Python's cvxopt module

?? The modules in Python that support convex optimization (convex planning) are cvxopt and are installed in the following ways:

1. Unloading the NumPy in the original Pyhon
2. Install cvxopt WHL file, link to: https://www.lfd.uci.edu/~gohlke/pythonlibs/
3. Install numpy+mkl WHL file, link to: https://www.lfd.uci.edu/~gohlke/pythonlibs/

This installation is chosen because of the incompatibility of Python's WHL and Pip Direct install.

?? Cvxopt Official documentation URL: http://cvxopt.org/index.html, now the latest version is 1.1.9, co-developed by Martin Andersen, Joachim Dahl and Lieven Vandenberghe , it can solve the problem of linear programming and two sub-type programming, and its application scenarios such as hard Margin SVM in SVM.

?? Examples of cvxopt use are:

Linear programming problems

Example 1:

Python Program code:

ImportNumPy asNp fromCvxoptImportMatrix, Solversa=Matrix ([[-1.0,-1.0,0.0,1.0], [1.0,-1.0,-1.0,-2.0]]) b=Matrix ([1.0,-2.0,0.0,4.0]) C=Matrix ([2.0,1.0]) Sol=SOLVERS.LP (C,A,B)Print(sol[' x '])Print(Np.dot (sol[' x ']. T, C))Print(sol[' Primal objective '])

Output Result:

     Pcost dcost Gap pres dres k/t 0:2.6471e+00-7.0588e-01 2e+01 8e-01 2e+00 1e+00 1:3.0726e+0 0 2.8437e+00 1e+00 1e-01 2e-01 3e-01 2:2.4891e+00 2.4808e+00 1e-01 1e-02 2e-02 5e-02 3:2.4999e+00 2.4998e+0 0 1e-03 1e-04 2e-04 5e-04 4:2.5000e+00 2.5000e+00 1e-05 1e-06 2e-06 5e-06 5:2.5000e+00 2.5000e+00 1e-07 1e -08 2e-08 5e-08optimal solution found. {' Primal objective ': 2.4999999895543072, ' s ': <4x1 matrix, tc= ' d ';, ' dual infeasibility ': 2.257878974569382e-08, ' Primal Slack ': 2.0388399547464153e-08, ' dual objective ': 2.4999999817312535, ' residual as dual infeasibility certificate ': None, ' dual slack ': 3.529915972607509e-09, ' x ': <2x1 matrix, tc= ' d ';, ' iterations ': 5, ' gap ': 1.3974945737723005e -07, ' residual as Primal infeasibility certificate ': None, ' z ': <4x1 matrix, tc= ' d ', ' Y ': <0x1 matrix, tc= ' d ' ;, ' status ': ' Optimal ', ' Primal infeasibility ': 1.1368786228004961e-08, ' relative gap ': 5.5899783359379607e-08}[5.00e-01][1.50e+00][[2.49999999]] 

Example 2

Python program code

ImportNumPy asNp fromCvxoptImportMatrix, Solversa=Matrix ([[1.0,0.0,-1.0], [0.0,1.0,-1.0]]) b=Matrix ([2.0,2.0,-2.0]) C=Matrix ([1.0,2.0]) d=Matrix ([-1.0,-2.0]) Sol1=SOLVERS.LP (C,A,B)min =Np.dot (sol1[' x ']. T, c) sol2=SOLVERS.LP (D,A,B)Max = -Np.dot (sol2[' x ']. T, D)Print(' min=%s, max=%s'%(min[0][0],Max[0][0]))

Output Result:

     pcost       dcost       gap    pres   dres   k/t 0:  4.0000e+00 -0.0000e+00  4e+00  0e+00  0e+00  1e+00 1:  2.7942e+00  1.9800e+00  8e-01  9e-17  7e-16  2e-01 2:  2.0095e+00  1.9875e+00  2e-02  4e-16  2e-16  7e-03 3:  2.0001e+00  1.9999e+00  2e-04  2e-16  6e-16  7e-05 4:  2.0000e+00  2.0000e+00  2e-06  6e-17  5e-16  7e-07 5:  2.0000e+00  2.0000e+00  2e-08  3e-16  7e-16  7e-09Optimal solution found.     pcost       dcost       gap    pres   dres   k/t 0: -4.0000e+00 -8.0000e+00  4e+00  0e+00  1e-16  1e+00 1: -5.2058e+00 -6.0200e+00  8e-01  1e-16  7e-16  2e-01 2: -5.9905e+00 -6.0125e+00  2e-02  1e-16  0e+00  7e-03 3: -5.9999e+00 -6.0001e+00  2e-04  1e-16  2e-16  7e-05 4: -6.0000e+00 -6.0000e+00  2e-06  1e-16  2e-16  7e-07Optimal solution found.min=2.00000000952,max=5.99999904803

Two sub-type planning problems

Where p,q,g,h,a,b is the input matrix, the problem is solved by using QP algorithm.

Example 1:

Python Program code:

 fromCvxoptImportMatrix, SOLVERSQ= 2*Matrix ([[2, .5], [.5,1]]) p=Matrix ([1.0,1.0]) G=Matrix ([[-1.0,0.0],[0.0,-1.0]]) H=Matrix ([0.0,0.0]) A=Matrix ([1.0,1.0], (1,2)) b=Matrix1.0) Sol=SOLVERS.QP (Q, p, G, H, A, B)Print(sol[' x '])Print(sol[' Primal objective '])

Output Result:

     pcost       dcost       gap    pres   dres 0:  1.8889e+00  7.7778e-01  1e+00  2e-16  2e+00 1:  1.8769e+00  1.8320e+00  4e-02  0e+00  6e-02 2:  1.8750e+00  1.8739e+00  1e-03  1e-16  5e-04 3:  1.8750e+00  1.8750e+00  1e-05  6e-17  5e-06 4:  1.8750e+00  1.8750e+00  1e-07  2e-16  5e-08Optimal solution found.[ 2.50e-01][ 7.50e-01]

Example 2:

Python Program code:

 fromCvxoptImportMatrix, SOLVERSP=Matrix ([[1.0,0.0], [0.0,0.0]]) Q=Matrix ([3.0,4.0]) G=Matrix ([[-1.0,0.0,-1.0,2.0,3.0], [0.0,-1.0,-3.0,5.0,4.0]]) H=Matrix ([0.0,0.0,-15.0,100.0,80.0]) Sol=SOLVERS.QP (P, Q, G, h)Print(sol[' x '])Print(sol[' Primal objective '])

Output results

     pcost       dcost       gap    pres   dres 0:  1.0780e+02 -7.6366e+02  9e+02  0e+00  4e+01 1:  9.3245e+01  9.7637e+00  8e+01  6e-17  3e+00 2:  6.7311e+01  3.2553e+01  3e+01  6e-17  1e+00 3:  2.6071e+01  1.5068e+01  1e+01  2e-17  7e-01 4:  3.7092e+01  2.3152e+01  1e+01  5e-18  4e-01 5:  2.5352e+01  1.8652e+01  7e+00  7e-17  3e-16 6:  2.0062e+01  1.9974e+01  9e-02  2e-16  3e-16 7:  2.0001e+01  2.0000e+01  9e-04  8e-17  5e-16 8:  2.0000e+01  2.0000e+01  9e-06  1e-16  2e-16Optimal solution found.[ 7.13e-07][ 5.00e+00]20.00000617311241

Python's cvxopt module