# 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:** 20\
**Page:** 2

<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 14, 2022, 4:08pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/22 "2022-03-14T16:08:21Z")

</div>

> [@lmiq](#):
>
> ```julia
> function f(x) 
> ...
> end
> function gradient!(x,g) 
> ...
> end
> x0 = initial_point(...)
> 
> ```

How about this?

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

const Ι = 250 #This is Greek Letter Iota
const J = 50
const p = 5
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)

function func(x) 
    res=sum( sum(x[j,i] for i = 1:Ι)
    ^((p-1)/p) for j = 1:J) - sum( sum( t[j,i]*x[j,i] for j =1:J )^(1/k) for i = 1:Ι) 
end

function gradient!(x,g) 
    X = sum(x,dims=2)
    Y = sum(t.*x,dims=1)'
    res = sum(X.^((p-1)/p)) - sum(Y.^(1/k))
    grad = zeros(J,Ι)

    for i= 1:Ι
        for j = 1:J
            grad[j,i] = (p-1)/p * X[j]^(-1/p) - t[j,i]/k * Y[i]^((1-k)/k)
        end
    end  

    g = vec(grad)

end

x0 = rand(J,Ι)

func(x0)

gradient!(x0,zeros(J*Ι))

```

---

<div class="post-metadata">

**Author:** ![cvanaret](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cvanaret/32/11594_2.png) [@cvanaret](https://discourse.julialang.org/u/cvanaret)\
**Post date:** [March 14, 2022, 5:21pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/23 "2022-03-14T17:21:43Z")

</div>

> [@jmcastro2109](#):
>
> For the advocates of Ipopt, @cvanaret @cgeoga I provided a MWE using Ipopt. Its performance, in my current code, is not dismal. It takes about one minute to solve the problem. However, the optimization speed for me is key, and sadly, one minute is too much.

Thanks for the details. A couple of questions:

- why did you disable function scaling (the `nlp_scaling_method` parameter)?
- what does the optimal x^\* look like? Lots of (near) 0 as expected?
- for a given instance, do you have an a priori idea of what the solution is going to look like?
- edit: can you post the log of Ipopt, where we see how the objective value evolves over the iterations?

---

<div class="post-metadata">

**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 14, 2022, 5:47pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/24 "2022-03-14T17:47:53Z")

</div>

Another thing to try is using a different linear solver. HSL etc can be much faster than mumps: [https://github.com/jump-dev/Ipopt.jl#linear-solvers](https://github.com/jump-dev/Ipopt.jl#linear-solvers)

---

<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 14, 2022, 7:11pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/25 "2022-03-14T19:11:07Z")

</div>

Sure,  
a) I am not sure why I am disabling the scaling. It is probably a legacy of a recycled code. However, if I comment that line of code the solution to the problem is garbage.

b) Yes, lot’s of 0s. Of the 12,500 entries, 98% are 0s.

c) I am not sure how to answer this question. I can characterize the solution sharply (in terms of getting a closed form) only if I assume knife-edge cases for the parameter k. I can characterize a little bit more sharply the sum of the X’s either across j or across i.

d) Yes, see below

```julia
This is Ipopt version 3.13.4, running with linear solver mumps.
NOTE: Other linear solvers might be more efficient (see Ipopt documentation).

Number of nonzeros in equality constraint Jacobian...: 0
Number of nonzeros in inequality constraint Jacobian.: 0
Number of nonzeros in Lagrangian Hessian.............: 1875000

Total number of variables............................: 12500
                     variables with only lower bounds: 12500
                variables with lower and upper bounds: 0
                     variables with only upper bounds: 0
Total number of equality constraints.................: 0
Total number of inequality constraints...............: 0
        inequality constraints with only lower bounds: 0
   inequality constraints with lower and upper bounds: 0
        inequality constraints with only upper bounds: 0

iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
   0 -2.6440765e+02 0.00e+00 7.97e+00 -1.0 0.00e+00 - 0.00e+00 0.00e+00 0
   1 -6.0569020e+02 0.00e+00 4.20e+00 -1.0 8.24e-02 - 1.00e+00 3.61e-01f 1
   2 -7.7536695e+02 0.00e+00 4.24e-02 -1.0 2.68e-02 - 1.00e+00 1.00e+00f 1
   3 -1.2437166e+02 0.00e+00 1.13e+00 -2.5 3.81e-02 - 1.00e+00 1.00e+00f 1
   4 -3.2839354e+01 0.00e+00 3.30e-01 -2.5 7.17e-03 - 1.00e+00 1.00e+00f 1
   5 -1.1183212e+01 0.00e+00 8.80e-02 -2.5 4.78e-03 - 1.00e+00 1.00e+00f 1
   6 -8.0100638e+00 0.00e+00 2.32e-03 -2.5 2.33e-02 - 1.00e+00 1.00e+00f 1
   7 1.0134570e+01 0.00e+00 7.14e-01 -3.8 7.75e-02 - 4.02e-01 1.00e+00f 1
   8 1.5443845e+01 0.00e+00 4.48e-01 -3.8 1.02e-01 - 3.63e-01 1.00e+00f 1
   9 1.7675529e+01 0.00e+00 1.30e-01 -3.8 8.33e-02 - 5.69e-01 1.00e+00f 1
iter objective inf_pr inf_du lg(mu) ||d|| lg(rg) alpha_du alpha_pr ls
  10 1.7987258e+01 0.00e+00 8.56e-04 -3.8 5.52e-02 - 1.00e+00 1.00e+00f 1
  11 1.9582324e+01 0.00e+00 2.92e-03 -5.7 2.99e-02 - 8.28e-01 8.88e-01f 1
  12 1.9798199e+01 0.00e+00 1.97e-05 -5.7 8.99e-03 - 1.00e+00 1.00e+00f 1
  13 1.9798484e+01 0.00e+00 2.09e-06 -5.7 4.92e-03 - 1.00e+00 1.00e+00f 1
  14 1.9820655e+01 0.00e+00 1.48e-05 -8.6 2.94e-03 - 9.76e-01 9.83e-01f 1
  15 1.9821050e+01 0.00e+00 1.74e-07 -8.6 1.50e-03 - 1.00e+00 1.00e+00f 1
  16 1.9821051e+01 0.00e+00 3.57e-08 -8.6 6.87e-04 - 1.00e+00 1.00e+00f 1
  17 1.9821051e+01 0.00e+00 4.37e-09 -8.6 2.41e-04 - 1.00e+00 1.00e+00f 1
  18 1.9821081e+01 0.00e+00 4.56e-09 -12.9 4.94e-05 - 1.00e+00 1.00e+00f 1

Number of Iterations....: 18

                                   (scaled) (unscaled)
Objective...............: -1.9821081353715069e+01 1.9821081353715069e+01
Dual infeasibility......: 4.5645540680538943e-09 4.5645540680538943e-09
Constraint violation....: 0.0000000000000000e+00 0.0000000000000000e+00
Complementarity.........: 4.0657970261467833e-10 -4.0657970261467833e-10
Overall NLP error.......: 4.5645540680538943e-09 4.5645540680538943e-09

Number of objective function evaluations = 19
Number of objective gradient evaluations = 19
Number of equality constraint evaluations = 0
Number of inequality constraint evaluations = 0
Number of equality constraint Jacobian evaluations = 0
Number of inequality constraint Jacobian evaluations = 0
Number of Lagrangian Hessian evaluations = 18
Total CPU secs in IPOPT (w/o function evaluations) = 60.696
Total CPU secs in NLP function evaluations = 1.161

