Memoizing finite horizon dynamic programming

question

#1

So I am trying to program a simple finite horizon dynamic programming problem.

I’m trying to use memoization to speed-up computation time.

using Optim

V2dict = Dict()

function V2(t, K)
    if t >= T
        return 0.0
    else
        if haskey(V2dict, (t, K))
            return V2dict[t, K]
        else
            opt = optimize(K′ -> -(log(K - K′) + β * V2(t+1, K′)), eps(), K, iterations = 100_000)
            V2dict[t, K] = Optim.minimum(opt)
            return V2dict[t, K]
        end
    end
end

T = 6
β = 0.95

@time V2(1, 100)
#-6.333197046721626
#  2.846427 seconds (10.56 M allocations: 611.703 MiB, 4.81% gc time)

I have two questions:
Is that how memoization is supposed to be implemented?
Why is V2dict saving many keys for each t? I am only trying to save the optimal values for V2:

V2dict
Dict{Any,Any} with 1799 entries:
  (4, 3.32187e-5)  => -24.3578
  (5, 5.22198e-15) => 32.9762
  (5, 4.50844e-16) => 36.4949
  (4, 8.69678e-5)  => -25.3202
  (5, 2.6052e-12)  => 26.6737
  (5, 2.19599e-9)  => 19.9366
  (5, 7.22726e-16) => 35.7118
  (5, 3.94054e-8)  => 17.0494
  (5, 0.118624)    => 2.1318
  (4, 1.3312e-14)  => -2.68559
  (4, 0.000596086) => -27.245
  (3, 4.50844e-16) => 35.5843
  (5, 6.72888e-16) => 35.8166
  (5, 0.0453104)   => 3.09422
  ⋮                => ⋮

#2

I have Finite horizon dynamic programming with memorization implemented here


Feel free to take a look for inspiration. The array memorymat is my memory.