# How do I find \`m\` and \`b\` such that \`some\_function((x) -\> m\*x + b)\` is maximized?

**URL:** <https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429>\
**Category:** Optimization (Mathematical)\
**Created:** [July 21, 2020, 12:43pm UTC](https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429 "2020-07-21T12:43:20Z")\
**Posts on this page:** 4\
**Page:** 1

<div class="post-metadata">

**Author:** ![NightMachinary](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nightmachinary/32/14196_2.png) [@NightMachinary](https://discourse.julialang.org/u/NightMachinary)\
**Post date:** [July 21, 2020, 12:43pm UTC](https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429/1 "2020-07-21T12:43:20Z")

</div>

I have the following Julia function which calculates the expected mean of some model. I want to find `m` and `b` such that `EBuy(4, (x) -> m*x + b)` is maximized.

```julia
using Distributions

function EBuy(λ, buy ; limDefault=10^2)

    function sell1(d1, m0)
        min(d1, buy(m0) + m0)
    end

    function trash1(d1, m0)
        max(m0 - d1, 0)
    end

    function remaining1(d1, m0)
        max(0, buy(m0) - max(0, d1 - m0))
    end

    function sell2(d1, m0, d2)
        r1 = remaining1(d1, m0)
        min(d2, buy(r1) + r1)
    end

    function trash2(d1, m0, d2)
        max(0, remaining1(d1, m0) - d2)
    end

    function profit(d1, m0, d2)
        profit_price = 20
        waste_price = -100

        profit_price * (sell1(d1, m0) + sell2(d1, m0, d2)) + waste_price * (trash1(d1, m0) + trash2(d1, m0, d2))
    end

    D1Rng = Poisson(λ)
    D2Rng = D1Rng
    function D1(x)
        pdf(D1Rng, x)
    end
    D2 = D1
    M0Rng = Poisson(λ - 2)
    function M0(x)
        pdf(M0Rng, x)
    end

    function EProfit(limI=limDefault, limJ=limDefault, limK=limDefault)
        sum = 0
        for i in 0:limI
            Sj = 0

            for j in 0:limJ
                Sk = 0

                for k in 0:limK
                    Sk += D2(k) * profit(j, i, k)
                end

                Sj += D1(j) * Sk
            end
            if i == 0
                println("With m0=0, EProfit = $(Sj)")
            end
            sum += M0(i) * Sj
        end

        return sum
    end

    EProfit()
end

```

I tried these this, but it did not lead me anywhere:

```julia
using ModelingToolkit
@parameters p_a p_b
sys1 = OptimizationSystem(EBuy(4, (meat) -> max(0, p_b - p_a * meat) ; limDefault=10^2), [], [p_a, p_b]

```

Update: I found a somewhat working solution (though I think a better one should be possible):

```julia
using JuMP, Ipopt
m = Model(optimizer_with_attributes(Ipopt.Optimizer, "max_cpu_time" => 60.0))
@variable(m, x1)
@variable(m, x2)
function f(m, b) 
    EBuy(4, (meat) -> max(0, b + m * meat), limDefault=10^2)
end
JuMP.register(m, :f, 2, f, autodiff=true)
@NLobjective(m, Max, f(x1, x2))
optimize!(m)
println("x1=$(value(x1)), x2=$(value(x2)), objective=$(objective_value(m)), direct objective=$(f(value(x1),value(x2)))")

```

---

<div class="post-metadata">

**Author:** ![johnmyleswhite](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/johnmyleswhite/32/31_2.png) [@johnmyleswhite](https://discourse.julialang.org/u/johnmyleswhite)\
**Post date:** [July 21, 2020, 1:43pm UTC](https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429/2 "2020-07-21T13:43:02Z")

</div>

I don’t really understand your problem. Is `x` fixed? AFAICT `(x) -> m*x + b` is a function, so it’s not clear to me what maximizing it means. I would think you want almost the opposite: you start with a function that maps `(x, m, b) -> m * x + b`, then construct a closure that holds `x` fixed and gives you a function of only `m` and `b`. Then you maximize with respect to the `m` and `b`.

---

<div class="post-metadata">

**Author:** ![NightMachinary](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/nightmachinary/32/14196_2.png) [@NightMachinary](https://discourse.julialang.org/u/NightMachinary)\
**Post date:** [July 21, 2020, 1:57pm UTC](https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429/3 "2020-07-21T13:57:41Z")

</div>

See my update, I found a partial solution. Do you understand my question based on that solution, or should I explain it more clearly?

---

<div class="post-metadata">

**Author:** ![tbeason](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/tbeason/32/15898_2.png) [@tbeason](https://discourse.julialang.org/u/tbeason)\
**Post date:** [July 21, 2020, 2:51pm UTC](https://discourse.julialang.org/t/how-do-i-find-m-and-b-such-that-some-function-x-m-x-b-is-maximized/43429/4 "2020-07-21T14:51:51Z")

</div>

This looks like something I’d use Optim.jl on rather than JuMP.jl.

You need to form your objective function to only be a function of the parameters `m` and `b`

```julia
objfun(m,b) = EBuy(4, (x) -> m*x + b)
result = optimize(p-> -1 * objfun(p[1],p[2]),p_guess,solverchoice)

```

where `p_guess` is your starting values for the parameters and `solverchoice` is one of the Optim.jl solvers. Note the sign flip because the tradition is to minimize not maximize.

I’d also say that you probably might want to even consider using a global solver like Particle Swarm because it looks like your objective function could take a weird shape with all of those `min` and `max`, but that’s for you to explore.