EXIT: Optimal Solution Found.
19.82108135371507
 66.078193 seconds (4.56 M allocations: 801.528 MiB, 0.21% gc time)

```

---

<div class="post-metadata">

**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 14, 2022, 9:01pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/26 "2022-03-14T21:01:35Z")

</div>

You should definitely switch linear solver. It looks like almost all of the time is spent in that.

---

<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 14, 2022, 9:09pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/27 "2022-03-14T21:09:07Z")

</div>

Thanks. I just sent the request. I am installing it on a Windows, so I’ll see how this goes.

---

<div class="post-metadata">

**Author:** ![cvanaret](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cvanaret/32/11594_2.png) [@cvanaret](https://discourse.julialang.org/u/cvanaret)\
**Post date:** [March 14, 2022, 9:23pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/28 "2022-03-14T21:23:00Z")

</div>

Indeed. Only 19 iterations, so if you find a good linear solver (I can recommend MA57 and Pardiso) it really should speed up the process.

> [@jmcastro2109](#):
>
> b) Yes, lot’s of 0s. Of the 12,500 entries, 98% are 0s.

Interesting. Is there a pattern you can identify?  
I would definitely follow my previous piece of advice: try an active-set solver (stuff like filterSQP, SNOPT, etc) and start from the (0, \ldots, 0) solution. With a bit of luck, it won’t have to do a lot of iterations to figure out the right active set.

---

<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 14, 2022, 9:25pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/29 "2022-03-14T21:25:25Z")

</div>

Yes, there is a clear pattern. However, in my _real_ problem I do not think this pattern is very useful.

Perfect, I will take a look and update you!

---

<div class="post-metadata">

**Author:** ![cvanaret](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cvanaret/32/11594_2.png) [@cvanaret](https://discourse.julialang.org/u/cvanaret)\
**Post date:** [March 14, 2022, 9:28pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/30 "2022-03-14T21:28:06Z")

</div>

Got it. Maybe a coarse estimate can be a good enough initial point 🙂

---

<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 15, 2022, 12:10am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/31 "2022-03-15T00:10:34Z")

</div>

Do you know if these have Julia implementations?

Your are right, for instance a good candidate point can be the one when the parameter k=1. In that case, for a given j there will be only one positive x\_{ij}, which is the one that minimizes t\_{ij} and can be computed analytically.

Edit: this did not work.

---

<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 15, 2022, 12:56pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/32 "2022-03-15T12:56:54Z")

</div>

Another potential solution is recasting everything as a linear program with Power Cone constraints and using Mosek in the following way

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

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)

function solution_numerical_Mosek(p,k,t)
    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,
        [j = 1:J],
        [Q[j], 1.0, 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)
end

