## Abstract

Three rejection tests for multi-objective optimization problems based on first order optimality conditions are proposed. These tests can certify that a box does not contain any local minimizer, and thus it can be excluded from the search process. They generalize previously proposed rejection tests in several regards: Their scope include inequality and equality constrained smooth or nonsmooth multiple objective problems. Reported experiments show that they allow quite efficiently removing the cluster effect in mono-objective and multi-objective problems, which is one of the key issues in continuous global deterministic optimization.

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## Notes

In the case of vector valued functions \(f=(f_1, \ldots , f_m),\,\partial f\) is a matrix whose columns are \(\partial f_i\), so \(\partial f(x_*) \, \lambda =\sum \nolimits _i\lambda _i\partial f_i(x_*)\).

Clusters of small boxes appear around local or global minimizers due to excessive splitting and failure to remove the resulting boxes because too close to these minimizers. This behavior is generic and one of the main issues in deterministic global optimization.

These inequality constraints are only potentially active because interval extensions are generally pessimistic. All rejection tests proposed remain correct when a potentially active constraint is actually inactive, although they are more efficient as inactive constraints are more accurately detected.

Typically, an approximate generalized inverse of the midpoint of \(\mathbf{G}_*(\mathbf{x})\).

Note that \(\varSigma (\mathbf{A},Ce)\) is preconditioned so the Gauss–Seidel iteration needs solving only diagonal entries of \(\mathbf{A}\). On the other hand, \(\varSigma (\mathbf{G}_*,e)\) is not preconditioned so all entries of \(\mathbf{G}_*\) need to be solved. This can be efficiently performed using inner subtraction.

Although not noted in [31], the angle between two gradients interval evaluations \(\mathbf{g}_1\) and \(\mathbf{g}_2\) can be proved not to contain \(\pi \) simply by checking that the scalar product \(\mathbf{g}_1\,\mathbf{g}_2\) does not intersect \(||\mathbf{g}_1||\,||\mathbf{g}_2||\). This is sufficient for rejecting the box, and easily computed for arbitrary dimensions.

The asymptotic analyses of the cluster effect provided in [6], in the context of unconstrained optimization, or in [34], in the context of a system of equations, lead to different models that do not hold here. In particular both [6] and [34] consider some pessimistic interval evaluations, while this academic problem suffers from the cluster effect in spite of exact interval evaluations of its objective function and constraints.

Experiments non reported here have shown that the constraint propagation can remove the cluster effect when \(n=2\), although the closer to the optimum the slower the convergence of the propagation, converging to infinitely slow convergence (which requires very expensive constraint propagation). This is generic in two variables, but not in higher dimensions where the constraint propagation is not able anymore to remove the cluster effect.

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## Acknowledgments

This work was partially funded by the *Région Pays de la Loire* of France, and the National Institute of Informatics of Japan. The authors are much grateful to Christophe Jermann for his valuable help in experimenting and analyzing the cluster effect of the presented academic examples.

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Goldsztejn, A., Domes, F. & Chevalier, B. First order rejection tests for multiple-objective optimization.
*J Glob Optim* **58**, 653–672 (2014). https://doi.org/10.1007/s10898-013-0066-x

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DOI: https://doi.org/10.1007/s10898-013-0066-x