Split Cuts for Two-Stage Stochastic Integer Programs Merve Bodur 1 - PowerPoint PPT Presentation

Split Cuts for Two-Stage Stochastic Integer Programs Merve Bodur 1 Sanjeeb Dash 2 Oktay Günlük 2 Jim Luedtke 1 1 University of Wisconsin-Madison 2 IBM T.J. Watson Research Center Aussois 2014 Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 1 / 22

Two-stage stochastic programs General formulation Two-stage stochastic programs There are uncertainties in some parameters ( ω ) Make first stage decisions ( x ) ց Observe realizations ց Make second stage (recourse) decisions ( y ) Objective: min (1 st stage cost) + E [2 nd stage cost] Assumptions: Finitely many scenarios ( K ) First stage: integer Second stage: continuous Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 2 / 22

Two-stage stochastic programs General formulation Extensive form A (very) large scale mixed integer program. min c T x + � q T k y k k ∈K s.t. T k x + W k y k ≥ h k , ∀ k ∈ K Ax ≥ b x ∈ Z n + , y ∈ R tK + Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 3 / 22

Two-stage stochastic programs General formulation Extensive form Introduce variables to represent objective in each scenario min c T x + � z k k ∈K s.t. z k ≥ q T ∀ k ∈ K k y k , T k x + W k y k ≥ h k , ∀ k ∈ K Ax ≥ b x ∈ Z n + , y ∈ R tK + , z ∈ R K := { ( z k , x , y k ) ∈ R × R n × R t Q LP + : T k x + W k y k ≥ h k , z k ≥ q T k y k } k X := { x ∈ Z n + : Ax ≥ b } Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 3 / 22

Two-stage stochastic programs General formulation Extensive form min c T x + � z k k ∈K s.t. ( z k , x , y k ) ∈ Q LP , ∀ k ∈ K k x ∈ X := { ( z k , x , y k ) ∈ R × R n × R t Q LP + : T k x + W k y k ≥ h k , z k ≥ q T k y k } k X := { x ∈ Z n + : Ax ≥ b } Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 3 / 22

Two-stage stochastic programs Decomposition methods Benders Decomposition (L-Shaped Method) LB (MP) LP : min � z , x c T x + z k Master k ∈K s.t. x ∈ X LP Cuts to enforce z k ≥ f k ( x ) , ∀ k ∈ K z ∈ R K ( S z ˆ T k (SP) k : f k (ˆ q T U , x ) := min k y k x ˆ C y k ) s.t. W k y k ≥ h k − T k ˆ x y k ∈ R t + Decomposes by scenario Subproblems Subproblems are small LP’s UB Embed within branch-and-cut Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 4 / 22

Two-stage stochastic programs Decomposition methods Dual decomposition (Carøe and Schultz, 1999) Copy the first-stage decision-variables � min c T x + q T k y k k ∈K s.t. x k = x , ∀ k ∈ K T k x k + W k y k ≥ h k , ∀ k ∈ K Ax k ≥ b x k ∈ Z n + , y ∈ R tK + Solve Lagrangian dual relaxing constraints x k = x Decomposes problem by scenario Subproblems are mixed-integer programs Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 5 / 22

Two-stage stochastic programs Decomposition methods Something in between? Benders Dual decomposition Fast solution of LP relaxation Expensive relaxation (many (LP subproblems) MIP subproblems) Potentially strong bounds Potentially weak bounds (Obvious) Idea Strengthen Benders with integrality-based cuts Integrality-based cuts in stochastic programming Carøe (1998), Sen and Higle (2005), Sen and Sherali (2006), Gade et al. (2012), Zhang and Küçükyavuz (2013), Ntaimo (2013) Focus has been on harder case of second-stage integer variables Common goal: Handle second-stage integrality with cuts only Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 6 / 22

Polyhedral Analysis Benders as a projection Extensive Form z , x , y c T x + � min z k k ∈K s.t. ( z k , x , y k ) ∈ Q LP , ∀ k ∈ K k x ∈ X Benders reformulates the problem in the space of first stage variables. z , x c T x + � min z k k ∈K s.t. ( z k , x ) ∈ Proj z k , x ( Q LP k ) , ∀ k ∈ K x ∈ X Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 7 / 22

Polyhedral Analysis Two options for using integrality-based cuts Project-and-cut: Cut-and-project: Generate Benders cuts to Use integrality information to obtain polyhedron B k with derive cuts valid for Q IP k := Q LP ∩ X to obtain k Proj z k , x ( Q LP k ) ⊆ B k polyhedron Q S k with: Use integrality information in Q IP k ⊆ Q S k ⊆ Q LP k X to derive cuts for the set B k ∩ X Project resulting polyhedra into ( x , z k ) space via Benders Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 8 / 22

