# Best Nonlinear Optimizer for a Continuous Convex Problem

**URL:** <https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771>\
**Category:** Optimization (Mathematical)\
**Tags:** optimization, convex-optimization\
**Created:** [March 11, 2022, 8:43pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771 "2022-03-11T20:43:18Z")\
**Posts on this page:** 5\
**Page:** 3

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**Author:** ![jmcastro2109](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jmcastro2109/32/38427_2.png) [@jmcastro2109](https://discourse.julialang.org/u/jmcastro2109)\
**Post date:** [March 21, 2022, 2:30am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/43 "2022-03-21T02:30:05Z")

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I need to make sensitivity analysis of the maximum to parameters that define the conic constraint.

Hence, I would like to know how can I use (if I can) the duals that I obtain to do this.

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**Author:** ![odow](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/odow/32/28685_2.png) [@odow](https://discourse.julialang.org/u/odow)\
**Post date:** [March 21, 2022, 3:08am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/44 "2022-03-21T03:08:00Z")

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As in, you want the change in optimal objective value with respect to p and k? No, I don’t think you cannot use the duals to compute this.

You could use finite differences though.

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**Author:** ![jmcastro2109](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jmcastro2109/32/38427_2.png) [@jmcastro2109](https://discourse.julialang.org/u/jmcastro2109)\
**Post date:** [March 21, 2022, 12:57pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/45 "2022-03-21T12:57:07Z")

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Hi @odow,

Yes, you are totally right that MWE is not well-suited to study this.

Here below, I provide another MWE in which I study the sensibility of the objective value to a vector of parameters `c`.

In this MWE, I compare the result of the duals computed by JuMP with the gradient computed by finite differences.

The good thing is that with JuMP I get this in a much faster way 😃 .

```julia
using JuMP, Ipopt, LinearAlgebra, Random, Gurobi, BenchmarkTools, MosekTools, NLopt, FiniteDifferences

const Ι = 250 #This is Greek Letter Iota

const J = 50

const p = 5.0

const k = 0.75

function distmat(J,Ι)

    Distances = zeros(J,Ι)

    Random.seed!(7777)

    coordinates_x = hcat([x for x = 1:J],fill(0.0,J))

    coordinates_y = hcat([x for x = 1:Ι].*J/Ι,fill(0.0,Ι))

    for j = 1:J, l=1:Ι

        Distances[j,l] = sqrt((coordinates_x[j,1]-coordinates_y[l,1])^2+(coordinates_x[j,2]-coordinates_y[l,2])^2)

    end

    return 1 .+ .1*Distances

end

const t = distmat(J,Ι)

const c = rand(J)

function solution_numerical_Mosek(p,k,t,c)

    opt = Model(Mosek.Optimizer)

    #set_silent(opt)

    @variable(opt, x[1:J,1:Ι] >= 0)

    @variable(opt, Q[1:J] >= 0)

    @variable(opt, Y[1:Ι] >= 0)

    @variable(opt, u[1:J] >= 0)

    @variable(opt, v[1:Ι] >= 0)

    @objective(

        opt,

        Min,

        -sum(u) + sum(v)  

    )

    @constraint(

        opt,

        [j = 1:J],

        sum(x[j,:])==Q[j],

    )

    @constraint(

        opt,

        [i = 1:Ι],

        sum(t[:,i].*x[:,i])==Y[i],

    )

    @constraint(

        opt,

        cons[j = 1:J],

        [Q[j], c[j], u[j]] in MOI.PowerCone((p-1)/p)

    )

    @constraint(

        opt,

        [i = 1:Ι],

        [v[i], 1.0, Y[i]] in MOI.PowerCone(k)

    )

    JuMP.optimize!(opt)

    return objective_value(opt), value.(x), value.(u), value.(v), value.(Q), value.(Y), dual.(cons)

end

sol_Mosek = @time solution_numerical_Mosek(p,k,t,c)

func_diff(c0)=solution_numerical_Mosek(p,k,t,c0)[1]

func_diff(c)

sol_Mosek[1]

sol_Mosek[7]

G=grad(central_fdm(5,1),func_diff,c)

sol_Mosek[7]

```

with results

```julia
G[1]
 -0.24245699759274053
 -0.22357152022579846
 -0.20679247906289683
 -0.21760034288078534
 -0.24328110841196968
 -0.30542630754893374
  ⋮
 -0.26832512482815
 -0.2494482578121521
 -0.2684235670955047
 -0.2358964325406763
 -0.29762939455915755

```

and

sol\_Mosek[7]

```julia
 [0.762515973362775, 0.24232273421054767, -0.9999999999456418]
 [0.7787216911409459, 0.22277212912850067, -0.9999999999370595]
 [0.7931027088597621, 0.20704855836748834, -0.999999999943988]
 [0.7828773176595666, 0.21807963027278515, -0.9999999999354549]
 [0.7617848184609916, 0.24325439205373628, -0.9999999999592387]
 [0.7194634527392707, 0.3057419236479007, -0.9999999999632796]
 ⋮
 [0.7431459759470492, 0.26859221583709564, -0.999999999962668]
 [0.7574372452326692, 0.24888763271384537, -0.9999999999499486]
 [0.7431459818699275, 0.26859220740933837, -0.9999999999963398]
 [0.7672181987625751, 0.23643641189661424, -0.9999999999282162]
 [0.7245949657845061, 0.29717255349696076, -1.0000000002071532]

```

so it seems to me that in each vector, the central element, measures the sensibility of the result to the parameter `c`.

My question, was exactly that, whether the order in which elements appear in the vector of the dual solutions correspond to the order in which we define the power cone.

---

<div class="post-metadata">

**Author:** ![miles.lubin](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/miles.lubin/32/279_2.png) [@miles.lubin](https://discourse.julialang.org/u/miles.lubin)\
**Post date:** [March 21, 2022, 2:26pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/46 "2022-03-21T14:26:10Z")

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> [@jmcastro2109](#):
>
> My question, was exactly that, whether the order in which elements appear in the vector of the dual solutions correspond to the order in which we define the power cone.

Yes.

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<div class="post-metadata">

**Author:** ![jmcastro2109](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jmcastro2109/32/38427_2.png) [@jmcastro2109](https://discourse.julialang.org/u/jmcastro2109)\
**Post date:** [March 21, 2022, 2:38pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/47 "2022-03-21T14:38:51Z")

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Thank you for the clarification.

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