Connectedness of Pareto solution sets for convex quadratic problems

Determine whether a convex quadratic multi-objective optimization problem can have a disconnected Pareto solution set.

Background

The paper proves connectedness and path-connectedness for weak Pareto solution sets of multi-objective problems with convex polynomial objectives and convex polynomial constraints, without requiring boundedness of the feasible set or a constraint qualification. However, the argument does not extend to the Pareto solution set: only inclusions between scalarization solution sets, the Pareto set, and the weak Pareto set are available.

The unresolved problem is identified as the Pareto half of Question 9.8 in Yen (2012), which asks whether a convex quadratic multi-objective optimization problem can have a disconnected solution set. The paper resolves the weak Pareto half negatively but leaves the Pareto case open.

References

Two questions accordingly remain open. The first is the Pareto half of Question~9.8; see Remark~\ref{rmk:pareto}.

— A connectedness result for convex polynomial multi-objective optimization without boundedness  (2610.01029 - Hieu, 1 Oct 2026) in Section 3, subsection “Scope of the hypotheses,” Remark “weak versus Pareto”