text:
### A Pluto.jl notebook ###
# v0.17.1

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ bb1927c5-54b7-4457-aa85-56e689bb38df
begin
	using DifferentialEquations
	using PlutoUI
	using Plots
end

# ╔═╡ 215136d6-9313-4849-81e1-b269f699aa33
md"""
## Bioprocesses I: Outside the Cell

Up to now, we have not considered chemical transformations and processes involving cells (called Bioprocesses). However, bioreactors and bioprocesses are important areas for Chemical Engineering because of the unique chemistries that can occur in the natural world. Many things that we take for granted every day are produced directly by or are associated with bioprocesses. For example, [the enzymes in laundry detergents](https://iubmb.onlinelibrary.wiley.com/doi/full/10.1002/bmb.20488), food products, wine or beer. Thus, let's begin to discuss some bioprocess basics. Toward this, by the end of this lecture you should be able to:

__Objectives__
* Write the species mass balances for substrate $S$, product $P$ and cells $X$ in a bioreactor
* Formulate _simple_ models for the rate of cell growth and product formation
* Solve _simple_ models of cell growth and product formation in different types of well-mixed continuous stirred tank reactors

"""

# ╔═╡ 4464cb96-2778-4946-978d-50961869d59a
md"""
##### Obj 1: Substrate, product and cell balances in a bioreactor
This is the easy part: we've sort of seen these balance equations before. Let's consider a bioprocess in a well-mixed continuous stirred tank reactor that may (or may not) be at a steady state. Substrates (starting materials) $S_{1},S_{2},\dots,S_{A}$ are transformed to product(s) $P_{1},P_{2},\dots,P_{\mathcal{P}}$ and cells $X$ in the bioreactor.

##### General case
Let's start with the species mass balances around substrates, products and cells where we number the substrate and product species $i=1,2,\dots,\mathcal{M}$ and denote cell concentration by X (units: gDW/L):

$$\begin{eqnarray}
V\frac{dM_{i}}{dt} & = & \sum_{s=1}^{\mathcal{S}}v_{s}\dot{F}_{s}M_{is} + \left(\sum_{j=1}^{\mathcal{T}}\tau_{ij}q_{j}X\right)V\qquad{i=1,2,\dots,\mathcal{M}}\\
V\frac{dX}{dt} & = & \sum_{s=1}^{\mathcal{S}}v_{s}\dot{F}_{s}X_{s} + \left(\tau_{X}\mu-d\right)XV
\end{eqnarray}$$

where $M_{i}$ denotes the _mass concentration_ of species $i$ in the reactor (units: g/L),
$M_{is}$ denotes the number of mass concentration of species $i$ in stream $s$ per time (units: g/L), where
$v_{s}$ denotes the direction parameter for stream $s$.
There are $s=1,2,\dots,\mathcal{S}$ total possible streams into and from the bioreactor.
The second set of terms on the right-hand side of the first equation are the (bio)reaction terms: the symbol $\tau_{ij}$ denotes the
_stoichiometric_ coefficient for species $i$ in rate process $j$, where $q_{j}$ denotes the _specific_ rate of biotransformation $j$ (units: $\star$/gDW-time), $X$ denotes the cell mass concentration (units: gDW/L) and $V$ denotes the volume of the reacting fluid in the reactor. The second equation is the cell mass balance, where the first term on the right-hand side if the transport of cells into and from the bioreactor in the streams, and the second term is the _net_ rate of cell growth, where $\mu$ denotes the specific growth rate (units: 1/time), $d$ denotes the specific rate of cell death (units: 1/time) and $X_{s}$ denotes the cell mass in the stream $s$ (units: gDW/L). Lastly, the term $\tau_{X}$ denotes the stoichiometric coefficient governing the growth of cells.

