On the convergence of SDDP and related algorithms

Speaker: Ziming Guan

Supervisor: A. B. Philpott

Sponsor: Fonterra New Zealand

Motivation

• Pereira and Pinto, Multi-Stage Stochastic Optimization Applied to Energy Planning, Mathematical Programming, 52, pp. 359-375, 1991.

Summary

• Description of problem class

• SDDP and its related algorithm

• Theoretical convergence

• Implementation issues

Properties for random quantities

• Random quantities appear only on the right-hand side of the linear constraints in each stage.

• The set of random outcomes is discrete and finite.

• Random quantities in different stages are independent.

• Can accommodate PARMA process for RHS uncertainty.

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Scenario tree, scenario outcome, scenario

Hydro-thermal scheduling

Stage problem

Cuts

Θ(t+1)

Reservoir storage, x(t+1)

Stochastic Dual Dynamic Programming

• [Pereira and Pinto, 1991]

• Initialization: Sample some scenarios and fix them through the course of the algorithm.

• Forward pass: For stage t=1,…,T, solve the stage t problem for each scenario.

• Calculate the lower bound and upper bound.• If not converge,

– Backward pass: For stage t=T-1,…,1, for the stage t problem in each scenario, solve all stage t+1 problems to calculate a cut for stage t problems.

– Back to Forward pass.

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Dynamic Outer Approximation Sampling Algorithm

No upper bound calculation until algorithm is terminated.

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• We have a convergence proof for DOASA.

• This can be used to understand the convergence behaviour of SDDP.

Sampling properties of DOASA

• Forward Pass Sampling Property (FPSP):

Each scenario is traversed infinitely many times with probability 1 in the forward pass.

How do we guarantee this?

Either • Independently sample a single outcome in each stage

with a positive probability for each scenario outcome in the forward pass.

• Repeat an exhaustive enumeration of each scenario in the forward pass.

Convergence Theorem

• Under FPSP, DOASA converges with probability 1 to an optimal solution to the stage 1 problem in a finite number of iterations.

Sampling in cut calculation

• Sample some stage problems.

• Keep a list of dual solutions, search the best one for the stage problem that are not sampled.

• Backward Pass Sampling Property (BPSP):

In any stage, each scenario outcome is visited infinitely many times with probability 1 in the backward pass.

Convergence Theorem

• Under FPSP and BPSP, the algorithm converges with probability 1 to an optimal solution to the stage 1 problem in a finite number of iterations.

Corollaries

• If every outcome is used in cut calculation we only need FPSP.

• We can bias sampling as long as FPSP is satisfied. (Note estimation of upper bound needs unbiased scenarios.)

Resampling

• SDDP does not resample the forward pass. It creates N scenarios of inflows at the start.

• FPSP is NOT satisfied.

• SDDP will terminate with probability 1.

• Cuts give a lower bound, but policy need not be optimal.

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Always Dry, when at convergence...

Negative inflows

• SDDP uses PARMA model for inflows.

• Negative inflows might result – not physically possible.

• Some implementations adjust random outcomes to make inflow non-negative – this destroys stage-wise independence.

• Cut sharing is no longer valid.

• Log-normal inflows not valid for convexity reasons.

Convexity matters in backward pass

• Transmission losses can make stage problem not convex if free disposal is not allowed.

• Unit commitment integer effects are not convex.

Convergence expectation

• We run DOASA on a problem at Fonterra NZ.

• Maximum size for convergence = 12 stages x 24 states.

• In revenue management application, 8 states, 5000 stages converge, 20 states, 5000 stages does not.

• Convergence is problem dependent.

Case study: NZ model

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demand TPO

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Computational results: NZ model

• 9 reservoirs• 52 weekly stages• 30 inflow outcomes per stage • Model written in AMPL/CPLEX

• Takes 100 iterations and 2 hours on a standard Windows PC to converge

2005-2006 policy simulated with historical inflow sequences

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