大规模优化问题，Scipy？Ceres？PyTorch！

背景：

优化问题一般通过scipy.optimize或者Ceres Solver优化器求解。但在参数量较大的优化问题上，scipy提供的BFGS、L-BFGS-B、CG、SLSQP等梯度优化算法其复杂度和存储需求指数级上升，无法满足计算效率；而Ceres需要额外的语言来支持，当前算法工程师一般以python为主，学习成本较大。且项目环境需要多语言混编，配置与部署难度较大。

本文使用PyTorch/TensorFlow等机器学习框架，通过自动求导高效计算梯度，避免手动计算梯度复杂性，利用CPU/CPU来加速求解优化问题。

PyTorch优化四部曲：

# 优化迭代次数，设为1000次
for _ in range(1000):# 1. 清零梯度optimizer.zero_grad()# 通过调用 zero_grad()，我们清零了所有被优化参数的梯度。这是因为在 PyTorch 中，# 每次进行反向传播时，梯度默认是累积的（即不会自动清零）。# 如果不清理梯度，可能会导致累积误差，从而影响下一步的梯度计算。# 2. 计算损失函数loss = objective_function(params)# 这里我们计算当前参数下的目标函数值（损失函数）。# 损失函数的值将作为后续反向传播计算梯度的基础。# 根据参数（params）的当前值，求出损失值（即预测和实际的误差）。# 3. 反向传播计算梯度loss.backward()# 调用 loss.backward() 会进行反向传播，计算损失函数对所有优化参数的梯度。# 这些梯度将存储在各个参数的 .grad 属性中，以便在优化步骤中使用。# 4. 更新参数optimizer.step()# 调用 optimizer.step() 会应用优化器来更新参数。# 优化器会利用刚才计算的梯度（存储在 .grad 属性中），按照指定优化算法（如 Adam）# 的规则更新参数。这样一来，参数会朝着能最小化损失函数的方向一步步调整。

开胃小菜：

在三维空间中，给定初始点（0,0,0），求解优化使其距离点(2,3,4)最近。

import torch# 定义目标函数，取任意数量优化变量
def objective_function(params):x, y, z = paramsreturn (x - 2)**2 + (y - 3)**2 + (z - 4)**2# 初始化优化变量
params = [torch.tensor([0.0], requires_grad=True),torch.tensor([0.0], requires_grad=True),torch.tensor([0.0], requires_grad=True)]# 优化器选择 Adam ，并传入列表参数
optimizer = torch.optim.Adam(params, lr=0.1)# 迭代优化
for _ in range(1000):optimizer.zero_grad()loss = objective_function(params)loss.backward()optimizer.step()# 输出优化结果
print(f"Optimized x: {params[0].item()}")
print(f"Optimized y: {params[1].item()}")
print(f"Optimized z: {params[2].item()}")

正餐：

假设场景：现有一张二维平面的游戏地图，并附带一份初始路网，请优化路网形状，使其平滑、美化、且与各个传送点保持挂接。

游戏地图如下，白色区域代表道路，黑色区域为地图外场景：

初始路网可视化见：

初始路网以link和node两张表表示：

node表

{
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{ "type": "Feature", "properties": { "nodeid": "90cbaa9a-3e8e-11ef-a98f-a13c99648c9a", "node_type": null }, "geometry": { "type": "Point", "coordinates": [ 576.0, 788.0 ] } },
{ "type": "Feature", "properties": { "nodeid": "90cbaaf4-3e8e-11ef-a98f-a13c99648c9a", "node_type": null }, "geometry": { "type": "Point", "coordinates": [ 621.0, 780.0 ] } },
{ "type": "Feature", "properties": { "nodeid": "90cbaa18-3e8e-11ef-a98f-a13c99648c9a", "node_type": null }, "geometry": { "type": "Point", "coordinates": [ 592.0, 810.0 ] } },
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}

