GEKKO
Developers	Logan Beal andJohn D. Hedengren
Stable release	1.3.1 / December 30, 2025; 48 days ago (2025-12-30)
Operating system	Cross-Platform
Type	Technical computing
License	MIT
Website	gekko.readthedocs.io/en/latest/
Repository	github.com/BYU-PRISM/GEKKO

TheGEKKO Python package^[1] solves large-scale mixed-integer and differential algebraic equations with nonlinear programming solvers (IPOPT,APOPT, BPOPT,SNOPT,MINOS). Modes of operation include machine learning, data reconciliation, real-time optimization, dynamic simulation, and nonlinearmodel predictive control. In addition, the package solvesLinear programming (LP),Quadratic programming (QP),Quadratically constrained quadratic program (QCQP),Nonlinear programming (NLP),Mixed integer programming (MIP), andMixed integer linear programming (MILP). GEKKO is available in Python and installed with pip from PyPI of the Python Software Foundation.

pipinstallgekko

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 (Raspberry Pi) processor options to solve without an Internet connection. GEKKO is an extension of theAPMonitor 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^[2] used to test the performance ofnonlinear programming solvers. This particular optimization problem has an objective function${\displaystyle \underset{x\in \mathbb{R}}{min}{x}_1{x}_4(x_1+x_2+x_3)+x_3}$ and subject to the inequality constraint${\displaystyle x_1{x}_2{x}_3{x}_4\ge 25}$ and equality constraint${\displaystyle {x_1}^2+{x_2}^2+{x_3}^2+{x_4}^2=40}$. The four variables must be between a lower bound of 1 and an upper bound of 5. The initial guess values are${\displaystyle x_1=1,x_2=5,x_3=5,x_4=1}$. This optimization problem is solved with GEKKO as shown below.

fromgekkoimportGEKKOm=GEKKO()# Initialize gekko# Initialize variablesx1=m.Var(value=1,lb=1,ub=5)x2=m.Var(value=5,lb=1,ub=5)x3=m.Var(value=5,lb=1,ub=5)x4=m.Var(value=1,lb=1,ub=5)# Equationsm.Equation(x1*x2*x3*x4>=25)m.Equation(x1**2+x2**2+x3**2+x4**2==40)m.Minimize(x1*x4*(x1+x2+x3)+x3)m.solve(disp=False)# Solveprint("x1: "+str(x1.value))print("x2: "+str(x2.value))print("x3: "+str(x3.value))print("x4: "+str(x4.value))print("Objective: "+str(m.options.objfcnval))

Applications includecogeneration (power and heat),^[3]drilling automation,^[4] severe slugging control,^[5] solar thermal energy production,^[6]solid oxide fuel cells,^[7][8] flow assurance,^[9]Enhanced oil recovery,^[10]Essential oil extraction,^[11] andUnmanned Aerial Vehicles (UAVs).^[12] There are many other references to APMonitor and GEKKO as a sample of the types of applications that can be solved. GEKKO is developed from the National Science Foundation (NSF) research grant #1547110^{[13][14][15][16]} and is detailed in a Special Issue collection on combined scheduling and control.^[17] Other notable mentions of GEKKO are the listing in the Decision Tree for Optimization Software,^[18] added support forAPOPT and BPOPT solvers,^[19] projects reports of the online Dynamic Optimization course from international participants.^[20] GEKKO is a topic in online forums where users are solving optimization and optimal control problems.^[21][22] GEKKO is used for advanced control in the Temperature Control Lab (TCLab)^[23] for process control education at 20 universities.^{[24][25][26][27]}

One application ofmachine 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${\displaystyle 1-\cos (x)}$. Anartificial 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 asbackpropagation. 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 satisfiesKarush–Kuhn–Tucker conditions. Using a gradient-based optimizer allows additional constraints that may be imposed with domain knowledge of the data or system.

fromgekkoimportbrainimportnumpyasnpb=brain.Brain()b.input_layer(1)b.layer(linear=3)b.layer(tanh=3)b.layer(linear=3)b.output_layer(1)x=np.linspace(-np.pi,3*np.pi,20)y=1-np.cos(x)b.learn(x,y)

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 (if available) to impose a structure that is valid over a wider range of conditions. In the example above, the hyperbolic tangent activation function (hidden layer 2) 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.

importmatplotlib.pyplotaspltxp=np.linspace(-2*np.pi,4*np.pi,100)yp=b.think(xp)plt.figure()plt.plot(x,y,"bo")plt.plot(xp,yp[0],"r-")plt.show()

Optimal control is the use ofmathematical optimization to obtain a policy that is constrained by differential${\displaystyle \left(\frac{d{x}_1}{dt}=u\right)}$, equality${\displaystyle \left(x_1(0)=1\right)}$, or inequality${\displaystyle \left(-1\le u(t)\le 1\right)}$ equations and minimizes an objective/reward function${\displaystyle \left(\underset{u}{min}\frac{1}{2}{\int }_0^2{x}_1^2(t)dt\right)}$. 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.

fromgekkoimportGEKKOimportnumpyasnpimportmatplotlib.pyplotaspltm=GEKKO()# initialize gekkont=101m.time=np.linspace(0,2,nt)# Variablesx1=m.Var(value=1)x2=m.Var(value=0)u=m.Var(value=0,lb=-1,ub=1)p=np.zeros(nt)# mark final time pointp[-1]=1.0final=m.Param(value=p)# Equationsm.Equation(x1.dt()==u)m.Equation(x2.dt()==0.5*x1**2)m.Minimize(x2*final)m.options.IMODE=6# optimal control modem.solve()# solveplt.figure(1)# plot resultsplt.plot(m.time,x1.value,"k-",label=r"$x_1$")plt.plot(m.time,x2.value,"b-",label=r"$x_2$")plt.plot(m.time,u.value,"r--",label=r"$u$")plt.legend(loc="best")plt.xlabel("Time")plt.ylabel("Value")plt.show()

• APMonitor andPython.

• GEKKO Overview with Machine Learning and Optimization
• GEKKO Documentation
• GEKKO Source Code
• GEKKO on PyPI for Python pip install
• GEKKO is open-source product ofNational Science Foundation (NSF) research grant 1547110
• References to APMonitor and GEKKO in the literature
• 18 examples of GEKKO: machine learning, optimal control, data regression
• Gekko Download Statistics

• Numerical programming languages
• Mathematical optimization software

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