Nonconvex.jl

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Toolbox for gradient-based and derivative-free non-convex constrained optimization with continuous and/or discrete variables.

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# Nonconvex

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`Nonconvex.jl` is an umbrella package over implementations and wrappers of a number of nonconvex constrained optimization algorithms and packages making use of automatic differentiation. Zero, first and second order methods are available. Nonlinear equality and inequality constraints as well as integer and nonlinear semidefinite constraints are supported. A detailed description of all the algorithms and features available in `Nonconvex` can be found in the [documentation](https://JuliaNonconvex.github.io/Nonconvex.jl/stable).

## The `JuliaNonconvex` organization

The `JuliaNonconvex` organization hosts a number of packages which are available for use in `Nonconvex.jl`. The correct package is loaded using the `Nonconvex.@load` macro with the algorithm or package name. See the [documentation](https://JuliaNonconvex.github.io/Nonconvex.jl/stable) for more details. The following is a summary of all the packages in the `JuliaNonconvex` organization.

| Package | Description | Tests | Coverage |
| ------- | ----------- | ----- | -------- |
| [Nonconvex.jl](https://github.com/JuliaNonconvex/Nonconvex.jl) | Umbrella package for nonconvex optimization | [![Actions Status](https://github.com/JuliaNonconvex/Nonconvex.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/Nonconvex.jl/actions) | [![codecov](https://codecov.io/gh/JuliaNonconvex/Nonconvex.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/Nonconvex.jl) |
| [NonconvexCore.jl](https://github.com/JuliaNonconvex/NonconvexCore.jl) | All the interface functions and structs | [![Build Status](https://github.com/JuliaNonconvex/NonconvexCore.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexCore.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexCore.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexCore.jl) |
| [NonconvexMMA.jl](https://github.com/JuliaNonconvex/NonconvexMMA.jl) | Method of moving asymptotes implementation in pure Julia | [![Build Status](https://github.com/JuliaNonconvex/NonconvexMMA.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexMMA.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexMMA.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexMMA.jl) |
| [NonconvexIpopt.jl](https://github.com/JuliaNonconvex/NonconvexIpopt.jl) | [Ipopt.jl](https://github.com/jump-dev/Ipopt.jl) wrapper | [![Build Status](https://github.com/JuliaNonconvex/NonconvexIpopt.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexIpopt.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexIpopt.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexIpopt.jl) |
| [NonconvexNLopt.jl](https://github.com/JuliaNonconvex/NonconvexNLopt.jl) | [NLopt.jl](https://github.com/JuliaOpt/NLopt.jl) wrapper | [![Build Status](https://github.com/JuliaNonconvex/NonconvexNLopt.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexNLopt.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexNLopt.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexNLopt.jl) |
| [NonconvexPercival.jl](https://github.com/JuliaNonconvex/NonconvexPercival.jl) | [Percival.jl](https://github.com/JuliaSmoothOptimizers/Percival.jl) wrapper (an augmented Lagrangian algorithm implementation) | [![Build Status](https://github.com/JuliaNonconvex/NonconvexPercival.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexPercival.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexPercival.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexPercival.jl) |
| [NonconvexJuniper.jl](https://github.com/JuliaNonconvex/NonconvexJuniper.jl) | [Juniper.jl](https://github.com/lanl-ansi/Juniper.jl) wrapper | [![Build Status](https://github.com/JuliaNonconvex/NonconvexJuniper.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexJuniper.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexJuniper.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexJuniper.jl) |
| [NonconvexPavito.jl](https://github.com/JuliaNonconvex/NonconvexPavito.jl) | [Pavito.jl](https://github.com/jump-dev/Pavito.jl) wrapper | [![Build Status](https://github.com/JuliaNonconvex/NonconvexPavito.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexPavito.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexPavito.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexPavito.jl) |
