Convergent upper bounds in global minimization with nonlinear equality constraints

Füllner, Christian; Kirst, Peter; Stein, Oliver

doi:10.1007/s10107-020-01493-2

Cited by 2 publications

(1 citation statement)

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“…For that reason, so-called upper bounding procedures have been developed that ensure Assumption 8.3 and, thus, termination of a branch-and-bound method by computing a convergent sequence of upper bounds, as for instance proposed in [20] for the purely inequality constrained case. For problems that also involve equality constraints we refer to [14]. These techniques can be adapted to multiobjective problems which is, however, beyond the scope of the present paper.…”

Section: Lemma 82mentioning

confidence: 99%

A general branch-and-bound framework for continuous global multiobjective optimization

et al. 2021

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Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.

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Section: Lemma 82mentioning

confidence: 99%

A general branch-and-bound framework for continuous global multiobjective optimization

et al. 2021

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Feasibility Verification and Upper Bound Computation in Global Minimization Using Approximate Active Index Sets

Füllner,

Kirst,

Otto

et al. 2024

INFORMS Journal on Computing

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We propose a new upper bounding procedure for global minimization problems with continuous variables and possibly nonconvex inequality and equality constraints. Upper bounds are crucial for standard termination criteria of spatial branch-and-bound (SBB) algorithms to ensure that they can enclose globally minimal values sufficiently well. However, whereas for most lower bounding procedures from the literature, convergence on smaller boxes is established, this does not hold for several methods to compute upper bounds even though they often perform well in practice. In contrast, our emphasis is on the convergence. We present a new approach to verify the existence of feasible points on boxes, on which upper bounds can then be determined. To this end, we resort to existing convergent feasibility verification approaches for purely equality and box constrained problems. By considering carefully designed modifications of subproblems based on the approximation of active index sets, we enhance such methods to problems with additional inequality constraints. We prove that our new upper bounding procedure finds sufficiently good upper bounds so that termination of SBB algorithms is guaranteed after a finite number of iterations. Our theoretical findings are illustrated by computational results on a large number of standard test problems. These results show that compared with interval Newton methods from the literature, our proposed method is more successful in feasibility verification for both, a full SBB implementation (42 instead of 26 test problems) and exhaustive sequences of boxes around known feasible points (120 instead of 29 test problems). History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0162 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0162 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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Convergent upper bounds in global minimization with nonlinear equality constraints

Cited by 2 publications

References 47 publications

A general branch-and-bound framework for continuous global multiobjective optimization

A general branch-and-bound framework for continuous global multiobjective optimization

Feasibility Verification and Upper Bound Computation in Global Minimization Using Approximate Active Index Sets

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