##### Single substrate $S$, growth product $P$ and cell mass $X$
Let's consider a simple case to better understand the structure and properties of the general balances shown above: a single substrate $S$ = 1, a single _growth associated_ product $P$ = 2, and cells $X$ in a well-mixed bioreactor with a single input ($s=1$) and a single output ($s=2$). The balances for $S$, $P$, and $X$, in this case, are given by:

$$\begin{eqnarray}
V\frac{dM_{1}}{dt} & = & \dot{F}_{1}M_{11} - \dot{F}_{2}M_{12} +\left(\tau_{11}q_{1}+\tau_{12}q_{2}\right)XV \\
V\frac{dM_{2}}{dt} & = & \dot{F}_{1}M_{21} - \dot{F}_{2}M_{22} +\tau_{22}q_{2}XV \\
V\frac{dX}{dt} & = & 	\dot{F}_{1}X_{1} - \dot{F}_{2}X_{2} +\left(\tau_{31}q_{1} - d\right)XV
\end{eqnarray}$$

To describe the chemistry that is occurring in reactions $q_{1}$ (which in this case equals the specific growth rate $\mu$) and $q_{2}$, the specific rate of product formation (which is proportional to the growth rate), we formulate a _pseudo_ reaction(s) of the form:

$$\tau_{11}S+X~{\longrightarrow}~\tau_{21}P + \left(1+\tau_{31}\right)X$$

However, this is __not__ an elementary reaction, meaning we do not know _a priori_ the values for the stoichiometric coefficients $\tau_{ij}$ _if_ cells are treated as _magical unknowable boxes_ in which we have no mechanistic knowledge or insight.

##### Where do the $\tau_{ij}$ in the bioreactor balances come from?
If we have no insight into what is going on inside the cells, then the values of the $\tau_{ij}$ (stoichiometric coefficients connecting bioprocess $j$ with substrate $i$) have to be estimated from data. These are called Yields in the bioprocess world and they have units of g-$\star$/g-$\star$. For example, the yield of cells on the substrate $S$ is defined as:

$$Y_{X/S} \equiv -\frac{\Delta{X}}{\Delta{S}} = \frac{1}{\tau_{11}}$$

where $\Delta{X}$ denotes the cell mass that is produced in the reactor, and $\Delta{S}$ denotes the amount of substrate consumed to produce the cells. In a similar way, we can define the yield of product on substrate $S$ as:

$$Y_{P/S} \equiv -\frac{\Delta{P}}{\Delta{S}} = \frac{1}{\tau_{12}}$$

where $\Delta{P}$ denotes the product produced per substrate consumed. Of course, these yields are _not_ independent: because total mass must be conserved, $Y_{X/S}$ and $Y_{P/S}$ must be related to one another. Furthermore, it's easy to show the relationship:

$$-Y_{X/S}{\times}-Y_{S/P}{\times}Y_{P/X} = 1$$

Lastly, by convention $\tau_{31} = 1$.

##### What is the specific rate of product formation $q_{2}$ and $\tau_{22}$?
The last issue that we need to consider is the rate of product formation $q_{2}$ and the value of $\tau_{22}$. Because product $P$ is growth associated (is only formed if the cells are growing e.g., ethanol or CO$_{2}$) we might expect $q_{2}$ to be proportional to the growth rate where the proportionality constant is a measurable yield:

$$\tau_{22}q_{2} = Y_{P/X}\mu$$

Putting all these ideas together gives the system of equations (including the volume balance):

$$\begin{eqnarray}
V\frac{dM_{1}}{dt} & = & \dot{F}_{1}M_{11} - \dot{F}_{2}M_{12} - \left(\frac{1}{Y_{S/X}}+\frac{1}{Y_{P/S}}\right)\mu XV \\
V\frac{dM_{2}}{dt} & = & \dot{F}_{1}M_{21} - \dot{F}_{2}M_{22} + \left(Y_{P/X}\right)\mu{XV} \\
V\frac{dX}{dt} & = & \dot{F}_{1}X_{1} - \dot{F}_{2}X_{2} +\left(\mu - d\right)XV\\
\rho\frac{dV}{dt} & = &  \dot{F}_{1}\rho_{11} - \dot{F}_{2}\rho_{12}
\end{eqnarray}$$

"""

# ╔═╡ 3a0df0cc-b0aa-4232-9275-389d378114aa
md"""
##### Obj 2: Formulate _simple_ models for the specific rate of cell growth and product formation
The _simple_ way to formulate the rate of substrate utilization and product formation in bioprocesses is to treat cells as _magical unknowable boxes_: substrate $S$ goes into the box and more cells and product(s) come out the box. How this happens, nobody knows!

