scripts/opt_tree1.py

import numpy as np
from scipy.optimize import minimize

# 参数
p = np.array([0.8, 0.5, 0.3, 0.2, 0.12, 0.07])
q = 1 - p  # 失效概率
N_total = 80

def objective(n):
    """目标函数：Σ n_i * [1 - q_i^{n_i}]"""
    return np.sum(n * (1 - q**n))

# 修正约束条件
constraints = [
    {'type': 'eq', 'fun': lambda n: np.sum(n) - N_total}
]

# 设置边界为至少1个节点
bounds = [(1, N_total)] * 6

# 初始猜测（均匀分配）
n0 = np.ones(6) * N_total / 6
n0 = np.maximum(n0, 1)  # 确保初始值 >=1

# 优化（使用连续实数）
res = minimize(lambda n: -objective(n), n0, 
               bounds=bounds,
               constraints=constraints,
               method='SLSQP',
               options={'maxiter': 50,
                        'ftol': 1e-6,
                        'disp': True})
n_opt = res.x
max_value = objective(n_opt)

print("最优节点分配（连续解）:")
for i in range(6):
    print(f"层 {i+1}: {n_opt[i]:.2f} 个节点")
print(f"\n最大值: {max_value:.4f}")

# 检查整数解
from itertools import product

def integer_search(center, radius=3):
    best_val = -1
    best_n = None
    
    # 生成附近整数组合（确保每个值至少为1）
    ranges = []
    for i in range(6):
        start = max(1, int(center[i]) - radius)  # 至少为1
        end = min(N_total, int(center[i]) + radius)  # 最多N_total
        ranges.append(range(start, end + 1))
    
    # 限制搜索范围，避免组合爆炸
    count = 0
    max_combinations = 100000  # 限制搜索组合数
    
    for combo in product(*ranges):
        count += 1
        if count > max_combinations:
            break
            
        if sum(combo) == N_total and all(x >= 1 for x in combo):
            val = objective(np.array(combo))
            if val > best_val:
                best_val = val
                best_n = combo
    
    # 如果没找到合适的解，尝试更简单的启发式搜索
    if best_n is None:
        print("直接搜索未找到合适解，使用四舍五入法...")
        # 四舍五入并调整到总和为80
        rounded = np.round(n_opt).astype(int)
        diff = N_total - np.sum(rounded)
        
        # 调整差值
        if diff > 0:
            # 从最小值开始加
            sorted_idx = np.argsort(n_opt - rounded)
            for i in range(diff):
                rounded[sorted_idx[i]] += 1
        elif diff < 0:
            # 从最大值开始减
            sorted_idx = np.argsort(rounded - n_opt)[::-1]
            for i in range(-diff):
                if rounded[sorted_idx[i]] > 1:  # 确保至少为1
                    rounded[sorted_idx[i]] -= 1
        
        # 确保所有值至少为1
        rounded = np.maximum(rounded, 1)
        best_n = tuple(rounded)
        best_val = objective(np.array(best_n))
    
    return best_n, best_val

int_n, int_val = integer_search(n_opt, radius=3)
print("\n近似最优整数解:")
for i in range(6):
    print(f"层 {i+1}: {int_n[i]} 个节点")
print(f"整数值: {int_val:.4f}")

# 验证
print(f"\n验证:")
print(f"总和: {sum(int_n)}")
print(f"所有节点 ≥ 1: {all(x >= 1 for x in int_n)}")

# 附加：使用更智能的整数搜索方法
print("\n\n备选：使用动态规划寻找最优整数解...")

# 由于节点数较少，可以尝试更系统的方法
def find_optimal_integer():
    from itertools import combinations_with_replacement
    import math
    
    best_val = -1
    best_n = None
    
    # 使用星棒法生成所有可能的组合
    # C(N_total-1, 6-1) = C(79, 5) ≈ 2.3 million 仍然很大
    # 使用更高效的方法：从连续解开始，在附近搜索
    
    # 生成所有6个数字总和为80的组合，每个至少为1
    # 这相当于找5个切割点
    
    # 由于组合数仍然较大，我们使用更智能的剪枝
    
    # 简化的搜索：先固定前5层，最后一层由总和决定
    candidates = []
    
    # 放宽搜索半径
    radius = 4
    center = np.round(n_opt).astype(int)
    
    for n1 in range(max(1, center[0]-radius), center[0]+radius+1):
        for n2 in range(max(1, center[1]-radius), center[1]+radius+1):
            for n3 in range(max(1, center[2]-radius), center[2]+radius+1):
                for n4 in range(max(1, center[3]-radius), center[3]+radius+1):
                    for n5 in range(max(1, center[4]-radius), center[4]+radius+1):
                        n6 = N_total - (n1+n2+n3+n4+n5)
                        if n6 >= 1:
                            combo = (n1, n2, n3, n4, n5, n6)
                            # 检查是否在合理范围内
                            if all(abs(combo[i] - center[i]) <= radius+2 for i in range(6)):
                                val = objective(np.array(combo))
                                if val > best_val:
                                    best_val = val
                                    best_n = combo
    
    return best_n, best_val

opt_int_n, opt_int_val = find_optimal_integer()
print("\n优化后的整数解:")
for i in range(6):
    print(f"层 {i+1}: {opt_int_n[i]} 个节点")
print(f"优化整数值: {opt_int_val:.4f}")