Keyboard shortcuts

Press ← or → to navigate between chapters

Press ? to show this help

Press Esc to hide this help

Hieu’s Weak Pareto Connectedness Refutation

Abstract

Continuous convex objectives on a closed convex strip can have a disconnected weak Pareto solution set.

Definition 1.1 (Weak Pareto solutions).

Formalization. D5/S3/Analytic/Convexity/WeakParetoDisconnected.weakPareto (✓ std3).

Citation. Vu Trung Hieu (2026). A connectedness result for convex polynomial multi-objective optimization without boundedness. URL: https://arxiv.org/abs/2610.01029v1.

Commentary.

In the Euclidean specialization E=EuclideanSpace ℝ (Fin n), weakPareto K f is a set of points in E. Membership means that the point is feasible and no feasible point is strictly smaller in every objective.

Definition 1.2 (Connectedness for continuous convex data).

Formalization. D5/S3/Analytic/Convexity/WeakParetoDisconnected.claim (✓ std3).

Citation. Vu Trung Hieu (2026). A connectedness result for convex polynomial multi-objective optimization without boundedness. URL: https://arxiv.org/abs/2610.01029v1.

Commentary.

Section 4.3 asks verbatim: “whether Sol^w(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”. The paper adds: “no explicit example of a disconnected weak Pareto solution set is known” and says that Theorem 1 “shows that no such example can have convex polynomial data”. The formal claim quantifies over nonempty closed convex feasible sets in finite-dimensional Euclidean spaces of dimension at least one, at least one objective, continuous convex objectives, and preconnected weak Pareto solutions.

Theorem 1.3 (A disconnected weak Pareto solution set).

Proof. Machine-checked in Lean as D5/S3/Analytic/Convexity/WeakParetoDisconnected.result (✓ std3). ∎

Resolves. Problems/hieu-2026-weak-pareto-connectedness (refuted) by D5/S3/Analytic/Convexity/WeakParetoDisconnected.result.

Source. Repository-derived.

Acknowledgement. Vu Trung Hieu (2026). A connectedness result for convex polynomial multi-objective optimization without boundedness. URL: https://arxiv.org/abs/2610.01029v1.

Commentary.

Take K=[0,∞)×[0,2], h(x)=√(x²+1)−x, f₁(x,y)=y², and f₂(x,y)=(max(h(x)−y/2,0))²−y/2. The Lean proof establishes nonemptiness, closedness and convexity of K, continuity and convexity of both objectives, weak Pareto boundary pieces y=0 and y≥2h(x), strict domination throughout 0<y<2h(x), and separation by the open sets y<h(x) and y>h(x). Thus the universal connectedness claim is false without boundedness or a constraint qualification.

References