GEKKO works on all platforms and with Python 2.7 and 3+. By default, the problem is sent to a public server where the solution is computed and returned to Python. There are Windows, MacOS, Linux, and ARM processor options to solve without an Internet connection. GEKKO is an extension of the APMonitor Optimization Suite but has integrated the modeling and solution visualization directly within Python. A mathematical model is expressed in terms of variables and equations such as the Hock & Schittkowski Benchmark Problem #71 used to test the performance of nonlinear programming solvers. This particular optimization problem has an objective function and subject to the inequality constraint and equality constraint. The four variables must be between a lower bound of 1 and an upper bound of 5. The initial guess values are. This optimization problem is solved with GEKKO as shown below. from gekko import GEKKO m = GEKKO # Initialize gekko
Applications include cogeneration, drilling automation, severe slugging control, solar thermal energy production, solid oxide fuel cells, flow assurance, Enhanced oil recovery, Essential oil extraction, and Unmanned Aerial Vehicles. There are many other references to as a sample of the types of applications that can be solved. GEKKO is developed from the National Science Foundation research grant #1547110 and is detailed in a Special Issue collection on combined scheduling and control. Other notable mentions of GEKKO are the listing in the Decision Tree for Optimization Software, added support for APOPT and BPOPT solvers, projects reports of the online Dynamic Optimization course from international participants. GEKKO is a topic in online forums where users are solving optimization and optimal control problems. GEKKO is used for advanced control in the Temperature Control Lab for process control education at 20 universities.
Machine Learning
One application of machine learning is to perform regression from training data to build a correlation. In this example, deep learning generates a model from training data that is generated with the function. An artificial neural network with three layers is used for this example. The first layer is linear, the second layer has a hyperbolic tangent activation function, and the third layer is linear. The program produces parameter weights that minimize the sum of squared errors between the measured data points and the neural network predictions at those points. GEKKO uses gradient-based optimizers to determine the optimal weight values instead of standard methods such as backpropagation. The gradients are determined by automatic differentiation, similar to other popular packages. The problem is solved as a constrained optimization problem and is converged when the solver satisfies Karush–Kuhn–Tucker conditions. Using a gradient-based optimizer allows additional constraints that may be imposed with domain knowledge of the data or system. from gekko import brain import numpy as np b = brain.Brain b.input_layer b.layer b.layer b.layer b.output_layer x = np.linspace y = 1 - np.cos b.learn
The neural network model is tested across the range of training data as well as for extrapolation to demonstrate poor predictions outside of the training data. Predictions outside the training data set are improved with hybrid machine learning that uses fundamental principles to impose a structure that is valid over a wider range of conditions. In the example above, the hyperbolic tangent activation function could be replaced with a sine or cosine function to improve extrapolation. The final part of the script displays the neural network model, the original function, and the sampled data points used for fitting. xp = np.linspace yp = b.think import matplotlib.pyplot as plt plt.figure plt.plot plt.plot plt.show
Optimal Control
is the use of mathematical optimization to obtain a policy that is constrained by differential, equality, or inequality equations and minimizes an objective/reward function. The basic optimal control is solved with GEKKO by integrating the objective and transcribing the differential equation into algebraic form with orthogonal collocation on finite elements. from gekko import GEKKO import numpy as np import matplotlib.pyplot as plt m = GEKKO # initialize gekko nt = 101 m.time = np.linspace
Variables
x1 = m.Var x2 = m.Var u = m.Var p = np.zeros # mark final time point p = 1.0 final = m.Param