Models for the kinetics of processes that occur in the _magical unknowable box_ are often _empirical_, meaning they describe data but are not necessarily mechanistic. A widely used, yet simple model, of cell growth and product formation, is the [Monod model](https://en.wikipedia.org/wiki/Monod_equation):

$$\mu = \frac{{\mu_{g}^{max}}S}{K_{S}+S}$$

where $\mu$ denotes the specific growth rate (units: 1/time), $\mu_{g}^{max}$ denotes the _maximum specific growth rate_ (units: 1/time), $S$ denote the _mass or mole concentration_ of growth substrate $S$, and $K_{S}$ denotes the saturation constant of growth on substrate $S$ (units: mass or mole concentration).

"""

# ╔═╡ 84091166-c516-4767-bf9c-cf6b6a80ceb8
md"""
##### Obj 3: Simulate cell growth and product formation in a well-mixed bioreactor
"""

# ╔═╡ d6a12732-fc18-43c8-85d3-72bdb4ebfe5f
md"""
Reactor operation: $(@bind case_flag Select(["first"=>"batch","second"=>"fed-batch", "third"=>"continuous"]))
"""

# ╔═╡ 705d9977-a155-478b-9a3f-5d3d749c1970
begin

	# initialize -
	model_parameters_dict = Dict{String,Any}()

	# general parameters -
	S_in = 40.0 # units: g/L
	P_in = 0.0  # units: g/L
	X_in = 0.0 	# units: g/L

	# setup initial conditions -
	initial_condition_array = [
		36.0 	# 1 S units: g/L
		0.0 	# 2 P units: g/L
		0.1 	# 3 X units: gDW/L
		8.0 	# 4 V units: L
	];

	# setup rate parameters -
	# Data taken from BIND: 102533 for K-12 E. coli
	μ_g_max = 0.76 								# units: 1/hr
	Kₛ = 0.00716 								# units: g/L

	# case specific parameters -
	if (isequal(case_flag, "first") == true)

		# setup batch-specific parameters -
		F1_dot = 0.0
		F2_dot = 0.0
	elseif (isequal(case_flag,"second") == true)

		# setup fed batch-specific parameters -
		F1_dot = 2.0
		F2_dot = 0.0
	elseif (isequal(case_flag, "third") == true)

		# setup continuous-specific parameters -
		F1_dot = 4.0
		F2_dot = 4.0
	end

	# Setup yeilds -
	Ypx = 1.961 # units g/g
	Yxs = 0.51 	# units g/g BIND: 105318
	Yps = Yxs*Ypx

	# set the death constant -
	k_d = 0.0;  # units: 1/hr

	# package -
	model_parameters_dict["S_in"] = S_in
	model_parameters_dict["P_in"] = P_in
	model_parameters_dict["X_in"] = X_in
	model_parameters_dict["initial_condition_array"] = initial_condition_array;
	model_parameters_dict["F1_dot"] = F1_dot
	model_parameters_dict["F2_dot"] = F2_dot
	model_parameters_dict["μ_g_max"] = μ_g_max
	model_parameters_dict["Kₛ"] = Kₛ
	model_parameters_dict["Ypx"] = Ypx
	model_parameters_dict["Yxs"] = Yxs
	model_parameters_dict["Yps"] = Yps
	model_parameters_dict["k_d"] = k_d

	# show -
	nothing
end

# ╔═╡ a0949df6-a328-489a-8e3b-acd28cb05128
with_terminal() do
	a = -(1/Yxs + 1/Yps)
	b = Ypx
	c = 1
	r_sum = a+b+c - (k_d/μ_g_max)
	println("Check: sum of generation terms = $(r_sum)")
end

# ╔═╡ 4b4bb0cf-5b53-408b-a719-02fc31eedf22
with_terminal() do
	yld_product = Yxs*(1/Yps)*Ypx
	println("Check: yield product = $(yld_product)")
end

