An upper bound on the Hausdorff distance between a Pareto set and its discretization in bi-objective convex quadratic optimization. Ondes, B. E. & Hunter, S. R. *Optimization Letters*, 17:45–74, 2023. Expand abstract below for update.Link Paper doi abstract bibtex We provide upper bounds on the Hausdorff distances between the efficient set and its discretization in the decision space, and between the Pareto set (also called the Pareto front) and its discretization in the objective space, in the context of bi-objective convex quadratic optimization on a compact feasible set. Our results imply that if $t$ is the dispersion of the sampled points in the discretized feasible set, then the Hausdorff distances in both the decision space and the objective space are $O(\sqrt{t})$ as $t$ decreases to zero. Update on May 31, 2023: There appears to be an error in the statement of Lemma 1. We think the respective upper bounds in the decision and objective spaces should be $t\sqrt{κ^*}$ and $h(t\sqrt{κ^*})$. The largest effect of this error would appear in Section 3 and, in particular, Lemma 3, where $t$ would be missing the multiplier of $\sqrt{κ^*}$ nearly everywhere. All results for $κ^*=1$ and the big-O results in the statement of the main theorem are unaffected.

@article{2023ondhun,
Year = {2023},
Author = {B. E. Ondes and S. R. Hunter},
Title = {An upper bound on the {Hausdorff} distance between a {Pareto} set and its discretization in bi-objective convex quadratic optimization},
journal = {Optimization Letters},
volume = {17},
pages = {45--74},
doi = {10.1007/s11590-022-01920-7},
url_Link = {https://rdcu.be/cWCJh},
url_Paper = {http://web.ics.purdue.edu/~hunter63/PAPERS/pre2022ondhun.pdf},
bibbase_note = {<span style="color: green">Expand abstract below for update.</span>},
abstract = {We provide upper bounds on the Hausdorff distances between the efficient set and its discretization in the decision space, and between the Pareto set (also called the Pareto front) and its discretization in the objective space, in the context of bi-objective convex quadratic optimization on a compact feasible set. Our results imply that if $t$ is the dispersion of the sampled points in the discretized feasible set, then the Hausdorff distances in both the decision space and the objective space are $O(\sqrt{t})$ as $t$ decreases to zero. Update on May 31, 2023: There appears to be an error in the statement of Lemma 1. We think the respective upper bounds in the decision and objective spaces should be $t\sqrt{\kappa^*}$ and $h(t\sqrt{\kappa^*})$. The largest effect of this error would appear in Section 3 and, in particular, Lemma 3, where $t$ would be missing the multiplier of $\sqrt{\kappa^*}$ nearly everywhere. All results for $\kappa^*=1$ and the big-O results in the statement of the main theorem are unaffected.},
keywords = {optimization > multi-objective > continuous}}

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