# Finding the right JuMP package to solve a mixed-integer convex program
**URL:** https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350
**Category:** Optimization (Mathematical)
**Created:** [January 28, 2022, 9:50am UTC](https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350 "2022-01-28T09:50:07Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@merlion](https://discourse.julialang.org/u/merlion)
#### Post date: [January 28, 2022, 9:50am UTC](https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350/1 "2022-01-28T09:50:07Z")
I have a fairly straightforward optimisation problem that looks like this:
\max \ \log(a \cdot x) + b \cdot x \\ s.t. \ x \in \{0,1\}^n \text{ and } \sum\_{i \in [n]}x\_i \leq 10,
for some a, b \in \mathbb{R}^n.
I’d like to solve this program with a package from the JuMP ecosystem, but there are so many and I don’t know how to proceed. Is [Pajarito.jl](https://github.com/JuliaOpt/Pajarito.jl) a good starting point? It doesn’t seem to have extensive documentation, so any pointers would be greatly appreciated.
---
### Author:  [@cvanaret](https://discourse.julialang.org/u/cvanaret)
#### Post date: [January 28, 2022, 10:56am UTC](https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350/2 "2022-01-28T10:56:51Z")
Have a look at SCIP: [https://github.com/scipopt/SCIP.jl](https://github.com/scipopt/SCIP.jl)
---
### Author:  [@mtanneau](https://discourse.julialang.org/u/mtanneau)
#### Post date: [January 29, 2022, 11:44pm UTC](https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350/3 "2022-01-29T23:44:54Z")
Hi!
You have several possibilities (more on solvers below):
1. Keep that `log` term as it is, and use a Mixed-Integer Nonlinear Programming (MINLP) solver (more on those below)
2. Convert the `log` term to conic form (see the [Mosek modeling cookbook](https://docs.mosek.com/modeling-cookbook/expo.html#logarithm) for how to do so), and use a mixed-integer conic solver.
3. Build your model using [Convex.jl](https://github.com/jump-dev/Convex.jl) ([this page](https://jump.dev/Convex.jl/stable/examples/general_examples/basic_usage/) should help get you started)
In terms of numerical stability, available tools and, ultimately, speed, I highly recommend the conic form. Note that, if you use Convex.jl, it will automatically reformulate the `log` term using a conic constraint.
In terms of solvers, you can find a list of supported solvers [here](https://jump.dev/JuMP.jl/stable/installation/#Supported-solvers).
Which solver you can use depends of how you modeled the problem (general nonlinear or conic)
- MINLP: Alpine, Bonmin, SCIP
- MI-Conic: Mosek, Pajarito
Mosek is a commercial product, but they have free academic licenses. SCIP is also free for academic research. The other ones are open-source. Note that both Alpine and Pajarito require external solvers.
* * *
Option D
If you’re OK with a relaxation / approximate solution, you can also build a polyhedral outer-approximation of the `log` term, and pass the resulting problem to your favorite MILP solver.
To do so:
- introduce an additional variable `t >= 0`…
- … and a constraint t = a^{T}x, so that the objective becomes \log(t) + b^{T}x.
- then linearize the \log(t) terms using a finite number of tangents. The more tangents you use, the better the approximation, but the larger the problem.
---
### Author:  [@merlion](https://discourse.julialang.org/u/merlion)
#### Post date: [January 30, 2022, 3:29pm UTC](https://discourse.julialang.org/t/finding-the-right-jump-package-to-solve-a-mixed-integer-convex-program/75350/4 "2022-01-30T15:29:51Z")
Thanks for your excellent suggestions, @mtanneau. I’m currently attempting your second approach, and I’m having difficulty converting the `log` term to conic form. Here’s my current attempt, in which I intend for y to stand for \log(a \cdot x). But if I’m not mistaken, (u, 1, z) \in K\_{exp} only guarantees z \leq \log(a \cdot x), whereas I need equality. Is there a clever way to achieve this?
\max y \\ \text{s.t.}\ \ y = z + x \cdot b, \\ \qquad \ \ \ (u, 1, z) \in K\_{exp}, \\ \ u = a \cdot x.
Edit: Since posting this reply, I’ve realised that an optimal solution (z,x) to the program above always satisfies z = \log(a \cdot x) (otherwise we could increase z which also increases the objective value).