# ╔═╡ aecf1437-4060-4560-ad79-cd3193b82e99
function μ(x, parameters_dict)

	# alias -
	S = x[1]

	# get parameters from dict -
	μ_g_max_val = parameters_dict["μ_g_max"]
	Kₛ_val = parameters_dict["Kₛ"]

	# compute μ -
	return (μ_g_max_val*S)/(Kₛ_val + S)
end

# ╔═╡ 6a7021ce-780f-4ee0-aeca-014208adcf3a
begin

	# setup a simulation of the Monod model -
	μ_array = Array{Float64,1}() 	# initialize some storage
	N = 10000

	# setup a substrate array -
	S_array = range(0.0,stop=0.5,length=N) |> collect

	# compute the rate at diff S values, put in array
	for S in S_array

		# compute -
		μ_value = μ(S, model_parameters_dict)

		# storage -
		push!(μ_array, μ_value)
	end

	# plot -
	plot(S_array, μ_array, legend=:right, label="specific growth rate μ", lw=2, color=:red)
	plot!(S_array, μ_g_max*ones(N), label="μ_G_max",lw=2,color=:blue)
	xlabel!("Substrate S concentration (g/L)",fontsize=18)
	ylabel!("Specific growth rate μ(S) (1/hr)",fontsize=18)
end

# ╔═╡ 11ec1381-560a-450c-a78c-d9f4b2485422
function balances(dx, x, parameter_dict, t)

	# alias -
	S = x[1] # 1 units: g/L
	P = x[2] # 2 units: g/L
	X = x[3] # 3 units: g/L
	V = x[4] # 1 units: L

	# get stuff from the parameter_dict -
	F1_dot = parameter_dict["F1_dot"]
	F2_dot = parameter_dict["F2_dot"]

	# input -
	S_in = parameter_dict["S_in"]
	P_in = parameter_dict["P_in"]
	X_in = parameter_dict["X_in"]

	# yields -
	Ypx = parameter_dict["Ypx"]
	Yxs = parameter_dict["Yxs"]
	Yps = parameter_dict["Yps"]

	# cell death -
	k_d = parameter_dict["k_d"]

	# setup rates -
	q₁ = μ(S, parameter_dict)
	q₂ = Ypx*μ(S, parameter_dict)

	# compute D -
	D₁ = F1_dot/V # units: 1/hr
	D₂ = F2_dot/V # units: 1/hr

	# compute the kinetics -
	μ_val = μ(x,parameter_dict)

	# setup balances -
	dx[1] = D₁*S_in - D₂*S - ((1/Yxs) + (1/Yps))*q₁*X
	dx[2] = D₁*P_in - D₂*P + q₂*X
	dx[3] = D₁*X_in - D₂*X + (q₁ - k_d)*X
	dx[4] = F1_dot - F2_dot # assumed constant density
end

# ╔═╡ e30672df-f442-4d23-af15-fbd93644f014
begin

	# setup the calculation -
	T_end_phase_1 = 36.0
	tspan = (0.0, T_end_phase_1)

	# get the initial conditions -
	xinitial = model_parameters_dict["initial_condition_array"]
	ode_prob = ODEProblem(balances,xinitial,tspan,model_parameters_dict)

	# solve the ODEs numerically -
	soln_phase_1 = solve(ode_prob, Tsit5(), reltol=1e-8, abstol=1e-8)

	# show -
	nothing
end

# ╔═╡ b71c2696-0d8c-4e4e-8b79-9a0e1cb41c4f
begin
	plot(soln_phase_1, vars=(1:4), label=["S (g/L)" "P (g/L)" "X (gDW/L)" "V (L)"], lw=2)
	xlabel!("Time (hr)", fontsize=18)
	ylabel!("Concentration (g/L -or- gDW/L)", fontsize=18)
end

# ╔═╡ a64ccefd-cda0-4147-951c-45e4fb344325
begin

	# convert soln to array -
	n_t_steps_phase_1 = length(soln_phase_1.t)
	total_number_t_steps = n_t_steps_phase_1

	# buld the T and X structures -
	T = vcat(soln_phase_1.t)
	X = vcat(soln_phase_1.u)

	# build the state array -
	x_soln_array = zeros(total_number_t_steps,5)
	for step_index = 1:total_number_t_steps

		x_soln_array[step_index,1] = T[step_index]
		x_soln_array[step_index,2] = X[step_index][1]
		x_soln_array[step_index,3] = X[step_index][2]
		x_soln_array[step_index,4] = X[step_index][3]
		x_soln_array[step_index,5] = X[step_index][4]
	end
end

