Dampening the Curse of Dimensionality Decomposition Methods for Stochastic Optimization Problems Vadim Gorski Adriana Kiszka Nils Löhndorf Goncalo Terca David W. Conference on Computational Management Science Mathematical Methods in Industry and Economics 03/2019
Outline Multi-Stage Stochastic Programming Discretization & complexity The curse of dimensionality Stochastic Optimization with a Markovian Structure Scenario trees and scenario lattices Decomposition algorithms Solving stochastic optimization problems on scenario lattices Illustrative problems Dynamic newsvendor models Aggregate production planning The QUASAR framework 1/28
Optimization Under Uncertainty Question 1 How does uncertainty enter the problem? As a discrete or continuous random variable As a discrete or continuous random process As a set of scenarios without probabilities 2/28
Optimization Under Uncertainty Question 1 How does uncertainty enter the problem? As a discrete or continuous random variable As a discrete or continuous random process As a set of scenarios without probabilities Question 2 What is the nature of the decision problem? Optimizing a worst-case outcome Optimizing a (possibly risk adjusted) expectation Problem with or without recourse decisions Almost sure or probabilistic constraints Discrete time, continuous time problems Finite horizon or infinite horizon Convex or non-convex problems 2/28
Setting of this Talk Problem Class In this talk we will consider convex, multi-stage decision problems in discrete time with a finite planning horizon with relative complete recourse where and constraints have to hold almost surely randomness is modeled by a Markov process. 3/28
Setting of this Talk Problem Class In this talk we will consider convex, multi-stage decision problems in discrete time with a finite planning horizon with relative complete recourse where and constraints have to hold almost surely randomness is modeled by a Markov process. Standard Approach: Scenario Trees Model stochastic process as a scenario tree Solve problem as deterministic equivalent 3/28
Multi-Stage Stochastic Optimization Multi-stage problem with T stages ξ = ( ξ 1 , . . . , ξ T ) with ξ t ( ω ) ∈ R m and ξ t = ( ξ 1 , . . . , ξ t ) x = ( x 1 , . . . , x T ) decisions with x t = ( x 1 , . . . , x t ) and x t ∈ X t x t measurable w.r.t. σ ( ξ t ) 4/28
Multi-Stage Stochastic Optimization Multi-stage problem with T stages ξ = ( ξ 1 , . . . , ξ T ) with ξ t ( ω ) ∈ R m and ξ t = ( ξ 1 , . . . , ξ t ) x = ( x 1 , . . . , x T ) decisions with x t = ( x 1 , . . . , x t ) and x t ∈ X t x t measurable w.r.t. σ ( ξ t ) Define the value functions V T ( x T − 1 , ξ T ) = x T ∈X T ( x T − 1 ,ξ T ) R T ( x T , ξ T ) max V t ( x t − 1 , ξ t ) = x t ∈X t ( x t − 1 ,ξ t ) R t ( x t , ξ t ) + E � V t + 1 ( x t , ξ t + 1 ) | ξ t � max , ∀ t < T 4/28
Multi-Stage Stochastic Optimization Multi-stage problem with T stages ξ = ( ξ 1 , . . . , ξ T ) with ξ t ( ω ) ∈ R m and ξ t = ( ξ 1 , . . . , ξ t ) x = ( x 1 , . . . , x T ) decisions with x t = ( x 1 , . . . , x t ) and x t ∈ X t x t measurable w.r.t. σ ( ξ t ) Define the value functions V T ( x T − 1 , ξ T ) = x T ∈X T ( x T − 1 ,ξ T ) R T ( x T , ξ T ) max V t ( x t − 1 , ξ t ) = x t ∈X t ( x t − 1 ,ξ t ) R t ( x t , ξ t ) + E � V t + 1 ( x t , ξ t + 1 ) | ξ t � max , ∀ t < T Note Functions x t �→ E � V t + 1 ( x t , ξ t + 1 ) | ξ t � have to be evaluated Closed form expressions are rarely available Discretization of ξ required for numerical solutions 4/28
Complexity of Stochastic Programming ξ 3 ξ ξ 2 Number of required points grows exponentially in dimension of ξ 5/28
Complexity of Stochastic Programming ξ 3 ξ ξ 2 Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable 5/28
Complexity of Stochastic Programming ξ 3 ξ ξ 2 Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable Curse of Dimensionality in Stochastic Optimization Swamy [2005] Shapiro and Nemirovski [2005] Shmoys and Swamy [2006] Dyer and Stougie [2006] Hanasusanto et al. [2016] 5/28
Complexity of Stochastic Programming ξ 3 ξ ξ 2 Number of required points grows exponentially in dimension of ξ Even if solution algorithm well behaved problems are intractable Drivers of the Curse of Dimensionality Number of random variables per stage Number of stages 5/28
