A deterministic algorithm for stochastic multistage problems or - PowerPoint PPT Presentation

C ( u ) u ¯ C ( u ) = max µ + ⟨ λ, u ⟩ µ ∈ R ,λ ∈ R n s . t . µ + ⟨ λ, ˆ u ⟩ ≤ G (ˆ u ) , ∀ ˆ u, || λ || ∗ ≤ α. The upper bound function | Construction

C ( u ) u ¯ C ( u ) = max µ + ⟨ λ, u ⟩ µ ∈ R ,λ ∈ R n s . t . µ + ⟨ λ, ˆ u ⟩ ≤ C (ˆ u ) , ∀ ˆ u, || λ || ∗ ≤ α. The upper bound function | Construction

Example 1 2 C 1 ( u ) + 1 min 2 C 2 ( u ) u ∈U The problem-child algorithm | Two-stage problem

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

1 2 C 1 + 1 2 C 2 C 2 C 1

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

The problem-child algorithm | Two-stage problem

Let’s fjx some notation: ∑ V n ( x n ) = min C n ( x n , u n ) + p ( n, m ) V m ( x m ) x m ,u n m ∈ R ( n ) s . t . x m = f m ( x n , u n ) , ∀ m ∈ R ( n ) , x m ∈ X m , ∀ m ∈ R ( n ) , u n ∈ U n ( x n ) . We want to compute V 0 ( x 0 ) . The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem

The problem-child algorithm | The multistage problem