) for rh = rg = 0, then these trust region methods will usually de ne rh and rg to be zero. In particular, these meth... |

26 |
A primal-dual trust-region algorithm for non-convex nonlinear programming
- Toint
(Show Context)
Citation Context ...recent approaches [16] aim at decreasing the cost per iteration. Interior point approaches provide a natural framework for deriving feasible methods for nonlinear programming. The methods proposed in =-=[1, 7, 12, 13]-=- either start with a feasible point or apply a phase-one procedure to compute one, and then generate strictly feasible iterates. Most other implementations of interior methods for nonlinear programmin... |

24 |
A globally and superlinearly convergent primal-dual interior point trust region method for large scale constrained optimization.
- Yamashita, Yabe, et al.
- 2005
(Show Context)
Citation Context |

22 |
Robust trust region methods for constrained optimization
- Byrd
- 1987
(Show Context)
Citation Context ...4:19b)-(4:19d) are compatible, and that the step makes su cient progress towards achieving feasibility. The methods of Vardi [21], Celis, Dennis and Tapia [6], Powell and Yuan [19], Byrd and Omojokun =-=[3, 17]-=-, and Yamashita and Yabe [22] can be described in this framework. If there is a step d satisfying (4:19b)-(4:19d) for rh = rg = 0, then these trust region methods will usually de ne rh and rg to be ze... |

19 | Nonlinear equality constraints in feasible sequential quadratic programming
- Lawrence, Tits
- 1996
(Show Context)
Citation Context ...er instances one may want to terminate an algorithm before optimality has been reached and be assured that the current approximate solution is feasible. Various feasible active set methods (see, e.g. =-=[15]-=-) have been developed by including de ections in the search directions which ensure that the total step points towards the interior of the feasible region. They typically require the solution of two o... |

16 | A feasible BFGS interior point algorithm for solving strongly convex minimization problems
- Armand, Gilbert, et al.
- 2000
(Show Context)
Citation Context |

9 |
A primal-dual algorithm for minimizing a nonconvex function subject to bound and linear equality constraints
- Toint
- 1996
(Show Context)
Citation Context ...s x� h� g applied to (3:8). Therefore, a primal-dual step for (3:7) is equivalent to a primal-dual step for (2:3), when the reset (3:6) is applied. This equivalence does not hold for interior meth=-=ods [5, 9, 23]-=- in which thestepd is only an approximate solution of (3:9), or is computed as the solution of a related problem with a trust region. However, as the iterates of those methods approach a solution, the... |

9 |
A catalogue of subroutines (HSL
- Library, Technology, et al.
- 2000
(Show Context)
Citation Context ...ctures and variables { and only dier in the two instructions enclosed in boxes. Our numerical experience indicates that the cost of solving the larger system (in (dx; ds)) using the HSL routine MA27 =-=[14]-=- is not signicantly larger (if at all) than solving a reduced system (in dx only). 5Making the substitution (3:6) has the eect of replacing ln(si) with ln(gi(x)) in the merit function, which is the ... |

4 |
Subroutine Library. Acatalogue of subroutines (HSL
- Harwell
- 2000
(Show Context)
Citation Context ...uctures and variables { and only di er in the two instructions enclosed in boxes. Our numerical experience indicates that the cost of solving the larger system (in (dx�ds)) using the HSL routine MA2=-=7 [14]-=- is not signi cantly larger (if at all) than solving a reduced system (in dx only). 4sMaking the substitution (3:6) has the e ect of replacing ln(si) with ln(gi(x)) in the merit function, which is the... |

3 |
A computationally ecient feasible sequential quadratic programming algorithm
- Lawrence, Tits
(Show Context)
Citation Context ...e that the total step points towards the interior of the feasible region. They typically require the solution of two or more linear systems of equations per iteration, although some recent approaches =-=[16]-=- aim at decreasing the cost per iteration. Interior point approaches provide a natural framework for deriving feasible methods for nonlinear programming. The methods proposed in [1, 7, 12, 13] either ... |

1 |
A computationally e cient feasible sequential quadratic programming algorithm
- Lawrence, Tits
- 1998
(Show Context)
Citation Context ...e that the total step points towards the interior of the feasible region. They typically require the solution of two or more linear systems of equations per iteration, although some recent approaches =-=[16]-=- aim at decreasing the cost per iteration. Interior point approaches provide a natural framework for deriving feasible methods for nonlinear programming. The methods proposed in [1, 7, 12, 13] either ... |