```

With very similar optimal value

Ipopt:

```julia
19.820920746692256

```

and with Mosek:

```julia
-19.820921011259458

```

and output:

```julia
Problem 
  Name :
  Objective sense : min
  Type : CONIC (conic optimization problem)
  Constraints : 1200
  Cones : 300
  Scalar variables : 14000
  Matrix variables : 0
  Integer variables : 0

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries : 1 time : 0.00
Lin. dep. - tries : 1 time : 0.00
Lin. dep. - number : 0
Presolve terminated. Time: 0.02
Problem
  Name :
  Objective sense : min
  Type : CONIC (conic optimization problem)
  Constraints : 1200
  Cones : 300
  Scalar variables : 14000
  Matrix variables : 0
  Integer variables : 0

Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 300
Optimizer - Cones : 300
Optimizer - Scalar variables : 13400 conic : 900
Optimizer - Semi-definite variables: 0 scalarized : 0
Factor - setup time : 0.00 dense det. time : 0.00
Factor - ML order time : 0.00 GP order time : 0.00
Factor - nonzeros before factor : 1.28e+04 after factor : 1.40e+04
Factor - dense dim. : 0 flops : 7.69e+05
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 1.3e+00 1.3e+00 6.7e+02 0.00e+00 3.307189139e+02 -3.342807529e+02 1.0e+00 0.02
1 7.0e-01 7.0e-01 2.4e+02 1.69e+00 1.285879661e+02 -1.641718015e+02 5.2e-01 0.02  
2 3.4e-01 3.4e-01 1.7e+02 4.81e-01 -4.664481087e+01 -3.639480472e+02 2.6e-01 0.02
3 1.5e-01 1.5e-01 1.9e+01 8.20e-01 -8.572806511e+01 -1.942027062e+02 1.1e-01 0.03
4 4.5e-02 4.5e-02 3.2e+00 6.70e-01 -6.992520893e+01 -1.044985663e+02 3.3e-02 0.03
5 2.0e-02 2.0e-02 1.2e+00 1.05e+00 -3.341086702e+01 -4.843598443e+01 1.5e-02 0.03
6 6.3e-03 6.3e-03 2.0e-01 1.29e+00 -2.647315351e+01 -3.067496218e+01 4.7e-03 0.03
7 2.5e-03 2.5e-03 5.2e-02 1.23e+00 -2.115278755e+01 -2.264358240e+01 1.8e-03 0.03  
8 5.1e-04 5.1e-04 5.8e-03 1.03e+00 -2.027652248e+01 -2.058739853e+01 3.8e-04 0.03
9 1.0e-04 1.0e-04 5.2e-04 1.00e+00 -1.991210073e+01 -1.997349925e+01 7.6e-05 0.05
10 1.2e-05 1.2e-05 2.1e-05 9.99e-01 -1.983120268e+01 -1.983837079e+01 8.8e-06 0.05
11 1.3e-06 1.3e-06 7.6e-07 1.00e+00 -1.982204670e+01 -1.982284111e+01 9.8e-07 0.05
12 1.6e-07 1.6e-07 3.3e-08 1.00e+00 -1.982105752e+01 -1.982115461e+01 1.2e-07 0.05
13 2.8e-08 2.7e-08 2.3e-09 1.00e+00 -1.982094383e+01 -1.982096048e+01 2.0e-08 0.05
14 8.7e-09 3.6e-09 1.1e-10 1.00e+00 -1.982092376e+01 -1.982092593e+01 2.7e-09 0.05  
15 2.2e-08 5.2e-10 6.2e-12 1.00e+00 -1.982092119e+01 -1.982092150e+01 3.9e-10 0.06
16 3.9e-09 5.2e-10 6.1e-12 1.00e+00 -1.982092118e+01 -1.982092150e+01 3.9e-10 0.06  
17 1.7e-08 5.0e-10 5.8e-12 1.00e+00 -1.982092117e+01 -1.982092148e+01 3.7e-10 0.08
18 1.7e-08 5.0e-10 5.8e-12 1.00e+00 -1.982092117e+01 -1.982092148e+01 3.7e-10 0.08
19 1.7e-08 5.0e-10 5.7e-12 9.99e-01 -1.982092117e+01 -1.982092147e+01 3.7e-10 0.08
20 2.0e-08 4.9e-10 5.5e-12 1.00e+00 -1.982092116e+01 -1.982092145e+01 3.6e-10 0.08  
21 2.0e-08 4.9e-10 5.5e-12 9.98e-01 -1.982092116e+01 -1.982092145e+01 3.6e-10 0.09
22 2.3e-08 4.9e-10 5.5e-12 9.97e-01 -1.982092116e+01 -1.982092145e+01 3.6e-10 0.09  
23 2.3e-08 4.8e-10 5.5e-12 9.98e-01 -1.982092115e+01 -1.982092145e+01 3.6e-10 0.11
24 1.5e-08 3.1e-10 2.9e-12 1.00e+00 -1.982092101e+01 -1.982092120e+01 2.3e-10 0.11
Optimizer terminated. Time: 0.11

  0.132441 seconds (655.88 k allocations: 52.483 MiB)

```

But it is 600x (?) faster.

EDIT: Typo in code corrected.

---

<div class="post-metadata">

**Author:** ![Oscar\_Smith](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/oscar_smith/32/25343_2.png) [@Oscar\_Smith](https://discourse.julialang.org/u/Oscar_Smith)\
**Post date:** [March 15, 2022, 1:29pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/33 "2022-03-15T13:29:10Z")

</div>

there’s a reason people like conic optimization.

---

<div class="post-metadata">

**Author:** ![lrnv](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/lrnv/32/19373_2.png) [@lrnv](https://discourse.julialang.org/u/lrnv)\
**Post date:** [March 15, 2022, 2:05pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/34 "2022-03-15T14:05:37Z")

</div>

If you can reformulate as a conic problem, try `Cosmo.jl`, works pretty damn well usually, and the structure of the problem can be reused with another configuration (since you seem to want to solve many problem like this one).

If the fact that the solution is worse is a _precision_ problem, COSMO will allow you to use higher preicsion such as DoubleFloats or BigFloats.

---

<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 15, 2022, 3:05pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/36 "2022-03-15T15:05:53Z")

</div>

> [@jmcastro2109](#):
>
> Hi, thanks so much! I will definitely look into it.
> 
> The solution now is almost the same as the one Ipopt, I had a typo in the code

Hi, thanks so much! I will definitely look into it.

