SLDP: example 1

This example is derived from Section 4.2 of the paper: Ahmed, S., Cabral, F. G., & da Costa, B. F. P. (2019). Stochastic Lipschitz Dynamic Programming. Optimization Online. PDF

using SDDP, HiGHS, Test

function sldp_example_one()
    model = SDDP.LinearPolicyGraph(
        stages = 8,
        lower_bound = 0.0,
        optimizer = HiGHS.Optimizer,
    ) do sp, t
        @variable(sp, x, SDDP.State, initial_value = 2.0)
        @variables(sp, begin
            x⁺ >= 0
            x⁻ >= 0
            0 <= u <= 1, Bin
            ω
        end)
        @stageobjective(sp, 0.9^(t - 1) * (x⁺ + x⁻))
        @constraints(sp, begin
            x.out == x.in + 2 * u - 1 + ω
            x⁺ >= x.out
            x⁻ >= -x.out
        end)
        points = [
            -0.3089653673606697,
            -0.2718277412744214,
            -0.09611178608243474,
            0.24645863921577763,
            0.5204224537256875,
        ]
        return SDDP.parameterize(φ -> JuMP.fix(ω, φ), sp, [points; -points])
    end
    SDDP.train(model, iteration_limit = 100, log_frequency = 10)
    @test SDDP.calculate_bound(model) <= 1.1675
    return
end

sldp_example_one()

------------------------------------------------------------------------------
          SDDP.jl (c) Oscar Dowson and SDDP.jl contributors, 2017-23

Problem
  Nodes           : 8
  State variables : 1
  Scenarios       : 1.00000e+08
  Existing cuts   : false
  Subproblem structure                      : (min, max)
    Variables                               : (7, 7)
    VariableRef in MOI.LessThan{Float64}    : (1, 2)
    VariableRef in MOI.ZeroOne              : (1, 1)
    AffExpr in MOI.GreaterThan{Float64}     : (2, 2)
    AffExpr in MOI.EqualTo{Float64}         : (1, 1)
    VariableRef in MOI.GreaterThan{Float64} : (4, 4)
Options
  Solver          : serial mode
  Risk measure    : SDDP.Expectation()
  Sampling scheme : SDDP.InSampleMonteCarlo

Numerical stability report
  Non-zero Matrix range     [1e+00, 2e+00]
  Non-zero Objective range  [5e-01, 1e+00]
  Non-zero Bounds range     [1e+00, 1e+00]
  Non-zero RHS range        [1e+00, 1e+00]
No problems detected

 Iteration    Simulation       Bound         Time (s)    Proc. ID   # Solves
       10    3.248004e+00   1.165036e+00   1.027749e-01          1        880
       20    2.212205e+00   1.165481e+00   2.037828e-01          1       1760
       30    2.554054e+00   1.165750e+00   3.110359e-01          1       2640
       40    2.987695e+00   1.165750e+00   4.208329e-01          1       3520
       50    3.682757e+00   1.167410e+00   5.331409e-01          1       4400
       60    3.633475e+00   1.167410e+00   6.788750e-01          1       5280
       70    3.483926e+00   1.167410e+00   7.975378e-01          1       6160
       80    3.198985e+00   1.167410e+00   9.145708e-01          1       7040
       90    3.497207e+00   1.167410e+00   1.030831e+00          1       7920
      100    4.070719e+00   1.167410e+00   1.151645e+00          1       8800

Terminating training
  Status         : iteration_limit
  Total time (s) : 1.151645e+00
  Total solves   : 8800
  Best bound     :  1.167410e+00
  Simulation CI  :  3.209543e+00 ± 8.791427e-02
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