# ╔═╡ 115aec36-898b-45af-b790-534b8703ab07
with_terminal() do

	if (isequal(case_flag,"first") == true)
		S_ic = x_soln_array[end,3] + x_soln_array[end,4] - x_soln_array[1,4]
		println("Batch check: Batch mass S initial = $(S_ic) g/L")
	else
		println("Test form not valid. Rewrite for this case")
	end
end

# ╔═╡ a0aa5eee-4601-11ec-1d3c-2d0230a4a0a3
html"""
<style>
main {
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}

a {
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}

.H1 {
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}
</style>"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
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# ╟─00000000-0000-0000-0000-000000000002


text:
"""
  smmoptimize!(sMMProblem::SMMProblem; verbose::Bool = true)

Function to launch an optimization. To be used after the following functions
have been called: (i) set_empirical_moments! (ii) set_priors!
(iii) set_simulate_empirical_moments! (iv) construct_objective_function!
"""
function smmoptimize!(sMMProblem::SMMProblem; verbose::Bool = true)

  # Initialize a BlackBoxOptim problem
  # this modifies sMMProblem.bbSetup
  #-----------------------------------
  set_global_optimizer!(sMMProblem)

  # If the global optimizer is using BlackBoxOptim
  #-----------------------------------------------
  if is_bb_optimizer(sMMProblem.options.globalOptimizer) == true

    # Run the optimization by "batches"
    numberBatches = sMMProblem.options.maxFuncEvals/sMMProblem.options.saveSteps
    # Store best fitness and best candidates
    listBestFitness = []
    listBestCandidates = []
    @showprogress for i=1:numberBatches

        if verbose == true
          info("-------------------------------------")
          info("Batch $(i) / $(numberBatches)")
          info("------------------------------------- \n")
        end

        # Run the optimization with BlackBoxOptim
        #----------------------------------------
        sMMProblem.bbResults = bboptimize(sMMProblem.bbSetup)

        # Save to disk
        #-------------
        saveSMMOptim(sMMProblem, verbose = verbose, saveName = sMMProblem.options.saveName);
        push!(listBestFitness, best_fitness(sMMProblem.bbResults))
        push!(listBestCandidates, best_candidate(sMMProblem.bbResults))
    end

  # In the future, we may use other global minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.globalOptimizer = $(sMMProblem.options.globalOptimizer) is not supported.")

  end

  return listBestFitness, listBestCandidates

end

"""
  function smm_minimizer(sMMProblem::SMMProblem)

Function to get the parameter value minimizing the objective function
"""
function smm_minimizer(sMMProblem::SMMProblem)

  # If the global optimizer is using BlackBoxOptim
  #-----------------------------------------------
  if is_bb_optimizer(sMMProblem.options.globalOptimizer) == true

    best_candidate(sMMProblem.bbResults)

  # In the future, we may use other global minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.globalOptimizer = $(sMMProblem.options.globalOptimizer) is not supported.")

  end

end

"""
  smm_refine_globalmin!(sMMProblem::SMMProblem; verbose::Bool = true)

Function to refine the global minimum using a local minimization routine.
To be used after the following functions have been called: (i) set_empirical_moments!
(ii) set_priors! (iii) set_simulate_empirical_moments! (iv) construct_objective_function!
(v) smmoptimize!
"""
function smm_refine_globalmin!(sMMProblem::SMMProblem; verbose::Bool = true)

  x0 = smm_minimizer(sMMProblem)

  # Let's use the result from the global maximizer as the starting value
  #---------------------------------------------------------------------
  if verbose == true
    info("Refining the global maximum using a local algorithm.")
    info("Using Fminbox = $(sMMProblem.options.minBox)")
    info("Starting value = $(x0)")
  end

  if is_local_optimizer(sMMProblem.options.localOptimizer) == true

    # If using Fminbox option is true
    #-------------------------------
    if sMMProblem.options.minBox == true

      lower = create_lower_bound(sMMProblem)
      upper = create_upper_bound(sMMProblem)

      sMMProblem.optimResults = optimize(sMMProblem.objective_function, x0, lower, upper,
                                        iterations = sMMProblem.options.maxFuncEvals,
                                        convert_to_fminbox(sMMProblem.options.localOptimizer))

    else
      sMMProblem.optimResults = optimize(sMMProblem.objective_function, x0,
                                        convert_to_optim_algo(sMMProblem.options.localOptimizer),
                                        Optim.Options(iterations = sMMProblem.options.maxFuncEvals))
    end

