Connectedness for general continuous convex data

Determine whether the weak Pareto solution set is connected for every multi-objective optimization problem with a nonempty closed convex feasible set and continuous convex objective functions, without assuming that the data are polynomial.

Background

The main theorem establishes connectedness of the weak Pareto solution set when the objectives and constraints are convex polynomials, and path-connectedness follows from semi-algebraicity. The paper notes that most of the connectedness argument applies to arbitrary continuous convex data, but polynomial structure is used for recession-direction properties, attainment of scalarized minima, convexity of the scalarization domain, and semi-algebraicity.

Consequently, it remains unresolved whether connectedness persists when the feasible set is merely closed and convex and the objective functions are continuous and convex, with no polynomial assumption. The paper emphasizes that no explicit disconnected weak Pareto solution set is known in this general setting, while its theorem rules out such examples for convex polynomial data.

References

The second is whether $\Solw(K,f)$ is still connected in the general convex case, with $K$ closed convex and the $f_i$ convex and continuous but not necessarily polynomial: no explicit example of a disconnected weak Pareto solution set is known Remark~6.1, and Theorem~\ref{thm:main} shows that no such example can have convex polynomial data.

— 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,” immediately after Remark “weak versus Pareto”