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 , 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.

                           p 12 p 11 p 13 p 21 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.

                  p 11 p 13          p 12

         p 11 p 13 p 12

                  p 11 p 13          p 12

Dynamic Outer Approximation Sampling Algorithm No upper bound calculation until algorithm is terminated.

                  p 11 p 13        p 12

         p 11 p 13 p 12

                  p 11 p 13        p 12

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.

Dry Wet   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 S N demand TPOHAWMAN

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

policy simulated with historical inflow sequences

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