| [NonconvexSemidefinite.jl](https://github.com/JuliaNonconvex/NonconvexSemidefinite.jl) | Nonlinear semi-definite programming algorithm | [![Build Status](https://github.com/JuliaNonconvex/NonconvexSemidefinite.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexSemidefinite.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexSemidefinite.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexSemidefinite.jl) |
| [NonconvexMultistart.jl](https://github.com/JuliaNonconvex/NonconvexMultistart.jl) | Multi-start optimization algorithms | [![Build Status](https://github.com/JuliaNonconvex/NonconvexMultistart.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexMultistart.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexMultistart.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexMultistart.jl) |
| [NonconvexBayesian.jl](https://github.com/JuliaNonconvex/NonconvexBayesian.jl) | Constrained Bayesian optimization implementation | [![Build Status](https://github.com/JuliaNonconvex/NonconvexBayesian.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexBayesian.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexBayesian.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexBayesian.jl) |
| [NonconvexSearch.jl](https://github.com/JuliaNonconvex/NonconvexSearch.jl) | Multi-trajectory and local search methods | [![Build Status](https://github.com/JuliaNonconvex/NonconvexSearch.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexSearch.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexSearch.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexSearch.jl) |
| [NonconvexAugLagLab.jl](https://github.com/JuliaNonconvex/NonconvexAugLagLab.jl) | Experimental augmented Lagrangian package | [![Build Status](https://github.com/JuliaNonconvex/NonconvexAugLagLab.jl/actions/workflows/CI.yml/badge.svg?branch=master)](https://github.com/JuliaNonconvex/NonconvexAugLagLab.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexAugLagLab.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexAugLagLab.jl) |
| [NonconvexUtils.jl](https://github.com/JuliaNonconvex/NonconvexUtils.jl) | Some utility functions for automatic differentiation, history tracing, implicit functions and more. | [![Build Status](https://github.com/JuliaNonconvex/NonconvexUtils.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/JuliaNonconvex/NonconvexUtils.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexUtils.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexUtils.jl) |
| [NonconvexTOBS.jl](https://github.com/JuliaNonconvex/NonconvexTOBS.jl) | Binary optimization algorithm called "topology optimization of binary structures" ([TOBS](https://www.sciencedirect.com/science/article/abs/pii/S0168874X17305619?via%3Dihub)) which was originally developed in the context of optimal distribution of material in mechanical components. | [![Build Status](https://github.com/JuliaNonconvex/NonconvexTOBS.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/JuliaNonconvex/NonconvexTOBS.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexTOBS.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexTOBS.jl) |
| [NonconvexMetaheuristics.jl](https://github.com/JuliaNonconvex/NonconvexMetaheuristics.jl) | Metaheuristic gradient-free optimization algorithms as implemented in [`Metaheuristics.jl`](https://github.com/jmejia8/Metaheuristics.jl). | [![Build Status](https://github.com/JuliaNonconvex/NonconvexMetaheuristics.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/JuliaNonconvex/NonconvexMetaheuristics.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexMetaheuristics.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexMetaheuristics.jl) |
| [NonconvexNOMAD.jl](https://github.com/JuliaNonconvex/NonconvexNOMAD.jl) | [NOMAD algorithm](https://dl.acm.org/doi/10.1145/1916461.1916468) as wrapped in the [`NOMAD.jl`](https://github.com/bbopt/NOMAD.jl). | [![Build Status](https://github.com/JuliaNonconvex/NonconvexNOMAD.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/JuliaNonconvex/NonconvexNOMAD.jl/actions) | [![Coverage](https://codecov.io/gh/JuliaNonconvex/NonconvexNOMAD.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/JuliaNonconvex/NonconvexNOMAD.jl) |