  # In the future, we may use other local minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.localOptimizer = $(sMMProblem.options.localOptimizer) is not supported.")

  end

  return Optim.minimizer(sMMProblem.optimResults)

end

"""
  function smm_minimizer(sMMProblem::SMMProblem)

Function to get the parameter value minimizing the objective function (local)
"""
function smm_local_minimizer(sMMProblem::SMMProblem)

  # If the global optimizer is using BlackBoxOptim
  #-----------------------------------------------
  if is_optim_optimizer(sMMProblem.options.localOptimizer) == true

    Optim.minimizer(sMMProblem.optimResults)

  # In the future, we may use other global minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.globalOptimizer = $(sMMProblem.options.globalOptimizer) is not supported.")

  end

end

"""
  function smm_minimum(sMMProblem::SMMProblem)

Function to get the local minimum value of the objetive function
"""
function smm_local_minimum(sMMProblem::SMMProblem)

  # If the global optimizer is using BlackBoxOptim
  #-----------------------------------------------
  if is_optim_optimizer(sMMProblem.options.localOptimizer) == true

    Optim.minimum(sMMProblem.optimResults)

  # In the future, we may use other global minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.globalOptimizer = $(sMMProblem.options.globalOptimizer) is not supported.")

  end

end

"""
  local_to_global!(sMMProblem::SMMProblem; x0 = Array{Float64}(0,0), nums::Int64 = nworkers(), verbose::Bool = true)

Function to run several local minimization algorithms in parallel, with different
starting values. The minimum is calculated as the minimum of the local minima.
Changes sMMProblem.optimResults. This function also returns
a list containing Optim results.
"""
function local_to_global!(sMMProblem::SMMProblem; x0 = Array{Float64}(0,0), nums::Int64 = nworkers(), verbose::Bool = true)

  # Safety checks
  #--------------
  if nums < nworkers()
    Base.errors("nums < nworkers()")
  elseif nums > nworkers()
    info("nums > nworkers(). Some starting values will be ignored.")
  end

  # To store minization results
  #----------------------------
  results = []

  # Look for valid starting values (for which convergence is reached)
  #-------------------------------------------------------------------
  if x0 == Array{Float64}(0,0)
    myGrid = search_starting_values(sMMProblem, nums, verbose = verbose)
  # Using starting values provided by the user
  #-------------------------------------------
  else

    # Check that enough starting values were provided
    if size(x0, 1) < nworkers()
      info("$(size(x0, 1)) starting value(s) were provided.")
      Base.error("The minimum number of starting value(s) to provide is $(nworkers()).")
    end

    myGrid = x0

  end

  # If the local optimizer is using Optim
  #--------------------------------------
  if is_optim_optimizer(sMMProblem.options.localOptimizer) == true

      # A. Starting tasks on available workers
      #---------------------------------------
      @sync for (workerIndex, w) in enumerate(workers())

        @async push!(results, @fetchfrom w wrap_smm_localmin(sMMProblem, myGrid[workerIndex,:], verbose = true))

      end

    # B. Looking for the minimum
    #----------------------------
    # Initialization
    minIndex = 0
    minValue = Inf
    minimizerValue = zeros(length(keys(sMMProblem.priors)))
    nbConvergenceReached = 0
    listOptimResults = []

    for (workerIndex, w) in enumerate(workers())

      push!(listOptimResults, results[workerIndex])

      try

        minimumValue = Optim.minimum(results[workerIndex])
        minimizer = Optim.minimizer(results[workerIndex])

        if minimumValue < minValue && Optim.converged(results[workerIndex]) == true

          minIndex = workerIndex
          minValue = minimumValue
          minimizerValue = minimizer
          nbConvergenceReached += 1

        end

      catch myError
        info("$(myError)")
      end

    end

    # D. If none of the optimization converged
    #-----------------------------------------
    if minIndex == 0
      info("None of the local optimizer algorithm converged.")
    else
      info("Convergence reached for $(nbConvergenceReached) worker(s).")
      info("Minimum value found with worker $(minIndex)")
      sMMProblem.optimResults = results[minIndex]
    end

