Data requirements for stochastic solvers H.I. Gassmann Dalhousie University Halifax, Canada

Overview b Introduction b Algorithms b Data structures b Conclusions

Multistage stochastic linear program Any data item with nonzero subscript may be random (including dimensions where mathematically sensible) ~ stands for arbitrary relation ( , =,  )

Constraints involving random elements  means ~ with probability 1 or with probability at least  or with expected violation at most v

Problem classes b Recourse problems All constraints hold with probability 1All constraints hold with probability 1 b Chance-constrained problems Typically single stageTypically single stage b Hybrid problems Recourse problems including probabilistic constraints (VaR) or integrated chance constraints (CVaR)Recourse problems including probabilistic constraints (VaR) or integrated chance constraints (CVaR) Regulatory necessityRegulatory necessity Often modelled using integer variables and/or linking constraintsOften modelled using integer variables and/or linking constraints

Algorithms b Direct solution of the deterministic equivalent “Curse of dimensionality”“Curse of dimensionality” b Decomposition Recognize structureRecognize structure Repeated calls to solver with different dataRepeated calls to solver with different data Sampling of scenarios during algorithmSampling of scenarios during algorithm –stochastic decomposition –successive approximation –“EVPI relaxation” Scenario generator between AML and solverScenario generator between AML and solver

Data Structures b Often O(10 6 ) variables and constraints b Most compact representation possible Packed matrix format is insufficientPacked matrix format is insufficient Blocks corresponding to nodes in the event treeBlocks corresponding to nodes in the event tree Change blocks if problem dimensions are deterministicChange blocks if problem dimensions are deterministic A stj = A st0 +  A stj (addition or replacement)A stj = A st0 +  A stj (addition or replacement)

Conclusions b Stochastic extensions are difficult b Time is right