Polyhedral Analysis Cut-and-project Master (MP) LP : min z , x c T x + � z k Benders Cuts k ∈K (ˆ z k , ˆ s.t. x ∈ X LP x ) Cuts to enforce z k ≥ f k ( x ) , ∀ k ∈ K z ∈ R K Subproblem k (SP) k : f k (ˆ q T x ) := min k y k y k s.t. W k y k ≥ h k − T k ˆ ( ˆ x Cuts x , ˆ y k ) y k ∈ R t + Note: Add cuts to subproblem, even though Separate cuts it’s an LP Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 9 / 22

Polyhedral Analysis Which is better? Project-and-cut Cut-and-project + Work in more compact space + Keep formulation structure ⇒ Can use problem-specific + Straightforward to generate cuts cuts using multiple scenarios + Easy to incorporate known - Lose structure cut separation routines - Working with B k may lead to weaker cuts - May be memory-intensive Can be overcome Question Does one approach yield stronger relaxations than the other when using a given class of cuts? We investigate this for split cuts Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 10 / 22

Split Cuts First Step : Single Split y 5 Q LP ⊆ R q + n + t (a polyhedron) Q LP \ S ′ P LP = Proj ( z , x ) ( Q LP ) 4 S = { ( z , x ) : γ + 1 > π x > γ } 3 S ′ = { ( z , x , y ) : γ + 1 > π x > γ } 2 Claim P LP \ S = proj ( z , x ) ( Q LP \ S ′ ) 1 x 0 Modaresi et al. (MIP 2012) showed 0 1 2 3 4 5 P LP \ S similar result for conic MIR Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 11 / 22

Split Cuts Intersectiong multiple splits � � conv ( Q LP \ S ′ � ? conv ( P LP \ S i ) � i ) = Question: Proj z , x i ∈ I i ∈ I Answer: ( ⊆ ) is easy to show. ( ⊇ ) is false. Counterexample: ( z , x , y ) where z , x ∈ Z , y ∈ R Q LP = conv ( { ( 1 / 2 , 0 , 0 ) , ( 1 / 2 , 1 , 0 ) , ( 0 , 1 / 2 , 1 ) , ( 1 , 1 / 2 , 1 ) } ) ⊆ R 3 P LP = Proj ( z , x ) ( Q LP ) z 1 SC ( P LP ) = { ( 1 / 2 , 1 / 2 ) } Again related to Modaresi et x al. (MIP 2012) 0 1 Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 12 / 22

Split Cuts Intersecting multiple splits conv ( Q LP \ S ′ conv ( P LP \ S i ) � � � ? � Question: Proj z , x i ) = i ∈ I i ∈ I Counterexample: ( z , x , y ) where z , x ∈ Z , y ∈ R Q LP = conv ( { ( 1 / 2 , 0 , 0 ) , ( 1 / 2 , 1 , 0 ) , ( 0 , 1 / 2 , 1 ) , ( 1 , 1 / 2 , 1 ) } ) ⊆ R 3 P LP = Proj ( z , x ) ( Q LP ) y SC ( P LP ) = { ( 1 / 2 , 1 / 2 ) } SC ( Q LP ) = ∅ z x Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 13 / 22

Split Cuts Can weakness in projected space be overcome? Let Q LP = { ( z , x , y ) ∈ R q + n + t : Hx + Kz + Ly ≥ g , x , y , z ≥ 0 } . i = { ( x , z , y ) ∈ R n + q + t : γ i + 1 > π i x > γ i } for i ∈ I S ′ conv ( Q LP \ S ′ � � � ˆ P = proj ( x , z ) i ) i ∈ I Is it possible to efficiently separate cuts for ˆ P ? I.e., simultaneously consider all splits in set I , and perform the projection Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 14 / 22

Split Cuts Valid inequality characterization Theorem The inequality cx + dz ≥ f is valid for the set ˆ P if and only if there exists a solution to : c i , d = d i , 0 = h i , f = f i � � � � c = i ∈ I i ∈ I i ∈ I i ∈ I c i λ i 1 H − µ i 1 π i c i λ i 2 H + µ i 2 π i ≥ ≥    d i λ i d i λ i  ≥ 1 K ≥ 2 K      h i λ i h i λ i ≥ 1 L ≥ 2 L ∀ i ∈ I 2 ( γ i + 1 )  f i λ i 1 g − µ i 1 γ i f i λ i 2 g + µ i  ≤ ≤     c i , d i , f i free  λ i 1 , λ i µ i 1 , µ i ≥ 0 ≥ 0 ,  2 2 Yields cut-generating LP that is 2 | I | times larger than original formulation For one split, overcomes limitation of working with Benders-cut based approximation Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 15 / 22

Numerical Illustration Is the difference significant? Two experiments: Compare lift-and-project closure (simple splits) using cut-and-project and project-and-cut Investigate impact of using split cuts in the cut-and-project approach to solve to optimality Test problems: CAP: Stochastic version of capacitated warehouse location problem SNIP: Stochastic network interdiction problem from Pan and Morton (2008) When solving to optimality, we use the Rank-1 GMI heuristic separation routine of Dash and Goycoolea (2010) Not restricted to simple disjunctions Bodur et al. (UW-Madison/IBM) Split Cuts for Stochastic IP Aussois 2014 16 / 22