link表

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{ "type": "Feature", "properties": { "linkid": "918089e2-3e8e-11ef-a98f-a13c99648c9a", "kind": "ske", "snode_type": null, "enode_type": null, "snodeid": "90cbb3dc-3e8e-11ef-a98f-a13c99648c9a", "enodeid": "90cbb0f8-3e8e-11ef-a98f-a13c99648c9a" }, "geometry": { "type": "LineString", "coordinates": [ [ 1187.0, 556.0 ], [ 1092.0, 623.0 ], [ 1078.0, 652.0 ], [ 1043.0, 654.0 ], [ 956.0, 708.0 ] ] } },
{ "type": "Feature", "properties": { "linkid": "91803c76-3e8e-11ef-a98f-a13c99648c9a", "kind": "ske", "snode_type": null, "enode_type": null, "snodeid": "90cbb04e-3e8e-11ef-a98f-a13c99648c9a", "enodeid": "90cbb0f8-3e8e-11ef-a98f-a13c99648c9a" }, "geometry": { "type": "LineString", "coordinates": [ [ 975.0, 767.0 ], [ 973.0, 761.0 ], [ 959.0, 754.0 ], [ 944.0, 722.0 ], [ 956.0, 708.0 ] ] } },
{ "type": "Feature", "properties": { "linkid": "7cb7a2b6-3e8e-11ef-bae9-7993d366a9e1", "kind": "ske", "snode_type": null, "enode_type": null, "snodeid": "90cbad1a-3e8e-11ef-a98f-a13c99648c9a", "enodeid": "90cbacf2-3e8e-11ef-a98f-a13c99648c9a" }, "geometry": { "type": "LineString", "coordinates": [ [ 862.0, 607.0 ], [ 785.0, 658.0 ], [ 769.0, 687.0 ] ] } },
{ "type": "Feature", "properties": { "linkid": "7cb7bada-3e8e-11ef-bae9-7993d366a9e1", "kind": "ske", "snode_type": null, "enode_type": null, "snodeid": "90cbacf2-3e8e-11ef-a98f-a13c99648c9a", "enodeid": "90cbac48-3e8e-11ef-a98f-a13c99648c9a" }, "geometry": { "type": "LineString", "coordinates": [ [ 769.0, 687.0 ], [ 801.0, 719.0 ], [ 861.0, 841.0 ] ] } }
]
}