## Design philosophy

`Nonconvex.jl` is a Julia package that implements and wraps a number of constrained nonlinear and mixed integer nonlinear programming solvers. There are 3 focus points of `Nonconvex.jl` compared to similar packages such as `JuMP.jl` and `NLPModels.jl`:

1. Emphasis on a function-based API. Objectives and constraints are normal Julia functions.
2. The ability to nest algorithms to create more complicated algorithms.
3. The ability to automatically handle structs and different container types in the decision variables by automatically vectorizing and un-vectorizing them in an AD compatible way.

## Installing Nonconvex

To install `Nonconvex.jl`, open a Julia REPL and type `]` to enter the package mode. Then run:
```julia
add Nonconvex
```

Alternatively, copy and paste the following code to a Julia REPL:
```julia
using Pkg; Pkg.add("Nonconvex")
```

## Loading Nonconvex

To load and start using `Nonconvex.jl`, run:
```julia
using Nonconvex
```

## Quick example

```julia
using Nonconvex
Nonconvex.@load NLopt

f(x) = sqrt(x[2])
g(x, a, b) = (a*x[1] + b)^3 - x[2]

model = Model(f)
addvar!(model, [0.0, 0.0], [10.0, 10.0])
add_ineq_constraint!(model, x -> g(x, 2, 0))
add_ineq_constraint!(model, x -> g(x, -1, 1))

alg = NLoptAlg(:LD_MMA)
options = NLoptOptions()
r = optimize(model, alg, [1.0, 1.0], options = options)
r.minimum # objective value
r.minimzer # decision variables
```

## Algorithms

A summary of all the algorithms available in `Nonconvex` through different packages is shown in the table below. Scroll right to see more columns and see a description of the columns below the table.

| Algorithm name | Is meta-algorithm? | Algorithm package | Order | Finite bounds | Infinite bounds | Inequality constraints | Equality constraints | Semidefinite constraints | Integer variables |
| ------- | ----------- | ----- | -------- | -------- | -------- | -------- | -------- | -------- | -------- |
| Method of moving asymptotes (MMA) | ❌ | `NonconvexMMA.jl` (pure Julia) or `NLopt.jl` | 1 | ✅ | ✅ | ✅ | ❌ | ❌ | ❌ |
| Primal dual interior point method | ❌ | `Ipopt.jl` | 1 or 2 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
|  DIviding RECTangles algorithm (DIRECT) | ❌ | `NLopt.jl` | 0 | ✅ | ❌ | ✅ | ❌ | ❌ | ❌ |
| Controlled random search (CRS) | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ | 
| Multi-Level Single-Linkage (MLSL) | Limited | `NLopt.jl` | Depends on sub-solver | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| StoGo | ❌ | `NLopt.jl` | 1 | ✅ | ❌ | ❌ | ❌ | ❌ | ❌ |
| AGS | ❌ | `NLopt.jl` | 0 | ✅ | ❌ | ✅ | ❌ | ❌ | ❌ |
| Improved Stochastic Ranking Evolution Strategy (ISRES) | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| ESCH | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| COBYLA | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| BOBYQA | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| NEWUOA | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| Principal AXIS (PRAXIS) | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| Nelder Mead | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| Subplex | ❌ | `NLopt.jl` | 0 | ✅ | ✅ | ❌ | ❌ | ❌ | ❌ |
| CCSAQ | ❌ | `NLopt.jl` | 1 | ✅ | ✅ | ✅ | ❌ | ❌ | ❌ |
| SLSQP | ❌ | `NLopt.jl` | 1 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| TNewton | ❌ | `NLopt.jl` | 1 | ❌ | ✅ | ❌ | ❌ | ❌ | ❌ |
| Shifted limited-memory variable-metric | ❌ | `NLopt.jl` | 1 | ❌ | ✅ | ❌ | ❌ | ❌ | ❌ |
| Augmented Lagrangian in `NLopt` | Limited | `NLopt.jl` | Depends on sub-solver | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Augmented Lagrangian in `Percival` | ❌ | `Percival.jl` | 1 or 2 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Multiple trajectory search | ❌ | `NonconvexSearch.jl` | 0 | ✅ | ❌ | ❌ | ❌ | ❌ | ❌ |
| Branch and bound for mixed integer nonlinear programming | ❌ | `Juniper.jl` | 1 or 2 | ✅ | ✅ | ✅ | ✅ | ❌ | ✅ |
| Sequential polyhedral outer-approximations for mixed integer nonlinear programming | ❌ | `Pavito.jl` | 1 or 2 | ✅ | ✅ | ✅ | ✅ | ❌ | ✅ |
| Evolutionary centers algorithm (ECA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Differential evolution (DE) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Particle swarm optimization (PSO) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Artificial bee colony (ABC) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Gravitational search algorithm (GSA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Simulated annealing (SA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Whale optimization algorithm (WOA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Machine-coded compact genetic algorithm (MCCGA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Genetic algorithm (GA) | ❌ | `Metaheuristics.jl` | 0 | ✅ | ✅ | ✅ | ✅ | ❌ | ❌ |
| Nonlinear optimization with the MADS algorithm (NOMAD) | ❌ | `NOMAD.jl` | 0 | ✅ | ✅ | ✅ | Limited | ❌ | ✅ |
| Topology optimization of binary structures (TOBS) | ❌ | `NonconvexTOBS.jl` | 1 | Binary | ❌ | ✅ | ❌ | ❌ | Binary |
| Hyperband | ✅ | `Hyperopt.jl` | Depends on sub-solver | ✅ | ❌ | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver |
| Random search | ✅ | `Hyperopt.jl` | Depends on sub-solver | ✅ | ❌ | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver |
| Latin hypercube search | ✅ | `Hyperopt.jl` | Depends on sub-solver | ✅ | ❌ | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver |
| Surrogate assisted optimization | ✅ | `NonconvexBayesian.jl` | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver |
| Log barrier method for nonlinear semidefinite constraint handling | ✅ | `NonconvexSemidefinite.jl` | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | Depends on sub-solver | ✅ | Depends on sub-solver |

