Initial cut set (Step 0)

Definition

StochasticProg_MultistageV2_Cuts_empty

by mikedeng1 · Oct 5, 2026 · Mathlib 0df444a (Lean v4.33.1)

The initial state of the nested L-shaped method (Step 0): $r^t_k=s^t_k=0$ for all $t,k$ , i.e. no feasibility and no optimality cuts at any node.

Definition code

import Mathlib
import Definitions.Def_StochasticProg_Multistage_Tree
import Definitions.Def_StochasticProg_MultistageV2_Cuts

namespace StochasticProg.MultistageV2

/-- The initial state of Step 0 (p. 267): `r^t_k = s^t_k = 0`, no feasibility or
optimality cuts at any node (so every NLDS(t,k) carries `θ^t_k = 0`). -/
def Cuts.empty {H : ℕ} (T : Multistage.Tree H) (n : ℕ) : Cuts T n :=
  ⟨fun _ => ∅, fun _ => ∅⟩

end StochasticProg.MultistageV2

Source

Birge & Louveaux, Introduction to Stochastic Programming, 2nd ed. (2011), Ch. 6, §6.1, Step 0, p. 267 (PDF p. 288)

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