  # In the future, we may use other local minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.localOptimizer = $(sMMProblem.options.localOptimizer) is not supported.")

  end

  if verbose == true
    if nbConvergenceReached != 0
      info("Best value found with starting values = $(myGrid[minIndex,:]).")
      info("Best value = $(minValue).")
      info("Minimizer = $(minimizerValue)")
    end
  end

  return listOptimResults

end

"""
  smm_localmin(sMMProblem::SMMProblem, x0::Array{Float64,1}; verbose::Bool = true)

Function find a local minimum using a local minimization routine, with starting value x0.
To be used after the following functions have been called: (i) set_empirical_moments!
(ii) set_priors! (iii) set_simulate_empirical_moments! (iv) construct_objective_function!
"""
function smm_localmin(sMMProblem::SMMProblem, x0::Array{Float64,1}; verbose::Bool = true)

  # Let's use the result from the global maximizer as the starting value
  #---------------------------------------------------------------------
  if verbose == true
    info("Starting value = $(x0)")
    info("Using Fminbox = $(sMMProblem.options.minBox)")
  end

  if is_local_optimizer(sMMProblem.options.localOptimizer) == true

    # If using Fminbox option is true
    #-------------------------------
    if sMMProblem.options.minBox == true

      lower = create_lower_bound(sMMProblem)
      upper = create_upper_bound(sMMProblem)

      optimResults = optimize(sMMProblem.objective_function, x0, lower, upper,
                                        iterations = sMMProblem.options.maxFuncEvals,
                                        convert_to_fminbox(sMMProblem.options.localOptimizer))
    else
      optimResults = optimize(sMMProblem.objective_function, x0,
                            convert_to_optim_algo(sMMProblem.options.localOptimizer),
                            Optim.Options(iterations = sMMProblem.options.maxFuncEvals))
    end

  # In the future, we may use other local minimizer
  # routines. For the moment, let's return an error
  #-------------------------------------------------
  else

    Base.error("sMMProblem.options.localOptimizer = $(sMMProblem.options.localOptimizer) is not supported.")

  end

  return optimResults

end

"""

"""
function wrap_smm_localmin(sMMProblem::SMMProblem, x0::Array{Float64,1}; verbose::Bool = true)

  try
    smm_localmin(sMMProblem, x0, verbose = verbose)
  catch myError
    info("$(myError)")
    info("Error with smm_localmin")
  end

end

"""
  search_starting_values(sMMProblem::SMMProblem; verbose::Bool = true, a::Array{Float64,1}, b::Array{Float64,1}, nums::Int64)

Search for nums valid starting values. To be used after the following functions have been called:
(i) set_empirical_moments! (ii) set_priors! (iii) set_simulate_empirical_moments!
(iv) construct_objective_function!
"""
function search_starting_values(sMMProblem::SMMProblem, numPoints::Int64; verbose::Bool = true)

  # Safety Check
  #-------------
  if is_optim_optimizer(sMMProblem.options.localOptimizer) == false
    Base.error("sMMProblem.options.localOptimizer = $(sMMProblem.options.localOptimizer) is not supported.")
  end

  if verbose == true
    info("Searching for $(numPoints) valid starting values")
  end

  # Generate upper and lower bounds vector readable by create_grid_stochastic
  #--------------------------------------------------------------------------
  lower_bound = zeros(length(keys(sMMProblem.priors)))
  upper_bound = zeros(length(keys(sMMProblem.priors)))

  for (kIndex, k) in enumerate(keys(sMMProblem.priors))
    lower_bound[kIndex] = sMMProblem.priors[k][2]
    upper_bound[kIndex] = sMMProblem.priors[k][3]
  end

  #Each row is a new point and each column is a dimension of this points.
  #---------------------------------------------------------------------
  Validx0 = zeros(numPoints, length(lower_bound))
  distanceValue = zeros(numPoints) #to store the distance associated to each point
  nbValidx0Found = 0

