I have demonstrated how to solve linear problems using SciPy in Python, and how to solve non-linear optimization problems in R (e.g. with nloptr for gradient-descent optimization). In this post I want to show how to solve quadratic optimization tasks in Python, using the cvxopt-module.

Using the cvxopt-module in Python I will model and solve the problem below:

In order to solve this problem using the cvxopt-module in Python, one must restate the problem in vector-matrix notation form – such that it satifies the template below:

In below coding example I solve the problem above, stating it according to the given template in vector-matrix notation form:

# preparing the modelling inputs, using cvxopt from cvxopt import matrix # setting matrices and vectors P = matrix([[2.0,0.0],[0.0,4.0]]) q = matrix([6.0,20.0]) G = matrix([[-1.0,-1.0],[-2.0,-3.0]]) h = matrix([-10.0,0.0]) # importing the solver from cvxopt-module, providing the solver function from cvxopt import solvers # using the qp-solver from cvxopt.solvers solution = solvers.qp(P=P, q=q, G=G, h=h)

pcost dcost gap pres dres 0: 3.1195e+00 1.0142e+02 2e+01 1e+00 3e-01 1: 1.1086e+02 1.1722e+02 4e+00 9e-02 2e-02 2: 1.1727e+02 1.1733e+02 4e-02 9e-04 2e-04 3: 1.1733e+02 1.1733e+02 4e-04 9e-06 2e-06 4: 1.1733e+02 1.1733e+02 4e-06 9e-08 2e-08 Optimal solution found.

# optimal x1 solution["x"][0]

4.6666665332235375

# optimal x2 solution["x"][1]

2.6666665168063117

# optimal objective function value solution["primal objective"] + 5

122.33332669148787

If interested, please go ahead and read my blog post on quadratic programming in R, using the quadprog R-package from CRAN.

Data scientist focusing on simulation, optimization and modeling in R, SQL, VBA and Python

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