radau伪谱法约束程序

Radau伪谱法是一种常用的求解优化问题的数值方法。它是一种基于伪谱方法的伪谱法，常用于求解约束优化问题。下面给出一个使用Radau伪谱法求解约束优化问题的示例程序。

from scipy.integrate import solve_ivp
from scipy.interpolate import CubicSpline
import numpy as np

def radau_pseudospectral(n, lb, ub, constraints, cost, guess=None, tol=1e-6, maxiter=100):
    # 计算伪谱节点和权重
    nodes, weights = np.polynomial.legendre.leggauss(n+1)
    nodes = 0.5*(nodes+1)*(ub-lb)+lb
    weights = 0.5*(ub-lb)*weights

    # 创建初始猜测
    if guess is None:
        guess = np.zeros((n+1,len(lb)))
        for i in range(n+1):
            guess[i] = lb+(ub-lb)*i/n
    
    # 定义状态和状态导数的函数
    def f(t, x):
        return cost(t, x)

    def jac(t, x):
        return np.concatenate((np.zeros((len(x),1)), np.eye(len(x))), axis=1)

    # 定义约束函数
    def g(x):
        return np.array([c(x) for c in constraints])

    def dg(x):
        return np.array([np.gradient(c(x), axis=0) for c in constraints]).T

    # 定义边界约束
    def bc(x0, xn):
        return np.concatenate((g(x0), g(xn)))

    def dbc(x0, xn):
        return np.concatenate((dg(x0), dg(xn)))

    # 定义目标函数
    def obj(x):
        t = np.zeros(n+1)
        for i in range(n+1):
            t[i] = nodes[i]
        fval = f(t, x)
        return np.sum(weights*fval)

    # 定义求解器
    def fun(t, z):
        x = z[:len(lb)]
        lam = z[len(lb):]
        return np.concatenate((f(t, x), np.dot(dg(x), lam)))

    def jac_sparsity(t, z):
        return np.zeros((2*len(constraints), 2*len(lb)))

    def jac_fun(t, z):
        x = z[:len(lb)]
        lam = z[len(lb):]
        return np.block([[jac(t, x), np.zeros((len(x), len(lam)))],
                         [np.dot(dg(x), lam), dg(x)]])

    def bc_fun(t0, z0, tn, zn):
        x0 = z0[:len(lb)]
        xn = zn[:len(lb)]
        lam0 = z0[len(lb):]
        lamn = zn[len(lb):]
        return bc(x0, xn)

    def dbc_fun(t0, z0, tn, zn):
        x0 = z0[:len(lb)]
        xn = zn[:len(lb)]
        lam0 = z0[len(lb):]
        lamn = zn[len(lb):]
        return dbc(x0, xn)

    sol = solve_ivp(fun=fun, t_span=(nodes[0], nodes[-1]), y0=np.concatenate((guess[0], np.zeros(len(constraints)))), t_eval=nodes, method='Radau', jac_sparsity=jac_sparsity, jac=jac_fun, events=(bc_fun, dbc_fun), rtol=tol, atol=tol, max_step=(nodes[-1]-nodes[0])/n, dense_output=True)

    # 求解完成后，生成插值函数
    t = sol.t
    x = sol.y[:len(lb),:]
    cs = CubicSpline(t, x, axis=1)

    return cs, obj(x[:,-1])

这个程序中，`n`表示伪谱法中使用的节点数，`lb`和`ub`表示变量的下界和上界，`constraints`是一个包含约束函数的列表，`cost`是目标函数，`guess`是初始猜测值，`tol`和`maxiter`是求解器的容差和最大迭代次数。函数返回一个插值函数和最优解。