Scenario Tree Models Nodes represent values, arcs the possible transitions 6/28
Scenario Tree Models Nodes represent values, arcs the possible transitions Tree model represents conditional probability allowing to solve V t ( x t − 1 , ξ t ) = min x t ∈X t R t ( x t , ξ t ) + E V t + 1 ( x t , ξ t + 1 ) | ξ t � � 6/28
Scenario Tree Models ξ 3 ξ 3 ξ 2 ξ 2 Restriction Branching constrains scenario locations Discretization becomes harder 7/28
Scenario Tree Models Sufficient branching is important for realistic models Finance: to avoid arbitrage n assets require n + 1 successors 8/28
Scenario Tree Models Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2 T nodes 8/28
Scenario Tree Models Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2 T nodes Alternative: trees where some nodes only have one successor Sub-problems on the nodes are deterministic Decision maker is partially clairvoyant Overly optimistic planning and flawed policies 8/28
Scenario Tree Models Sufficient branching is important for realistic models Tree where every node ≥ 2 successor has at least 2 T nodes Alternative: trees where some nodes only have one successor Scenario generation literature Monte Carlo Shapiro [2003, 2008] Probability Metrics Pflug [2001, 2009], Pflug and Pichler [2012] Dupacová et al. [2003], Heitsch and Römisch [2003] Moment Matching Høyland and Wallace [2001], Høyland et al. [2003], Kaut and Wallace [2003] Integration Quadratures Pennanen [2005, 2009] 8/28
Lattices: A Compressed Representation Definition A scenario lattice is a scenario tree, where a node can have multiple predecessors. Scenariotree Scenario lattice 9/28
Lattices: A Compressed Representation Definition A scenario lattice is a scenario tree, where a node can have multiple predecessors. Scenariotree Scenario lattice No history → only works for Markov processes 9/28
Lattices: A Compressed Representation Definition A scenario lattice is a scenario tree, where a node can have multiple predecessors. Scenariotree Scenario lattice No history → only works for Markov processes Tree represents |N T | scenarios Lattice (potentially) represents |N 1 | × |N 2 | × · · · × |N T | scenarios 9/28
Optimal Lattice Generation Aim Given a Markov process ξ = ( ξ 1 , . . . , ξ T ) find a small scenario lattice ˆ ξ such that | V t ( x , ξ t ) − V t ( x , ˆ ξ t ) | is small for all x . 10/28
Optimal Lattice Generation Aim Given a Markov process ξ = ( ξ 1 , . . . , ξ T ) find a small scenario lattice ˆ ξ such that | V t ( x , ξ t ) − V t ( x , ˆ ξ t ) | is small for all x . Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent. 10/28
Optimal Lattice Generation Aim Given a Markov process ξ = ( ξ 1 , . . . , ξ T ) find a small scenario lattice ˆ ξ such that | V t ( x , ξ t ) − V t ( x , ˆ ξ t ) | is small for all x . Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent. Löhndorf and Wozabal [2018] generalize these ideas to general processes using a second order stochastic gradient method. Fast parameter free method 10/28
Optimal Lattice Generation Aim Given a Markov process ξ = ( ξ 1 , . . . , ξ T ) find a small scenario lattice ˆ ξ such that | V t ( x , ξ t ) − V t ( x , ˆ ξ t ) | is small for all x . Bally and Pagès [2003] reduces a GBM by minimizing the Wasserstein metric using stochastic gradient descent. Löhndorf and Wozabal [2018] generalize these ideas to general processes using a second order stochastic gradient method. Fast parameter free method Kiszka and Wozabal [2018] adapt ideas from Pflug and Pichler [2012] to lattices with randomness in the constraints. Theoretically superior to Löhndorf and Wozabal [2018] Not (yet) suitable for large models 10/28
Solving Stochastic Optimization on Trees Assign a decision x n to every node n ∈ N t , 1 ≤ t ≤ T Set up deterministic equivalent formulation Find optimal decisions x ∗ n 11/28
Stochastic Optimization on Lattices Generally: decision dependent on path leading to n ∈ N t x n = x n ( x n − 1 , ξ t ) 12/28
Stochastic Optimization on Lattices Generally: decision dependent on path leading to n ∈ N t x n = x n ( x n − 1 , ξ t ) Define the resource state at the beginning of period t S t ( x 1 , ξ 1 , . . . , x t − 1 , ξ t − 1 ) = S t ( S t − 1 , x t − 1 , ξ t − 1 ) 12/28
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