The following is an explanation of all the columns in the table:
- Algorithm name. This is the name of the algorithm and/or its acronym. Some algorithms have multiple variants implemented in their respective packages. When that's the case, the whole family of algorithms is mentioned only once.
- Is meta-algorithm? Some algorithms are meta-algorithms that call a sub-algorithm to do the optimization after transforming the problem. In this case, a lot of the properties of the meta-algorithm are inherited from the sub-algorithm. So if the sub-algorithm requires gradients or Hessians of functions in the model, the meta-algorithm will also require gradients and Hessians of functions in the model. Fields where the property of the meta-algorithm is inherited from the sub-solver are indicated using the "Depends on sub-solver" entry. Some algorithms in `NLopt` have a "Limited" meta-algorithm status because they can only be used to wrap algorithms from `NLopt`.
- Algorithm package. This is the Julia package that either implements the algorithm or calls it from another programming language. `Nonconvex` wraps all these packages using a consistent API while allowing each algorithm to be customized where possible and have its own set of options.
- Order. This is the order of the algorithm. Zero-order algorithms only require the evaluation of the objective and constraint functions, they don't require any gradients or Hessians of objective and constraint functions. First-order algorithms require both the value and gradients of objective and/or constraint functions. Second-order algorithms require the value, gradients and Hessians of objective and/or constraint functions.
- Finite bounds. This is true if the algorithm supports finite lower and upper bound constraints on the decision variables. One special case is the `TOBS` algorithm which only supports binary decision variables so an entry of "Binary" is used instead of true/false.
- Infinite bounds. This is true if the algorithm supports unbounded decision variables either from below, above or both.
- Inequality constraints. This is true if the algorithm supports nonlinear inequality constraints.
- Equality constraints. This is true if the algorithm supports nonlinear equality constraints. Algorithms that only support linear equality constraints are given an entry of "Limited".
- Semidefinite constraints. This is true if the algorithm supports nonlinear semidefinite constraints.
- Integer variables. This is true if the algorithm supports integer/discrete/binary decision variables, not just continuous. One special case is the `TOBS` algorithm which only supports binary decision variables so an entry of "Binary" is used instead of true/false.

## How to contribute?

**A beginner?** The easiest way to contribute is to read the documentation, test the package and report issues.

**An impulsive tester?** Improving the test coverage of any package is another great way to contribute to the `JuliaNonconvex` org. Check the coverage report of any of the packages above by clicking the coverage badge. Find the red lines in the report and figure out tests that would cover these lines of code.

**An algorithm head?** There are plenty of optimization algorithms that can be implemented and interfaced in `Nonconvex.jl`. You could be developing the next big nonconvex semidefinite programming algorithm right now! Or the next constraint handling method for evolutionary algorithms!

**A hacker?** Let's figure out how to wrap some optimization package in Julia in the unique, simple and nimble `Nonconvex.jl` style.

**A software designer?** Let's talk about design decisions and how to improve the modularity of the ecosystem.

You can always reach out by opening an issue.

## How to cite?

If you use Nonconvex.jl for your own research, please consider citing the following publication: Mohamed Tarek. Nonconvex.jl: A Comprehensive Julia Package for Non-Convex Optimization. 2023. doi: 10.13140/RG.2.2.36120.37121.

```
@article{MohamedTarekNonconvexjl,
  doi = {10.13140/RG.2.2.36120.37121},
  url = {https://rgdoi.net/10.13140/RG.2.2.36120.37121},
  author = {Tarek,  Mohamed},
  language = {en},
  title = {Nonconvex.jl: A Comprehensive Julia Package for Non-Convex Optimization},
  year = {2023}
}
```

Nonconvex.jl

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