(optimization_guide)=
# Optimization
One of the main applications of **CADET-Process** is performing optimization studies.
Optimization refers to the selection of a solution with regard to some criterion.
In the simplest case, an optimization problem consists of minimizing some function $f(x)$ by systematically varying the input values $x$ and computing the value of that function.

$$
\min_x f(x)
$$

In the context of physico-chemical processes, examples for the application of optimization studies include scenarios such as process optimization (see {ref}`batch_elution_optimization_single`) and parameter estimation (see {ref}`fit_column_transport`).
Here, often many variables are subject to optimization, multiple criteria have to be balanced, and additional linear and nonlinear constraints need to be considered.

$$
\min_x f(x) \\

s.t. \\
    &g(x) \le 0, \\
    &h(x) = 0, \\
    &x \in \mathbb{R}^n \\
$$

where $g$ summarizes all inequality constraint functions, and $h$ equality constraints.


In the following, the optimization module of CADET-Process is introduced.
To decouple the problem formulation from the problem solution, two classes are provided:
An {class}`~CADETProcess.optimization.OptimizationProblem` class to specify optimization variables, objectives and constraints.
And an {class}`~CADETProcess.optimization.OptimizerBase` class which allows interfacing different external optimizers to solve the problem.

```{toctree}
:maxdepth: 2

optimization_problem
optimizer
```

## Installation of different Optimizers
To maintain the manageability and efficiency of CADET-Process, some optimizers that come with a substantial number of dependencies are made optional.
This approach ensures that the core package remains lightweight, while providing users the flexibility to install additional optimizers if needed.
By default, scipy and pymoo are installed.
Below, we provide instructions on how to install these optional dependencies.

### Ax/BoTorch
Ax is an adaptable machine learning optimization library developed by Facebook.
At its core, it uses BoTorch, a Bayesian optimization framework also developed by Facebook.
Ax/BoTorch leverages Gaussian Processes to model the objective function and applies Bayesian optimization techniques to find the optimal parameters.

To install Ax as an optional dependency of CADET-Process, use the following command:
```bash
pip install cadet-process[ax]
```

## Advanced Configuration
```{toctree}
:maxdepth: 2

parallel_evaluation
evaluator
multi_objective_optimization
variable_indices
variable_normalization
variable_dependencies
```