NeuM4 / Expensive-Black-Box-Optim-ChemEng

Expensive Black-Box Optimization for Chemical Engineering applications

This repository provides state-of-the-art black box optimization techniques for chemical engineering problems.

Here, we consider a constrained optimization problem of the following form:

This includes one black-box objective f, black-box constraints g_k, continuous variables x, and input box-bounds x^L and x^U. ω denotes the potential presence of stochasticity.

Methods

The methods examined in this work come from the model-based, direct search and finite difference philosophies and involve well known implementations. For the existing implementations, we created a wrapper so that the solvers take the same input as our in-house implementations.

The algorithms implemented can be found in the algorithms folder and include:

• Bayesian optimization: In-house, model-based
• CUATRO (local and global): In-house, model-based
• DIRECT-L randomized: direct search, adapted from the NLopt library
• Newton, BFGS, Adam: Finite differences, In-house
• Py-BOBYQA: model-based, as found here
• SQSnobfit: model-based, as found here
• Nesterov: smoothed finite differences, in-house
• Simplex/Nelder-Mead: direct, in-house

Applications

The case studies are intended to present the behaviour of typical chemical/process engineering problems while remaining tractable. These include two constrained test functions (test_functions) and five chemical engineering applications (case_studies):

• Quadratic constrained: 2-dimensional (2-d), deterministic and stochastic, convex
• Rosenbrock constrained: 2-d, deterministic and stochastic
• Real-Time Optimization: 2-d, deterministic and stochastic, constrained, convex
• Self-Optimizing Reactor: 2-d, deterministic and stochastic, constrained, convex
• Model-Based Design of Experiments: 4-d, determinsitic
• PID controller tuning: 4-d, deterministic and stochastic
• Control Law Synthesis: 10-d, stochastic

Aim

The aim is to equip the practitioner with the information to choose a DFO algorithm based on their problem's size and how well-behaved, and stochastic their application is.

The subsequent comparison is based on the assumption that the black boxes that are queried are very expensive, meaning that the optimum should be found in the lowest number of function evaluations. Emphasis is placed on the discussion of how different algorithms can handle increasing levels of stochasticity inherent to the problems and on constraint satisfaction.

Dependencies and Installation

• environment.yml: Lists all packages plus the version used. Using conda env create --file environment.yml in the anaconda prompt will recreate the dependencies

• requirements.txt: Lists all packages plus the version used. If the reader prefers pip, a virtual environment should be created and the author's dependencies recreated with pip install -r requirements.txt

Implementation

The functions that are used are defined as follows:

def problem(x):
    f = @(x) ..
    g1 = @(x) ..
    g2 = @(x) ..

    return f(x), [g1(x), g2(x)]

where the first output denotes the objective function, and where the second output denotes the list of constraint functions. If the problem has no black-box constraints, then the list should be empty return f(x), [].

The solver (wrappers) require only three essential inputs, the problem function, the initial guess, and the input bounds. Methods that rely on penalty functions to map the constraints also require the number of constraints. The function evaluation budget has a default value of 100.

x0 = np.array([0.1, 0.2, 0.3]) # for 3-d problems
bounds = np.array([[0,1], [0,1], [0,1]])
max_f_eval = 100

# Example of solver call
dictionary = SQSnobFitWrapper().solve(problem, x0, bounds, \
                                    maxfun = max_f_eval, constraints=2)

The input argument formats can change between solvers and examples of each use can be found within problem_comparisons.py for each case study comparison problem_comp folder.

Plots

The comparisons within each folder generate a number of plots, the most important of which are stored within the Publication plots folder of each case study.

For the real-time optimization case study for example, this folder contains the function evaluation versus number of function evaluations convergence plot solution bands for the model-based, and other methods for the deterministic and randomized case study. For the random case, the model-based convergence plot looks as follows (RTORand_Model.svg):

It also contains the convergence of the best methods in the 2-d solution space if applicable (RTORand_2D_convergence_best.svg):

Finally, it contains box plots of how the optimality gap changes for the best methods at increasing levels of stochasticity if applicable (RTO_feval50ConvergenceLabel.svg):

Other plot folders of the RTO comparison contain detailed convergence plots for each method on each case study.