SciPy优化未返回预期最优解：双T截面惯性矩问题排查

我正在为硕士论文优化双T截面的惯性矩函数，通过SciPy的minimize函数最大化double_T_Ixx，但优化输出与已知解析解不符。

目标函数定义 #

def double_T_Ixx(lengths, thicknesses):
    A_top_l = lengths[0] * thicknesses[0]
    A_top_r = lengths[1] * thicknesses[1]
    A_mid   = lengths[2] * thicknesses[2]
    A_bot_l = lengths[3] * thicknesses[3]
    A_bot_r = lengths[4] * thicknesses[4]

    y_top = lengths[2]
    y_mid = lengths[2] / 2
    y_bot = 0

    Ixx_top_l = (lengths[0] * thicknesses[0]**3) / 12
    Ixx_top_r = (lengths[1] * thicknesses[1]**3) / 12
    Ixx_mid   = (thicknesses[2] * lengths[2]**3) / 12
    Ixx_bot_l = (lengths[3] * thicknesses[3]**3) / 12
    Ixx_bot_r = (lengths[4] * thicknesses[4]**3) / 12

    A_total = A_top_l + A_top_r + A_mid + A_bot_l + A_bot_r
    y_composite = (A_top_l*y_top + A_top_r*y_top + A_mid*y_mid + A_bot_l*y_bot + A_bot_r*y_bot) / A_total

    Ixx = (Ixx_top_l + 
           Ixx_top_r + 
           Ixx_mid + 
           Ixx_bot_l + 
           Ixx_bot_r +
           A_top_l*(y_top - y_composite)**2 +
           A_top_r*(y_top - y_composite)**2 +
           A_mid  *(y_mid - y_composite)**2 +
           A_bot_l*(y_bot - y_composite)**2 +
           A_bot_r*(y_bot - y_composite)**2)
    return Ixx

优化目标与约束定义 #

MIN_LENGTHS   = 1
MAX_LENGTH    = 10
MIN_THICKNESS = 0.5
MAX_THICKNESS = 2.5
MAX_AREA      = 10

def objective(x):
    num_segments = len(x) // 2
    lengths = x[:num_segments]
    thicknesses = x[num_segments:]
    
    return -double_T_Ixx(lengths, thicknesses)
    
def constraint_area(x):
    lengths = x[:len(x)//2]
    thicknesses = x[len(x)//2:]
    A_top_l = lengths[0] * thicknesses[0]
    A_top_r = lengths[1] * thicknesses[1]
    A_mid   = lengths[2] * thicknesses[2]
    A_bot_l = lengths[3] * thicknesses[3]
    A_bot_r = lengths[4] * thicknesses[4]
    return MAX_AREA - (A_top_l + A_top_r + A_mid + A_bot_l + A_bot_r)

预期最优解 #

根据领域知识，该问题存在多组最优解（目标值一致但参数不同），其中一组代表性解为：

c = [l1 = 1, 
     l2 = 1, 
     l3 = 10, 
     l4 = 1, 
     l5 = 1,
 
     t1 = 1.25, 
     t2 = 1.25, 
     t3 = 0.5, 
     t4 = 1.25, 
     t5 = 1.25]   

核心规律：l3取最大值、t3取最小值，l1/l2/l4/l5取最小值，t1/t2/t4/t5取最大值，且满足面积约束。

SciPy优化尝试 #

使用SLSQP算法进行优化：

constraints = [{'type':'eq', 'fun':lambda x: constraint_area(x)}]
initial_guess = [(b[0] + b[1] / 2) for b in bounds]

result = minimize(
    fun=objective,
    x0=initial_guess,
    method='SLSQP',
    bounds=bounds,
    constraints=constraints,
    tol=1e-12,
    options={'disp': True}
)

当前问题 #

优化得到的目标值接近解析解，但参数值不符合预期：
[2.265, 2.265, 10.000, 2.321, 2.270, 0.532, 0.567, 0.500, 0.562, 0.531]
预期中l1/l2/l4/l5应等于1，t1/t2/t4/t5应取最大值。

疑问 #

• 如何修改优化配置以得到预期最优解？
• 该问题是否不适配SciPy优化器？
• 还是我在建模或设置中存在遗漏？

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