# EAGO - Easy Advanced Global Optimization

EAGO is an open-source development environment for robust and global optimization in Julia. See the full README for more information.

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EAGO is a deterministic global optimizer designed to address a wide variety of optimization problems, emphasizing nonlinear programs (NLPs), by propagating McCormick relaxations along the factorable structure of each expression in the NLP. Most operators supported by modern automatic differentiation (AD) packages are supported by EAGO and a number utilities for sanitizing native Julia code and generating relaxations on a wide variety of user-defined functions have been included. Currently, EAGO supports problems that have a priori variable bounds defined and have differentiable constraints. That is, problems should be specified in the generic form below:

\begin{aligned} f^{*} = & \min_{\mathbf y \in Y \subset \mathbb R^{n_{y}}} f(\mathbf y) \\ {\rm s.t.} \; \; & \mathbf h(\mathbf y) = \mathbf 0 \\ & \mathbf g(\mathbf y) \leq \mathbf 0 \\ & Y = [\mathbf y^{L}, \mathbf y^{U}] \in \mathbb{IR}^{n} \\ & \qquad \mathbf y^{L}, \mathbf y^{U} \in \mathbb R^{n} \end{aligned}

For each nonlinear term, EAGO makes use of factorable representations to construct bounds and relaxations.

For example, given the function

$$$f(x) = x (x - 5) \sin(x),$$$

a list is generated and rules for constructing McCormick relaxations are used to formulate relaxations in the original decision space, $X$ [1]:

\begin{aligned} v_{1} & = x \\ v_{2} & = v_{1} - 5 \\ v_{3} & = \sin(v_{1}) \\ v_{4} & = v_{1} v_{2} \\ v_{5} & = v_{4} v_{3} \\ f(x) & = v_{5} \\ \end{aligned}

Either these original relaxations, differentiable McCormick relaxations [2], or affine relaxations thereof can be used to construct relaxations of optimization problems useful in branch and bound routines for global optimization. Utilities are included to combine these with algorithms for relaxing implicit functions [3] and forward-reverse propagation of McCormick arithmetic [4].

## Installation

EAGO is a registered Julia package that can be installed using the Julia package manager:

import Pkg
Pkg.add("EAGO")

## Use with JuMP

EAGO makes use of JuMP to improve the user's experience in setting up optimization models. Consider the "process" problem instance from [5]:

\begin{aligned} \max_{\mathbf x \in X} & 0.063 x_{4} x_{7} - 5.04 x_{1} - 0.035 x_{2} - 10 x_{3} - 3.36 x_{2} \\ {\rm s.t.} \; \; & x_{1} (1.12 + 0.13167 x_{8} - 0.00667 x_{8}^{2}) + x_{4} = 0 \\ & -0.001 x_{4} x_{9} x_{6} / (98 - x_{6}) + x_{3} = 0 \\ & -(1.098 x_{8} - 0.038 x_{8}^{2}) - 0.325 x_{6} + x_{7} = 0 \\ & -(x_{2} + x_{5}) / x_{1} + x_{8} = 0 \\ & -x_{1} + 1.22 x_{4} - x_{5} = 0 \\ & x_{9} + 0.222 x_{10} - 35.82 = 0 \\ & -3.0 x_{7} + x_{10} + 133.0 = 0 \\ & X = [10, 2000] \times [0, 16000] \times [0, 120] \times [0, 5000] \\ & \qquad \times [0, 2000] \times [85, 93] \times [90,9 5] \times [3, 12] \times [1.2, 4] \times [145, 162] \end{aligned}

This model can be formulated in Julia as:

using JuMP
import EAGO
# Build model using EAGO's optimizer
model = Model(EAGO.Optimizer)
# Define bounded variables
xL = [10.0, 0.0, 0.0, 0.0, 0.0, 85.0, 90.0, 3.0, 1.2, 145.0]
xU = [2000.0, 16000.0, 120.0, 5000.0, 2000.0, 93.0, 95.0, 12.0, 4.0, 162.0]
@variable(model, xL[i] <= x[i=1:10] <= xU[i])
# Define nonlinear constraints
@NLconstraints(model, begin
-x[1]*(1.12 + 0.13167*x[8] - 0.00667*(x[8])^2) + x[4] == 0.0
-0.001*x[4]*x[9]*x[6]/(98.0 - x[6]) + x[3] == 0.0
-(1.098*x[8] - 0.038*(x[8])^2) - 0.325*x[6] + x[7] == 57.425
-(x[2] + x[5])/x[1] + x[8] == 0.0
end)
# Define linear constraints
@constraints(model, begin
-x[1] + 1.22*x[4] - x[5] == 0.0
x[9] + 0.222*x[10] == 35.82
-3.0*x[7] + x[10] == -133.0
end)
# Define nonlinear objective
@NLobjective(
model,
Max,
0.063*x[4]*x[7] - 5.04*x[1] - 0.035*x[2] - 10*x[3] - 3.36*x[5],
)
# Solve the optimization problem
optimize!(model)

## Documentation

EAGO has numerous features: a solver accessible from JuMP/MathOptInterface (MOI), domain reduction routines, McCormick relaxations, and specialized nonconvex semi-infinite program solvers. A full description of all features can be found on the documentation website.

A series of examples have been provided in the documentation and in the form of Jupyter Notebooks in the separate EAGO-notebooks repository.

## A Cautionary Note on Global Optimization

As a global optimization platform, EAGO's solvers can be used to find solutions of general nonconvex problems with a guaranteed certificate of optimality. However, global solvers suffer from the curse of dimensionality and therefore their performance is outstripped by convex/local solvers.

For users interested in large-scale applications, be warned that problems generally larger than a few variables may prove challenging for certain types of global optimization problems.

## Citing EAGO

Please cite the following paper when using EAGO. In plain text form this is:

Wilhelm, M.E. and Stuber, M.D. EAGO.jl: easy advanced global optimization in Julia.
Optimization Methods and Software. 37(2): 425—450 (2022). DOI: 10.1080/10556788.2020.1786566

As a BibTeX entry:

@article{doi:10.1080/10556788.2020.1786566,
author = {Wilhelm, M.E. and Stuber, M.D.},
title = {EAGO.jl: easy advanced global optimization in Julia},
journal = {Optimization Methods and Software},
volume = {37},
number = {2},
pages = {425-450},
year  = {2022},
publisher = {Taylor & Francis},
doi = {10.1080/10556788.2020.1786566},
URL = {https://doi.org/10.1080/10556788.2020.1786566},
eprint = {https://doi.org/10.1080/10556788.2020.1786566}
}

## References

1. Mitsos, A., Chachuat, B., and Barton, P.I. McCormick-based relaxations of algorithms. SIAM Journal on Optimization. 20(2): 573—601 (2009).
2. Khan, K.A., Watson, H.A.J., and Barton, P.I. Differentiable McCormick relaxations. Journal of Global Optimization. 67(4): 687—729 (2017).
3. Stuber, M.D., Scott, J.K., and Barton, P.I.: Convex and concave relaxations of implicit functions. Optimization Methods and Software 30(3): 424—460 (2015).
4. Wechsung, A., Scott, J.K., Watson, H.A.J., and Barton, P.I. Reverse propagation of McCormick relaxations. Journal of Global Optimization 63(1): 1—36 (2015).
5. Bracken, J., and McCormick, G.P. Selected Applications of Nonlinear Programming. John Wiley and Sons, New York (1968).