  # Create many grids (stochastic draws) with many potential points
  #----------------------------------------------------------------
  if verbose == true
    info("Creating $(sMMProblem.options.maxTrialsStartingValues) potential grids")
    info("gridType = $(sMMProblem.options.gridType)")
  end

  listGrids = []
  for i=1:sMMProblem.options.maxTrialsStartingValues
    # sampling from uniform distribution or normal distribution
    if sMMProblem.options.gridType == :uniform || sMMProblem.options.gridType == :normal
      push!(listGrids, create_grid_stochastic(lower_bound, upper_bound, numPoints, gridType = sMMProblem.options.gridType))
    # latin hypercube
    elseif sMMProblem.options.gridType == :latin
      push!(listGrids, latin_hypercube_sampling(lower_bound, upper_bound, numPoints))
    else
      Base.err("sMMProblem.options.gridType = $(sMMProblem.options.gridType) is not a valid sampling procedure.")
    end

  end

  listGridsIndex = 0

  # Looping until all the points have been found
  #---------------------------------------------
  while nbValidx0Found < numPoints

    results = []
    listGridsIndex += 1

    if listGridsIndex > sMMProblem.options.maxTrialsStartingValues
      Base.error("Maximum number of attempts reached without success. maxTrialsStartingValues = $(sMMProblem.options.maxTrialsStartingValues)")
    end

    # Use available workers to simulate moments
    #------------------------------------------
    @sync for (workerIndex, w) in enumerate(workers())

      @async push!(results, @fetchfrom w sMMProblem.objective_function(listGrids[listGridsIndex][workerIndex, :]))

    end

    # Check for convergence
    #----------------------
    for (workerIndex, w) in enumerate(workers())

      # Set penalty value by default
      distanceValue[workerIndex] = sMMProblem.options.penaltyValue

      try

        distanceValue[workerIndex] = results[workerIndex]

        # discard inf distances, values equal to penaltyValue and values above the threshold
        if isinf(distanceValue[workerIndex]) == false && distanceValue[workerIndex] != sMMProblem.options.penaltyValue && distanceValue[workerIndex] < sMMProblem.options.thresholdStartingValue

          nbValidx0Found +=1

          if nbValidx0Found <= numPoints

            Validx0[nbValidx0Found,:] = listGrids[listGridsIndex][workerIndex, :]
            info("Valid starting value = $(Validx0[nbValidx0Found,:]), distance = $(distanceValue[workerIndex])")
          end

        end

      catch myError
        info("$(myError)")
      end

    end

  end

  # sorting starting values according to distance value (in ascending order)
  p = sortperm(distanceValue) #get the ascending order
  Validx0 = Validx0[p,:]  #re-order rows

  if verbose == true
    info("Found $(nbValidx0Found) valid starting values")
  end

  # If requested, save (valid) starting values generated
  if sMMProblem.options.saveStartingValues == true
    if verbose == true
      info("Saving starting values to disk.")
    end

    tempfilename = "starting_values_"* sMMProblem.options.saveName * ".jld2"
    JLD2.@save tempfilename Validx0

    tempfilename = "starting_distances_"* sMMProblem.options.saveName * ".jld2"
    JLD2.@save tempfilename distanceValue

  end

  return Validx0

end

ext: jl


text:
# ==================================
# International Space Station (ISS)
#
# system type: continuous LTI system
# state dimension: 270
# input dimension: 3
# ==================================
using ReachabilityBenchmarks, MathematicalSystems, LazySets, MAT

function iss_model()
    file = matopen(@relpath "iss.mat")

    # system matrix
    A = read(file, "A")

    # input matrix
    B = read(file, "B")

    # state domain
    X = Universe(270)

    # input domain
    U = Hyperrectangle([0.05, 0.9, 0.95], [0.05, 0.1, 0.05])

    # continuous LTI system
    S = @system(x' = Ax + Bu, x ∈ X, u ∈ U)

    return S
end

max_stars_repo_name: JuliaReach/ReachabilityBenchmarks

max_issues_repo_name: JuliaReach/ReachabilityBenchmarks

max_forks_repo_name: JuliaReach/ReachabilityBenchmarks