MadNLP is an open-source nonlinear programming solver, purely implemented in Julia. MadNLP implements a filter line-search interior-point algorithm, as used in Ipopt. MadNLP seeks to streamline the development of modeling and algorithmic paradigms in order to exploit structures and to make efficient use of high-performance computers. Notably, MadNLP is part of MadSuite, a suite of GPU-accelerated optimization solvers.

Design

MadNLP's problem structure

MadNLP targets the solution of constrained nonlinear problems, formulating as

\[ \begin{aligned} \min_{x} \; & f(x) \\ \text{subject to} \quad & g_\ell \leq g(x) \leq g_u \\ & x_\ell \leq x \leq x_u \end{aligned}\]

where $x \in \mathbb{R}^n$ is the decision variable, $f: \mathbb{R}^n \to \mathbb{R}$ and $g: \mathbb{R}^n \to \mathbb{R}^m$ two smooth nonlinear functions. MadNLP makes the distinction between the bound constraints $x_\ell \leq x \leq x_u$ and the generic constraints $g_\ell \leq g(x) \leq g_u$. No other structure is assumed a priori.

MadNLP is built on top of NLPModels.jl, a generic package to represent optimization models in Julia. In addition, MadNLP is interfaced with ExaModels, JuMP and Plasmo.

MadNLP's performance

In any interior-point algorithm, the two computational bottlenecks are

1. The evaluation of the first- and second-order derivatives.
2. The factorization of the primal-dual KKT system.

The first point is problem-dependent and is often related to the automatic differentiation backend being employed. The second point is more problematic, as the primal-dual KKT system is symmetric indefinite and ill-conditioned.

Linear solvers

By default, MadNLP is using MUMPS to solve the primal-dual KKT system. Other efficient linear solvers are interfaced, all listed in the table below.

Citing MadNLP.jl

If you use MadNLP.jl in your research, we would greatly appreciate your citing it.

@article{shin2024accelerating,
  title     = {Accelerating optimal power flow with {GPU}s: {SIMD} abstraction of nonlinear programs and condensed-space interior-point methods},
  author    = {Shin, Sungho and Anitescu, Mihai and Pacaud, Fran{\c{c}}ois},
  journal   = {Electric Power Systems Research},
  volume    = {236},
  pages     = {110651},
  year      = {2024},
  publisher = {Elsevier}
}

@article{shin2021graph,
  title     = {Graph-based modeling and decomposition of energy infrastructures},
  author    = {Shin, Sungho and Coffrin, Carleton and Sundar, Kaarthik and Zavala, Victor M},
  journal   = {IFAC-PapersOnLine},
  volume    = {54},
  number    = {3},
  pages     = {693--698},
  year      = {2021},
  publisher = {Elsevier}
}