python - 仅依赖于 NumPy/SciPy 的二次规划 (QP) 求解器？

Question

我希望学生在作业中解决二次程序，而不必安装 cvxopt 等额外软件。是否有仅依赖于 NumPy/SciPy 的 python 实现？

Answer 1

我对二次规划不是很熟悉，但我认为你可以使用 scipy.optimize 的约束最小化算法来解决这类问题。这是一个例子:

import numpy as np
from scipy import optimize
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d.axes3d import Axes3D

# minimize
#     F = x[1]^2 + 4x[2]^2 -32x[2] + 64

# subject to:
#      x[1] + x[2] <= 7
#     -x[1] + 2x[2] <= 4
#      x[1] >= 0
#      x[2] >= 0
#      x[2] <= 4

# in matrix notation:
#     F = (1/2)*x.T*H*x + c*x + c0

# subject to:
#     Ax <= b

# where:
#     H = [[2, 0],
#          [0, 8]]

#     c = [0, -32]

#     c0 = 64

#     A = [[ 1, 1],
#          [-1, 2],
#          [-1, 0],
#          [0, -1],
#          [0,  1]]

#     b = [7,4,0,0,4]

H = np.array([[2., 0.],
              [0., 8.]])

c = np.array([0, -32])

c0 = 64

A = np.array([[ 1., 1.],
              [-1., 2.],
              [-1., 0.],
              [0., -1.],
              [0.,  1.]])

b = np.array([7., 4., 0., 0., 4.])

x0 = np.random.randn(2)

def loss(x, sign=1.):
    return sign * (0.5 * np.dot(x.T, np.dot(H, x))+ np.dot(c, x) + c0)

def jac(x, sign=1.):
    return sign * (np.dot(x.T, H) + c)

cons = {'type':'ineq',
        'fun':lambda x: b - np.dot(A,x),
        'jac':lambda x: -A}

opt = {'disp':False}

def solve():

    res_cons = optimize.minimize(loss, x0, jac=jac,constraints=cons,
                                 method='SLSQP', options=opt)

    res_uncons = optimize.minimize(loss, x0, jac=jac, method='SLSQP',
                                   options=opt)

    print '\nConstrained:'
    print res_cons

    print '\nUnconstrained:'
    print res_uncons

    x1, x2 = res_cons['x']
    f = res_cons['fun']

    x1_unc, x2_unc = res_uncons['x']
    f_unc = res_uncons['fun']

    # plotting
    xgrid = np.mgrid[-2:4:0.1, 1.5:5.5:0.1]
    xvec = xgrid.reshape(2, -1).T
    F = np.vstack([loss(xi) for xi in xvec]).reshape(xgrid.shape[1:])

    ax = plt.axes(projection='3d')
    ax.hold(True)
    ax.plot_surface(xgrid[0], xgrid[1], F, rstride=1, cstride=1,
                    cmap=plt.cm.jet, shade=True, alpha=0.9, linewidth=0)
    ax.plot3D([x1], [x2], [f], 'og', mec='w', label='Constrained minimum')
    ax.plot3D([x1_unc], [x2_unc], [f_unc], 'oy', mec='w',
              label='Unconstrained minimum')
    ax.legend(fancybox=True, numpoints=1)
    ax.set_xlabel('x1')
    ax.set_ylabel('x2')
    ax.set_zlabel('F')

输出:

Constrained:
  status: 0
 success: True
    njev: 4
    nfev: 4
     fun: 7.9999999999997584
       x: array([ 2.,  3.])
 message: 'Optimization terminated successfully.'
     jac: array([ 4., -8.,  0.])
     nit: 4

Unconstrained:
  status: 0
 success: True
    njev: 3
    nfev: 5
     fun: 0.0
       x: array([ -2.66453526e-15,   4.00000000e+00])
 message: 'Optimization terminated successfully.'
     jac: array([ -5.32907052e-15,  -3.55271368e-15,   0.00000000e+00])
     nit: 3