# Welcome to oppy’s documentation!

Heat development of a heating pipe model with a defect cooling pipe.

The optimization package oppy is implemented in the programming language Python. Besides algorithms for solving constrained, unconstrained and non-linear optimization problems, the package contains built-in iterative methods for solving linear systems.

Advanced methods for optimization are included such as SQP (Square Quadratic Programming), Augmented Lagrangian and different newton-type methods. Furthermore certain Krylov methods are implemented for solving linear systems in a stable way.

The goal is to provide a straightforward integration of the library to other applications such that other methods benefit from it. Currently we are working on a combination of optimal control problems with the model order reduction method.

The idea behind oppy was to provide some optimization methods which are used in the group of Prof. Dr. Volkwein quite often. After a while oppy grew up to a whole optimization package.

The package is still in develop mode. If you want to install oppy use

pip install git+https://gitlab.inf.uni-konstanz.de/ag-volkwein/oppy

## Available subpackages

### conOpt

Subpackage which provide some methods for constraint optimization. For problems which are subject to equality and inequality constraints like

$\min f(x)$
$\text{s. t. } e(x) = 0$
$g(x) \leq 0$

we can use

1. Penalty Method
2. Augmented Lagrangian Method
3. SQP with a BFGS update strategy (at the moment only equality constraint)

and for box constraint problems like

$\min f(x)$
$\text{s. t. } x_a \leq x \leq x_b$

we can use

2. The L-BFGS-B Method
3. Projected Newton-Krylov Method (if you can provide the action of the second derivative)

### itMet

Iterative methods for solving linear systems like

$Ax = b.$

Here we can use either stationary methods like

1. Jacobi
2. Gauß-Seidel
3. SOR

or we use krylov methods like

1. steepest descent
2. CG
3. GMRES

For future release we are planing to add preconditioning in the krylov methods. There of course you will be able to use the stationary methods as precondition method.

### leastSquares

Subpackage which provide some methods for linear and nonlinear least squares problems, e.g:

$\text{min} ||Ax - b||_2$

and

$\text{min} \frac{1}{2}||f(x)||_2^2$

Right now we can solve this kind of problems with the following methods.

• Linear Least Squares
1. linear least squares (solving normal equation)
• Optimization Methods
1. Gauss-Newton algorithm with several choices.

### linOpt

Linear optimization methods. With the methods in this subpackage you can solve linear programming

$\text{max } c^T x$
$\text{s. t. } Ax \leq b$
$x \geq 0$

with or without integer constraints. For that kind of problems we have the following methods:

1. simplex
2. branch and bound

### multOpt

Scalarization methods for solving (possibly box-constrained) multiobjective optimization problems of the form

$\min (f_1(x), \ldots, f_k(x)),$
$\text{s. t. } x_a \leq x \leq x_b.$

The general idea of scalarization methods is to transform the multiobjective optimization problem into a series of scalar optimization problems. which can then be solved by using methods from unconstrained or constrained optimization (see the subpackages unconOpt or conOpt). Here we can use the following three scalarization methods

1. Weighted-Sum Method (WSM)
2. Euclidean Reference Point Method (ERPM)
3. Pascoletti-Serafini Method (PSM)

### unconOpt

Subpackage which provide some methods for unconstrained optimization, e.g:

$\min_{x \in \mathbb{R}^n} f(x)$

Right now we can solve this kind of problems with line search based first- and second-order methods.