A conjugate directions-type procedure for quadratic multiobjective optimization

Abstract

We propose an extension of the real-valued conjugate directions method for unconstrained quadratic multiobjective problems. As in the single-valued counterpart, the procedure requires a set of directions that are simultaneously conjugate with respect to the positive definite matrices of all quadratic objective components. Likewise, the multicriteria version computes the steplength by means of the unconstrained minimization of a single-variable strongly convex function at each iteration. When it is implemented with a weakly-increasing (strongly-increasing) auxiliary function, the scheme produces weak Pareto (Pareto) optima in finitely many iterations.

Keywords:

• Multiobjective optimization
• weak Pareto optimality
• Pareto optimality
• conjugate directions method

Acknowledgments

We would like to thank the anonymous referees for their suggestions, which improved the original version of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We point out that in [5, Subsection 3.2] there are examples of w- or s-increasing continuous auxiliary functions Φ such that $\mathrm{\Phi }\circ f$ is strongly convex. For instance, $\mathrm{\Phi }\left(u\right)=\underset{i=1,\dots ,m}{max}\left\{u_i\right\}$ is a w-increasing continuous function such that $\mathrm{\Phi }\circ f$ is strongly convex.

Additional information

Funding

The first author received support by the Grant-in-Aid for Scientific Research (C) (19K11840) from Japan Society for the Promotion of Science. The second author received support from FAPERJ/CNPq through PRONEX-Optimization E-26/010.001247/2016. The third author received support by a PSC-CUNY grant (611157-00 49), jointly funded by The Professional Staff Congress and The City University of New York. This paper was completed in part with support from A Room of One's Own.