N. D. Yen’s weak Pareto connectedness question

Consider a convex multi-objective optimization problem $\min (f_1(x),\ldots,f_m(x))$ over $X=\{x\in\mathbb{R}^n:g_j(x)\le 0\ \text{for }j=1,\ldots,r\}$ , where the objective functions $f_i$ and constraint functions $g_j$ are convex. No boundedness assumption on $X$ and no constraint qualification are imposed. The weak Pareto solution set is $W=\{x\in X:\nexists y\in X\text{ such that }f_i(y)<f_i(x)\text{ for every }i=1,\ldots,m\}$ . Is $W$ necessarily connected whenever it is nonempty?

References

Primary source

arXiv

Additional references

A connectedness result for convex polynomial multi-objective optimization without boundedness — arXiv — Vu Trung Hieu

optimization real-algebraic-geometry

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new preprint claims the question is settled for convex polynomial optimization, but the broader problem remains open.

N. D. Yen’s question asks whether weak Pareto solution sets remain connected after dropping boundedness and constraint-qualification assumptions. The new result concerns convex polynomial data rather than arbitrary multi-objective optimization.

Known results

Hieu (2020): weak Pareto and Pareto solution sets are semi-algebraic, with finitely many connected, path-connected components, without closedness or constraint qualifications.
Lee and Yen: connectedness was known when the constraint set is bounded.
Hieu: for monotone vector variational inequalities, disconnected weak Pareto sets have only unbounded components; hence nonempty bounded sets are connected.

October 2026 convex-polynomial result

Vu Trung Hieu’s preprint claims connectedness of the weak Pareto set under convex polynomial assumptions, and therefore path-connectedness when it is nonempty. This is a claimed advance, not independently verified here, and does not settle the unrestricted question.

Current status (as of October 2026): The convex-polynomial weak Pareto case is claimed settled by Hieu’s preprint, while the broader question for arbitrary multi-objective problems remains open.

Sources

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