Two tools used in this course are MATLAB and Python. Python Gekko is a package within Python to provide support for solution of dynamic simulation, estimation, and control.
There are many tools for simulation and optimization of dynamic systems. There are also many factors to consider when selecting the appropriate tool for a particular task. Some of the considerations for a dynamic optimization tool are size (scaling with number equations and degrees of freedom), speed (use in real-time estimation or control), usability (training requirements and ease of adaptation), extensibility (ability to modify for new or complex systems), support (commercial sustainability or open-source community), and portability (ability to embed on a system or communicate with a particular platform).
Here is a list of some of the most popular optimization packages in Matlab:
fmincon: This is a built-in function in Matlab that implements a variety of optimization algorithms for constrained nonlinear optimization.
fminunc: This is another built-in function in Matlab that implements a variety of optimization algorithms for unconstrained nonlinear optimization.
linprog: This is a built-in function in Matlab that implements linear programming algorithms for solving optimization problems with linear constraints.
quadprog: This is a built-in function in Matlab that implements quadratic programming algorithms for solving optimization problems with quadratic objective functions and linear constraints.
Global Optimization Toolbox: This is a toolbox in Matlab that provides a variety of algorithms for global optimization, including genetic algorithms, particle swarm optimization, and simulated annealing.
Here is a list of some of the most popular optimization packages in Python:
scipy.optimize: This is a module in the SciPy library that provides a variety of optimization algorithms for solving nonlinear optimization problems.
cvxpy: This is a package that provides a modeling language for convex optimization problems, and it has interfaces to several solvers including CVXOPT, ECOS, and SCS.
gekko: This is a package that provides a modeling language for linear, nonlinear, and mixed integer programming problems, and it has interfaces to several solvers including IPOPT, APOPT, and BPOPT.
pygmo: This is a package that provides a variety of optimization algorithms for global optimization, including genetic algorithms, particle swarm optimization, and simulated annealing.
PyOpt: This is a package that provides a variety of optimization algorithms for solving nonlinear optimization problems, including gradient-based and derivative-free methods.
The objective of this section is to introduce optimization in MATLAB and Python. This is not intended to be a general tutorial but more specifically to cover topics that are relevant for dynamic optimization. Topics of interest for dynamic optimization are listed below.
Introduction to Nonlinear Optimization
This course uses principles from nonlinear programming but does not delve into the details of solution strategies. Instead of detailing how a nonlinear programming solver iterates to find a solution, this course focuses on formulating problems and the use of differential constraints in describing and optimizing dynamic systems. For more background on how nonlinear programming solvers work, refer to the Course on Optimization and the associated online Design Optimization Textbook. A good illustrative example is the Two Bar Truss optimization or the example below.
GEKKO (Python)
GEKKO Python is designed for large-scale optimization and accesses solvers of constrained, unconstrained, continuous, and discrete problems. Problems in linear programming, quadratic programming, integer programming, nonlinear optimization, systems of dynamic nonlinear equations, and multi-objective optimization can be solved. The platform can find optimal solutions, perform tradeoff analyses, balance multiple design alternatives, and incorporate optimization methods into external modeling and analysis software. It is free for academic and commercial use under the MIT license.
APMonitor is a tool for nonlinear optimization and dynamic optimization. Source files are available for Julia, Python, and MATLAB. Users can select either a public server or a local host if the APMonitor Server is installed locally. The user must also specify an application name.
apm_meas(server,app,name,value) - Specify a measurement
apm_option(server,app,name,value) - Set an option
apm_sol(server,app) - Retrieve the solution
apm_tag(server,app,name) - Retrieve an NLC, FV, MV, SV, or CV option
apm_web(server,app) - Open a web interface to the dashboard
apm_web_var(server,app) - Open a web interface to the variable values
csv_load(server,app,filename) - Load data file
An example script is below. The same commands are used in Python, MATLAB, and Julia with minor modifications such as the ';' in MATLAB to suppress console output or change in the comment characters.