#自建link and node
import numpy as np
import cv2
from skimage.morphology import skeletonize
import torch
import torch.optim as optim
import torch.nn.functional as F
import geopandas as gpd 
import shapely.geometry as G 
from scipy.interpolate import interp1ddef read_binary_image(path):"""读取二值图像"""img = cv2.imread(path, cv2.IMREAD_GRAYSCALE)_, binary_img = cv2.threshold(img, 200, 255, cv2.THRESH_BINARY)return binary_img #向量V单位化
def normed(v):return v/np.maximum(np.linalg.norm(v, axis=-1, keepdims=True), 1e-9)#插值
def resample(xy, interval = 8.0, tol = 0.1):xy = np.asarray(G.LineString(xy).simplify(tol, True).coords, np.double)norms = np.linalg.norm(xy[1:] - xy[:-1], axis=1)offs = np.empty(len(xy), xy.dtype)offs[0] = 0offs[1:] = np.add.accumulate(norms)n = max(int(np.ceil(offs[-1]/interval)), 1)kwargs = dict(copy=False, fill_value='extrapolate', assume_sorted=True)xy = interp1d(offs, xy, axis=0, **kwargs)(np.linspace(0, offs[-1], n + 1, dtype=xy.dtype))return xydef generate_distance_transform(road_mask):"""生成距离变换图（反映每个像素到最近非路面像素的距离）;为了保证修型后的线段不压盖边界，设计损失函数，当其里边界越近，损失越大；压盖边界时，损失极大"""grid = cv2.distanceTransform(255 - road_mask, cv2.DIST_L2, 5)# cv2.imwrite("grid.jpg",grid)grid2 = cv2.distanceTransform(road_mask, cv2.DIST_L2, 5)# cv2.imwrite("grid2.jpg",grid2)grid = np.where(grid2 > 0, -grid2, grid)grid = np.round(np.sqrt(np.tanh(grid/3) + 1)*45000).astype(np.int32)# cv2.imwrite("tmp2.jpg",grid.astype(np.uint8))# print(grid)# print(np.max(grid),np.min(grid))return griddef smooth_skeleton(nodes, links_dict, nodes_dict, dist_transform, fi, follow,lambda1=1.0, lambda2=1.0, lambda3=1.0, lambda4=1.0, lambda5=1.0, lambda6=1.0, max_iterations=50):"""使用 PyTorch 优化骨架节点，使骨架尽量平滑且不与非路面相交"""device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')nodes_tensor = torch.tensor(nodes, dtype=torch.float32, requires_grad=True, device=device)dist_transform_tensor = torch.tensor(dist_transform, dtype=torch.float32, device=device).unsqueeze(0).unsqueeze(0)print(nodes_tensor.shape)optimizer = optim.Adam([nodes_tensor], lr=0.03)H, W = dist_transform.shape#同一link相邻两个点links_cons1 = [] for d in links_dict.values(): (b, e) = d['rng']links_cons1.extend([(fi(i), fi(i + 1))for i in range(b, e - 1)])links_cons1 = np.array(links_cons1) initial_diffs = nodes[links_cons1[:, 0]] - nodes[links_cons1[:, 1]]initial_distances = np.linalg.norm(initial_diffs, axis=1)initial_distances = np.maximum(initial_distances, 1)#同一link相邻三个点links_cons2 = [] for d in links_dict.values(): (b, e) = d['rng']if e-b <= 2:continuelinks_cons2.extend([(fi(i), fi(i + 1), fi(i + 2))for i in range(b, e - 2)])links_cons2 = np.array(links_cons2) #固定点 nods_cons1 = [] for d in links_dict.values(): mode = d["mode"] (b, e) = d['rng'] if mode < 2 : nods_cons1.append([fi(b)]) nods_cons1 = np.array(nods_cons1) initial_solid = nodes[nods_cons1[:, 0]]  #固定点射出方向vecs_cons1 = [] for d in links_dict.values(): mode = d["mode"] (b, e) = d['rng'] if mode == 1: vecs_cons1.append([fi(b), fi(b+1)]) vecs_cons1 = np.array(vecs_cons1) initial_vecs1 = nodes[vecs_cons1[:, 0]] - nodes[vecs_cons1[:, 1]] #多link挂接时，角度均匀vecs_cons2 = [] #多link挂接时，近似180度的两个射出角保持直线vecs_cons3 = [] for d in nodes_dict.values(): node_couple_list = d["vecs"] if len(node_couple_list) >= 2: for idx in range(len(node_couple_list)):(s,e) = node_couple_list[idx] if len(node_couple_list) == idx+1:(ss,ee) = node_couple_list[0] else:(ss,ee) = node_couple_list[idx+1] vecs_cons2.append([fi(s), fi(e), fi(ss), fi(ee)]) if