cvxpy+ecos:问题不可行，如何正确扩展

Question

我有以下代码：

import numpy as np
import cvxpy as cp
import math
import sys

def solve05( p, a ):
    m,n,ids,inv,k = 0,len(p),{},{},0
    for i in range(n):
        for j in range(n):
            ids[(i,j)] = k
            inv[k] = (i,j)
            k = k+1
    # Problem data
    A = np.zeros((2*n,n*n+n))
    D = np.zeros((2*n,n*n+n))
    b = np.zeros(2*n)
    B = np.zeros(2*n)
    c = np.zeros(2*n)
    for j in range(n):
        for i in range(n):
            idx = ids[(i,j)]
            A[j,idx] = 1
        b[j] = 1
    for i in range(n):
        for j in range(n):
            idx = ids[(i,j)]
            A[i+n,idx] = p[j]
        A[i+n,n*n+i] = -1
        b[i+n] = p[i]
    # Construct the problem
    x = cp.Variable(n*n+n)
    print("M = ",A)
    print("b = ",b)
    CF = 1e3
    print("Now scaling M by ",CF)
    A = A*CF
    print(A)
    b = b*CF
    constraints = [0 <= x, A*x == b]
    pex = x[n*n]+x[n*n+1]+x[n*n+2]+1
    constraints.append(x[n*n] <= a[0]*CF)
    constraints.append(x[n*n+1] <= a[1]*CF)
    constraints.append(x[n*n+2] <= a[2]*CF)
    constraints.append(x[n*n] >= 0.01)
    constraints.append(x[n*n+1] >= 0.01)
    constraints.append(x[n*n+2] >= 0.01)
    ex = pex.__pow__(-1)
    print("Dummy variables: ",x[n*n],x[n*n+1],x[n*n+2])
    print("Objective function: ",ex)
    print("[should be convex] Curvature: ",ex.curvature)
    objective = cp.Minimize(ex)
    prob = cp.Problem(objective,constraints)
    result = prob.solve(verbose=True)
    print('problem state: ', prob.status)
    alpha = np.zeros((n,n))
    for i in range(n):
        for j in range(n):
            alpha[i,j] = x.value[ids[(i,j)]]
    dummy = [x.value[j] for j in range(n*n,n*n+n)]
    return (x,alpha)


if __name__ == '__main__':
    p = [0.0005,0.0001,0.0007]
    a = [900,500,700]
    n = len(a)
    (sl,alpha) = solve05(p,a)
    for row in alpha:
        for x in row:
            print("%.4f " % (x), end=" "),
        print("")

它失败了，因为“问题不可行”的结论，我渴望知道为什么。有没有办法知道更多？我不是凸编程专家，所以任何关于为什么这是一个糟糕的模型的评论都是值得赞赏的。我也尝试过扩展这个问题，因为我认为一些数值不稳定可能是导致问题的原因，但是天啊。

Answer 1

ecos+cvxpy给出的答案是正确的。这个问题是不可行的，这可以通过总结所有的方程并观察到，对于某些e > 0，LHS是一些量F，而RHS是F+e。