基于Lagrange-Newton法的SQP局部算法python实现

序列二次规划（SQP）是解决约束优化问题中较好的一种算法，其流程为

在实现算法的过程中，使用了scipy.optimize模块：

scipy.optimize.minimize(fun, x0, args=(), method=None, jac=None, hess=None, hessp=None, bounds=None, constraints=(), tol=None, callback=None, options=None)
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minimize f(x) subject to

g_i(x) >= 0,  i = 1,...,m
h_j(x)  = 0,  j = 1,...,p

import sympy,numpy
from numpy import *
from scipy.optimize import fsolve, minimize

def SQP(x,w,v):
    x0=numpy.array([1,1])
    w0=5;v0=2
    W=numpy.array([[2+2*w0,0],[0,2]])
    x = x0
    w,v=w0,v0
    for i in range(30):
     f = x[0]**2+(x[1]-4)**2
     f_d = numpy.array([2*x[0],2*x[1]-8])
     g = x[1] - (x[0]-4)**2
     g_d = numpy.array([-2*(x[0]-4),1])
     h = 2*x[0]-x[1]-1
     h_d = numpy.array([2,-1])
     #res = SQP(numpy.array([[1.0],[-3.0]]))
     #********求lambda最小值************
     Q =  lambda d: 0.5 * (numpy.dot(numpy.dot(d.T, W), d)) + numpy.dot(f_d.T, d)
     cons = ({'type': 'ineq', 'fun': lambda x:  x[1] - (x[0]-4)**2+numpy.dot(g_d.T, x)},
         {'type': 'eq', 'fun': lambda x:2*x[0]-x[1]-1+numpy.dot(h_d.T, x)})
     bnds = ((0, None), (0, None))
     res = minimize(Q, (0, 0), method='SLSQP', bounds=bnds, constraints=cons)
     print res['x']
     #********end************
     x = x+numpy.array(res['x'])
     A=[[g_d[0],h_d[0]],[g_d[1],h_d[1]]] #线性方程组求解
     A = numpy.array(A)
     Q_d = numpy.dot(W, res['x'])
     b=Q_d
     b = numpy.array(b)
     solu = numpy.linalg.solve(A, b)
     w=solu[0]
     v=solu[1]
     print "d=%s,x=%s"%(res['x'],x)