## Improved and simplified validation of feasible points: Inequality and equality constrained problems (2005)

Venue: | Mathematical Programming, submitted |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Kearfott05improvedand,

author = {R. Baker Kearfott},

title = {Improved and simplified validation of feasible points: Inequality and equality constrained problems},

booktitle = {Mathematical Programming, submitted},

year = {2005}

}

### OpenURL

### Abstract

Abstract. In validated branch and bound algorithms for global optimization, upper bounds on the global optimum are obtained by evaluating the objective at an approximate optimizer; the upper bounds are then used to eliminate subregions of the search space. For constrained optimization, in general, a small region must be constructed within which existence of a feasible point can be proven, and an upper bound on the objective over that region is obtained. We had previously proposed a perturbation technique for constructing such a region. In this work, we propose a much simplified and improved technique, based on an orthogonal decomposition of the normal space to the constraints. In purely inequality constrained problems, a point, rather than a region, can be used, and, for equality and inequality constrained problems, the region lies in a smaller-dimensional subspace, giving rise to sharper upper bounds. Numerical experiments on published test sets for global optimization provide evidence of the superiority of the new approach within our GlobSol environment. 1.

### Citations

310 |
Rigorous Global Search: Continuous Problems
- Kearfott
- 1996
(Show Context)
Citation Context ... we can still attempt to apply our techniques. Let A (ι) ∈ R n×mi be that matrix whose j-th column is the approximate gradient ∇gij (ˇx), 1 ≤ j ≤ mi, and form a QR factorization A (ι) = Q (ι) R (ι) . =-=(2)-=- We then use the following algorithm to produce a direction v at ˇx that is likely to point into the feasible region. Algorithm 1 (Producing a direction into the feasible region) INPUT: The matrix Q (... |

25 | T.: A comparison of complete global optimization solvers
- Neumaier, Shcherbina, et al.
- 2005
(Show Context)
Citation Context ... all solutions have been found, and part of the difficulty has been in validation of feasible points. For a more systematic test, we used those problems from the “Tiny 1” Neumaier–Shcherbina test set =-=[7]-=- for which we both had correct Fortran 90 input files 9 and such that we had already implemented rigorous linear relaxations in GlobSol for the standard functions that occurred. For testing Algorithm ... |

18 | An SQP Algorithm for Finely Discretized Continuous Minimax Problems and Other Minimax Problems With Many Objective Functions
- Zhou, Tits
- 1996
(Show Context)
Citation Context ...cal parameters and configuration, except in the first set of runs, we did not use Algorithm 2 whereas we did in the second set of runs. As an initial test, we used the problem designated as “OET5” in =-=[10]-=- and which we formulated as Equation 4 in [4] using Lemaréchal’s conditions. Although approximate optimizers easily find Kuhn–Tucker points for this problem and although the BARON global optimization ... |

11 |
Rigorous global search: industrial applications
- Corliss, Kearfott
- 1999
(Show Context)
Citation Context ...hin which an interval Newton method can prove existence of feasible points. Since then, we have obtained significant additional experience with this technique within our GlobSol (see [4, §2], [6] and =-=[1]-=-) algorithmic environment. In particular, we have found that, for a significant number of problems, the interval Newton method fails to prove existence of a feasible point within x, or the widths of t... |

9 | On proving existence of feasible points in equality constrained optimization problems
- Kearfott
(Show Context)
Citation Context ...mate optimizer ˇx. However, because of likely singularity of this system, this technique is likely to fail; see the discussion in [5, §2] to gain insight into the case where the NLP (1) is linear. In =-=[3]-=-, we proposed and provided test results for a technique for perturbing the approximate optimizing point ˇx and by constructing a box about the perturbed feasible point within which an interval Newton ... |

8 |
2004, ‘Empirical Comparisons of Linear Relaxations and Alternate Techniques in Validated Deterministic Global Optimization’. preprint, http://interval.louisiana.edu/preprints/ validated global optimization search comparisons.pdf
- Kearfott
- 2004
(Show Context)
Citation Context ...ove, except that we place the columns of A (ι) first, then form a QR decomposition of the resulting matrix. That is, we form Ã = [A (ι) , A (eq) ] = ˜ Q ˜ R = [ ˜ Q (ι) , ˜ Q (eq) , ˜ Q (null) ] ˜ R. =-=(4)-=-sImproved and Simplified Validation of Feasible Points 11 As with Q (eq) in Equation 3, the columns of ˜ Q (eq) form a basis for the space spanned by the columns of A (eq) , except that, here, the col... |

6 |
Homepage of IPOPT
- Wächter
- 2002
(Show Context)
Citation Context ...er of times a perturbed point ˆx from Algorithm 2 was proven to be feasible. 9 There were some difficulties with the conversion process from AMPL format, for some files. 10 We used a version of IPOPT =-=[9]-=- dating from early 2003 as a floating point optimizer.s8 R. Baker Kearfott N¬v is the number of times a point ˆx was found in Algorithm 1 but could not be proven to be feasible in the interval evaluat... |

5 |
interactive systems for verified computations: Four case studies
- Libraries
- 2004
(Show Context)
Citation Context ...oint within which an interval Newton method can prove existence of feasible points. Since then, we have obtained significant additional experience with this technique within our GlobSol (see [4, §2], =-=[6]-=- and [1]) algorithmic environment. In particular, we have found that, for a significant number of problems, the interval Newton method fails to prove existence of a feasible point within x, or the wid... |

1 |
Validated probing with linear relaxations, 2004. submitted to the Journal of Global Optimization, http://interval.louisiana.edu/preprints/ 2004 probing and Lagrange multipliers.pdf
- Kearfott
(Show Context)
Citation Context ...fine ˜c : Rn → Rme to be the function whose i-th component is the i-th equality constraint that has been identified as active, and form f : Rmi mi → R by � � f(u) = ˜c me � ˆx + i=1 ui ˜ Q (eq) :,i , =-=(5)-=- where ˆx will have been previously obtained by perturbing an approximate optimizing point using Algorithm 1, but with Q (ι) computed from Equation (3). The algorithm for validating existence of a fea... |