## A Merit Function for Inequality Constrained Nonlinear Programming Problems (1993)

Venue: | Internal Report 4702, National Institute of Standards and Technology |

Citations: | 5 - 4 self |

### BibTeX

@INPROCEEDINGS{Boggs93amerit,

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

title = {A Merit Function for Inequality Constrained Nonlinear Programming Problems},

booktitle = {Internal Report 4702, National Institute of Standards and Technology},

year = {1993}

}

### OpenURL

### Abstract

We consider the use of the sequential quadratic programming (SQP) technique for solving the inequality constrained minimization problem min x f(x) subject to: g i (x) 0; i = 1; : : : ; m: SQP methods require the use of an auxiliary function, called a merit function or line-search function, for assessing the steps that are generated. We derive a merit function by adding slack variables to create an equality constrained problem and then using the merit function developed earlier by the authors for the equality constrained case. We stress that we do not solve the slack variable problem, but only use it to construct the merit function. The resulting function is simplified in a certain way that leads to an effective procedure for updating the squares of the slack variables. A globally convergent algorithm, based on this merit function, is suggested, and is demonstrated to be effective in practice. Contribution of the National Institute of Standards and Technology and not subject to copyr...

### Citations

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(Show Context)
Citation Context ... program is infeasible. February 10,sNumerical Results 19 4. Numerical Results The algorithm described in this paper was implemented in Fortran and tested on a selection of problems from [HocS81] and =-=[Sch87]-=-. The problems chosen were the higher dimensional ones with a large number of nonlinear inequality constraints. (The low dimensional problems converged too quickly to be of much interest.) Because equ... |

133 |
A fast algorithm for nonlinearly constrained optimization calculations
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(Show Context)
Citation Context ...mportant to update the approximation of the Hessian of the Lagrangian at each iteration, while still maintaining positive definiteness. Accordingly we use the numerically proven damped BFGS update of =-=[Pow78]-=-, defined as follows. In the standard BFGS one uses the vectors s = x k+1 \Gamma x k y = r x L(x k+1 ;sk+1 ) \Gamma r x L(x k ;sk+1 ) in the update formula. Since s T y need not be positive, Powell re... |

38 |
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(Show Context)
Citation Context ...ns. It is well known that a strong local solution is isolated and is stable with respect to small perturbations in the data of the problem. It can also be inferred, following the lines of argument in =-=[Rob74]-=- that if fx k g is a sequence converging to x , a strong local solution to (1.1), and fB k g is a sequence of uniformly bounded positive definite matrices, then the active constraints of (1.1) at x an... |

36 | On the Convergence of a Sequential Quadratic Programming Method with an Augmented Lagrangian Line Search Function
- Schittkowski
- 1982
(Show Context)
Citation Context ...ifferentiability of the merit function on the boundary of the feasible region. Work that has been directed to, or encompasses, the inequality-constrained problem includes that of [Han77], [ChaLLP82], =-=[Sch83], and [Gil-=-MSW86]. Han, in his initial papers on the SQP methods employed an l 1 penalty function for his merit function. Chamberlain et al. [ChaLLP82] developed their "watchdog" technique to correct a... |

24 |
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- 1978
(Show Context)
Citation Context ...ry 10,s20 Numerical Results A pre-update scaling strategy suggested for unconstrained optimization [OreS76] is used. This pre-update scaling is only performed on the initial iteration as suggested by =-=[ShaP78]. The ma-=-trix B 0 is set to I and the first step is computed; s and y are calculated. Before the above updating procedure is used, however, B 0 is set to ��I where �� = ( s T y=s T s if s T y ? 0 1 oth... |

24 | Diagonalized multiplier methods and quasi-Newton methods for constrained optimization - Tapia - 1977 |

16 |
Optimal conditioning of self-scaling and variable metric algorithms
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(Show Context)
Citation Context ...g schemes exist, e.g., [Tap88], but because of their local nature, they are not employed here. February 10,s20 Numerical Results A pre-update scaling strategy suggested for unconstrained optimization =-=[OreS76]-=- is used. This pre-update scaling is only performed on the initial iteration as suggested by [ShaP78]. The matrix B 0 is set to I and the first step is computed; s and y are calculated. Before the abo... |

13 |
A recursive quadratic programming algorithm that uses differentiable penalty functions
- Powell, Yuan
- 1986
(Show Context)
Citation Context ...tion into their NPSOL program. Our work on equality constrained problems employs an exact penalty function originally due to Fletcher [Fle72], but derived differently. (See [BogT84].) Powell and Yuan =-=[PowY86]-=- use a similar function. While both approaches are effective, Shanno and Phua [ShaP89] have studied the class and prefer our method in numerical computations. The merit function proposed here differs ... |

8 |
On secant updates for use in general constrained optimization
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- 1988
(Show Context)
Citation Context ...is chosen as ` = ( 1 if s T ysis T B k s (1 \Gamma i)s T B k s=(s T B k s \Gamma s T y) otherwise. and the value of i was chosen to be (.1). More theoretically justified updating schemes exist, e.g., =-=[Tap88]-=-, but because of their local nature, they are not employed here. February 10,s20 Numerical Results A pre-update scaling strategy suggested for unconstrained optimization [OreS76] is used. This pre-upd... |

4 | The performance of two subroutines for constrained optimization on some difficult test problems - Powell - 1985 |

1 |
Test examples for nonlinear programming codes
- unknown authors
- 1981
(Show Context)
Citation Context ...the quadratic program is infeasible. February 10,sNumerical Results 19 4. Numerical Results The algorithm described in this paper was implemented in Fortran and tested on a selection of problems from =-=[HocS81]-=- and [Sch87]. The problems chosen were the higher dimensional ones with a large number of nonlinear inequality constraints. (The low dimensional problems converged too quickly to be of much interest.)... |

1 |
Numerical experience with sequential quadratic programming algorithms for equality constrained nonlinear programming, acm Trans
- Shanno, Phua
- 1989
(Show Context)
Citation Context ...act penalty function originally due to Fletcher [Fle72], but derived differently. (See [BogT84].) Powell and Yuan [PowY86] use a similar function. While both approaches are effective, Shanno and Phua =-=[ShaP89]-=- have studied the class and prefer our method in numerical computations. The merit function proposed here differs from most, if not all, others in that it does not have an unconstrained local minimum ... |