The solution now is almost the same as the one Ipopt, I had a typo in the code

---

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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 20, 2022, 6:27pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/37 "2022-03-20T18:27:21Z")

</div>

Building in this same problem,

What is in the interpretation of the Lagrange multipliers that `Jump.jl` spits from the Conic constraints?

For instance, say I run this code

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

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)

function solution_numerical_Mosek(p,k,t)

    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,

        [j = 1:J],

        [Q[j], 1.0, u[j]] in MOI.PowerCone((p-1)/p)

    )

    @constraint(

        opt,

        cons[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)

sol_Mosek[7]

```

Looking at the output, for the last array that the function produces, which are the duals for the last constraint. I get:

```julia
250-element Vector{Vector{Float64}}:
 [0.9999999851948024, 0.26745049727837067, -0.5742708009654043]
 [0.9999999855626119, 0.2737730991899862, -0.5851060990300552]
 [0.9999999859286929, 0.2803699276341491, -0.5963581393283741]
 [0.9999999862930139, 0.2872585002963847, -0.6080514361094072]
 [0.9999999866555548, 0.294457827370603, -0.6202124647621234]
 [0.9999999862930065, 0.2872585002868521, -0.6080514361094109]
 ⋮
 [0.9999999875163498, 0.3130998007092086, -0.6514307285877697]
 [0.9999999871825755, 0.3055915575179915, -0.638903238124342]
 [0.9999999875105084, 0.31295508254897175, -0.6511898386528795]
 [0.9999999878368773, 0.32064425895101034, -0.663958266795144]
 [0.9999999881616518, 0.32868030617677824, -0.6772374320635317]

```

Say that we take the second element in the first row? What does this mean? Is this the change in the objective function coming from of changing the 1.0 in that conic constraint infinitesimally?

I am checking the Mosek documentation here [8 Duality in conic optimization — MOSEK Modeling Cookbook 3.3.0](https://docs.mosek.com/modeling-cookbook/duality.html) , and I understand how the dual cone is defined. However, I do not understand to what corresponds the value that Julia is giving me and what is the interpretation?

---

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**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 20, 2022, 11:15pm UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/38 "2022-03-20T23:15:09Z")

</div>

> [@jmcastro2109](#):
>
> What is in the interpretation of the Lagrange multipliers that `Jump.jl` spits from the Conic constraints?

These are conic duals, not lagrange multipliers. See [Duality · JuMP](https://jump.dev/JuMP.jl/stable/moi/background/duality/).

You’ll have to refer to a textbook for the full story, but here’s one perturbation interpretation.

A conic constraint:

A\_i x + b\_i \in C\_i

has a dual multiplier y\_i \in C\_i^\*. The dual objective has a term -b\_i^Ty\_i. This means that if y\_i is a dual solution with objective value T and you perturb \tilde b\_i = b\_i + \epsilon then there exists a dual feasible solution with objective value T - \epsilon^Ty\_i. Assuming minimization, if T is the optimal objective value and - \epsilon^Ty\_i is positive then you learn that the optimal objective value _must_ increase by at least - \epsilon^Ty\_i in response to the perturbation.

---

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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, 1:34am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/39 "2022-03-21T01:34:45Z")

</div>

Hi @miles.lubin, thanks so much for the answer. It is very clear conceptually. However, in terms of the output of ‘JuMP’ what is the interpretation of that triplet that I am getting for each constraint?

Trying to be more precise. In my MWE, for constraint ‘cons1’ I get

```julia

[0.9999999851948024, 0.26745049727837067, -0.5742708009654043]

```

To what does each of these numbers correspond to?

I could not find anything in the documentation that speaks to 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, 1:52am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/40 "2022-03-21T01:52:30Z")

</div>

Each vector is the dual solution of the corresponding power cone constraint.

---

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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, 1:59am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/41 "2022-03-21T01:59:33Z")

</div>

But can the elements of this vector be interpreted as the perturbation of the corresponding entry on the power cone or not?

---

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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, 2:16am UTC](https://discourse.julialang.org/t/best-nonlinear-optimizer-for-a-continuous-convex-problem/77771/42 "2022-03-21T02:16:52Z")

</div>

A better question might be: what do you want to do with the dual solution?

If you only care about the primal solution, you can ignore the dual solution. There’s no need to interpret it.

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