使用CVXPY调用CVXOPT求解LP问题失败，求故障原因

我尝试用CVXPY结合CVXOPT求解一个简单的线性规划问题，代码如下：

import cvxpy as cp
import numpy as np

n = 2
x = cp.Variable(n)

prob = cp.Problem(cp.Maximize(cp.sum(x)), [cp.sum(x) <= 10])
prob.solve(solver=cp.CVXOPT, verbose=True)

print("\nThe optimal value is", prob.value)
print("A solution x is")
print(x.value)
print("A dual solution is")
print(prob.constraints[0].dual_value)

运行后得到错误输出：

===============================================================================
                                     CVXPY                                      
                                     v1.4.3                                     
===============================================================================
(CVXPY) Apr 17 08:41:27 PM: Your problem has 2 variables, 1 constraints, and 0 parameters.
(CVXPY) Apr 17 08:41:27 PM: It is compliant with the following grammars: DCP, DQCP
(CVXPY) Apr 17 08:41:27 PM: (If you need to solve this problem multiple times, but with different data, consider using parameters.)
(CVXPY) Apr 17 08:41:27 PM: CVXPY will first compile your problem; then, it will invoke a numerical solver to obtain a solution.
(CVXPY) Apr 17 08:41:27 PM: Your problem is compiled with the CPP canonicalization backend.
-------------------------------------------------------------------------------
                                  Compilation                                   
-------------------------------------------------------------------------------
(CVXPY) Apr 17 08:41:27 PM: Compiling problem (target solver=CVXOPT).
(CVXPY) Apr 17 08:41:27 PM: Reduction chain: FlipObjective -> Dcp2Cone -> CvxAttr2Constr -> ConeMatrixStuffing -> CVXOPT
(CVXPY) Apr 17 08:41:27 PM: Applying reduction FlipObjective
(CVXPY) Apr 17 08:41:27 PM: Applying reduction Dcp2Cone
(CVXPY) Apr 17 08:41:27 PM: Applying reduction CvxAttr2Constr
(CVXPY) Apr 17 08:41:27 PM: Applying reduction ConeMatrixStuffing
(CVXPY) Apr 17 08:41:27 PM: Applying reduction CVXOPT
(CVXPY) Apr 17 08:41:27 PM: Finished problem compilation (took 2.427e-03 seconds).
-------------------------------------------------------------------------------
                                Numerical solver                                
-------------------------------------------------------------------------------
(CVXPY) Apr 17 08:41:27 PM: Invoking solver CVXOPT  to obtain a solution.
Traceback (most recent call last):
  File "/home/me/test2.py", line 8, in <module>
    prob.solve(solver=cp.CVXOPT, verbose=True)
  File "/home/me/venv/lib/python3.10/site-packages/cvxpy/problems/problem.py", line 503, in solve
    return solve_func(self, *args, **kwargs)
  File "/home/me/venv/lib/python3.10/site-packages/cvxpy/problems/problem.py", line 1086, in _solve
    self.unpack_results(solution, solving_chain, inverse_data)
  File "/home/me/venv/lib/python3.10/site-packages/cvxpy/problems/problem.py", line 1411, in unpack_results
    raise error.SolverError(
cvxpy.error.SolverError: Solver 'CVXOPT' failed. Try another solver, or solve with verbose=True for more information.

切换为默认求解器ECOS后能返回正确解，为何CVXOPT无法解决该问题？已尝试重装相关包并查阅文档，未解决问题。

核心原因分析 #

这个问题的根源在于你的线性规划问题存在无穷多最优解，CVXOPT与ECOS对这种退化情况的处理逻辑存在差异：

• 目标是最大化sum(x)，约束为sum(x) ≤10，所有满足sum(x)=10的x都是最优解（比如[10,0]、[5,5]等），属于典型的退化LP问题。
• CVXOPT的内部收敛判定逻辑对这类无唯一最优解的情况鲁棒性较弱，容易触发求解失败；而ECOS对退化问题的数值稳定性处理更友好，能够正常返回可行的最优解。
• 此外，CVXPY将LP问题转换为锥形式时，可能给CVXOPT传递了不符合其默认收敛阈值的问题形式，进一步加剧了求解失败的概率。

解决办法 #

1. 添加约束消除解的退化性：给变量增加简单的边界约束，让问题拥有唯一或更稳定的最优解，示例代码：

   prob = cp.Problem(cp.Maximize(cp.sum(x)), [cp.sum(x) <= 10, x >= 0])
   

   此时CVXOPT通常能正常求解，最优解会是[10,0]这类顶点解。

2. 调整CVXOPT的求解参数：放宽收敛阈值（如abstol绝对误差、reltol相对误差），让CVXOPT接受退化问题的解，示例代码：

   prob.solve(solver=cp.CVXOPT, verbose=True, abstol=1e-6, reltol=1e-6)
   

3. 直接使用CVXOPT原生接口：绕开CVXPY的转换层，直接用CVXOPT的LP接口定义问题，避免锥转换带来的额外问题，示例代码：

   from cvxopt import matrix, solvers
   # CVXOPT默认求解最小化问题，因此将目标取反
   c = matrix([-1.0, -1.0])
   A = matrix([[1.0, 1.0]])
   b = matrix([10.0])
   sol = solvers.lp(c, A, b)
   print("最优解：", sol['x'])
   print("最优值：", -sol['primal objective'])
   

内容的提问来源于stack exchange，提问作者asfiwefewrno