Technical-condition question for asymptotic optimality in two-stage sample robust optimization

The problem asks for a precise condition ensuring asymptotic optimality in two-stage sample robust optimization. The underlying 2019 work established convergence under assumptions including a feasibility condition, but left that condition only partly characterized.

Known results

• Theorem $2$: under assumptions $[A1]$--$[A4]$, optimal costs and first-stage decisions converge almost surely (Bertsimas, Shtern, and Sturt, 2019).
• Proposition $2$: a common linear decision rule gives a polynomial-time sufficient test for $[A4]$, but the condition is not necessary (Bertsimas, Shtern, and Sturt, 2019).
• An example satisfies $[A4]$ despite having no feasible linear decision rule (Bertsimas et al., 2019).

August 24, 2026 characterization

An arXiv report claims that the technical condition is characterized by a simple-versus-nonsimple support dichotomy and supplies a polynomial-time vertex-certification algorithm. It further claims that failure for nonsimple supports is existential rather than universal. This claimed resolution remains unverified.

Current status (as of August 2026): The earlier sufficient and nonnecessary criteria are established, while the claimed complete support-dichotomy characterization and its algorithm remain unverified.