abs(np.arctan2(np.cross(nodes[fi(s)] - nodes[fi(e)] , nodes[fi(ss)] - nodes[fi(ee)]), (nodes[fi(s)] - nodes[fi(e)] )@(nodes[fi(ss)] - nodes[fi(ee)])))>2.8:vecs_cons3.append([fi(e), fi(s), fi(ee)])vecs_cons2 = np.array(vecs_cons2) vecs_cons3 = np.array(vecs_cons3) links_cons2 = np.concatenate([links_cons2,vecs_cons3],axis=0)for _ in range(max_iterations):optimizer.zero_grad()# 平滑项# smoothness_term = torch.sum(torch.norm(nodes_tensor[links[:, 0]] - nodes_tensor[links[:, 1]], dim=1))# 保持与初始距离的比例，且最小长度为1个像素length__start_points = nodes_tensor[links_cons1[:, 0]]length__end_points = nodes_tensor[links_cons1[:, 1]]length__distances = torch.norm(length__start_points - length__end_points, dim=1)smoothness_term = torch.sum((length__distances - torch.tensor(initial_distances, device=device))**2)  # 平滑项，角度均匀first_points = nodes_tensor[links_cons2[:, 0]]second_points = nodes_tensor[links_cons2[:, 1]]third_points = nodes_tensor[links_cons2[:, 2]]smoothness_term2 = (1 - F.cosine_similarity((second_points-first_points).unsqueeze(0), (third_points-second_points).unsqueeze(0), dim=1)).sum()# print("debug",smoothness_term2)#固定点solid_points = nodes_tensor[nods_cons1[:, 0]]solid_term = torch.sum(torch.norm(solid_points - torch.tensor(initial_solid, device=device), dim=1))#固定射出角度solid_first_points = nodes_tensor[vecs_cons1[:, 0]]solid_second_points = nodes_tensor[vecs_cons1[:, 1]]solid_vecs_term = (1 - F.cosine_similarity((solid_first_points-solid_second_points).unsqueeze(0), torch.tensor(initial_vecs1, device=device).unsqueeze(0), dim=1)).sum()# 距离项# 将节点坐标转换为插值中的标准化坐标并取样grid = nodes_tensor.unsqueeze(0).unsqueeze(0)  # [1, 1, N, 2]grid = torch.cat((grid[:, :, :, 0:1] * 2 / (W - 1) - 1, grid[:, :, :, 1:2] * 2 / (H - 1) - 1), dim=3)sampled_dist = F.grid_sample(dist_transform_tensor, grid, mode='bilinear', padding_mode='border', align_corners=True)distance_term = sampled_dist.view(-1).sum() #多度点均匀挂接 one_vec = nodes_tensor[vecs_cons2[:, 0]] - nodes_tensor[vecs_cons2[:, 1]] the_other_vec = nodes_tensor[vecs_cons2[:, 2]] - nodes_tensor[vecs_cons2[:, 3]] # 计算点积 dot_products = torch.sum(one_vec * the_other_vec, dim=1)cross_products = one_vec[:, 0] * the_other_vec[:, 1] - one_vec[:, 1] * the_other_vec[:, 0]# 计算夹角 nodes_angles = torch.atan2(cross_products, dot_products)# 将结果张量的形状调整为 (100, 1)nodes_angles = nodes_angles.view(-1, 1) nodes_vecs_term = torch.sum(torch.max(- nodes_angles + torch.full_like(nodes_angles, torch.tensor(torch.pi / 2)),torch.full_like(nodes_angles, torch.tensor(0))))# #近似180度挂接的保持直线# vecs_cons3loss = (lambda1 * smoothness_term +  #保持原有长度损失lambda2 * distance_term +  #grid损失lambda3 * smoothness_term2 +  #角度平滑损失lambda4 * solid_term +  #固定点不动约束lambda5 * solid_vecs_term +#固定点射出角约束lambda6 * nodes_vecs_term #多度挂接角度均匀 )loss.backward() optimizer.step()print(loss)optimized_coords = nodes_tensor.cpu().detach().numpy()for (i, j) in follow.items():optimized_coords[i] = optimized_coords[j]return optimized_coordsdef visualize_skeleton(binary_img, nodes, links, optimized_nodes,fi):"""可视化优化前后的骨架"""img = cv2.cvtColor(binary_img, cv2.COLOR_GRAY2BGR) view_links = []for d in links.values(): (b, e) = d['rng']view_links.extend([(fi(i), fi(i + 1)) for i in range(b, e - 1)]) links = np.array(view_links) # 绘制原始骨架（红色）for x, y in nodes: cv2.circle(img, (x, y), 2, (0, 0, 255), -1)for i, j in links: cv2.line(img, tuple(nodes[i]), tuple(nodes[j]), (0, 0, 255), 1)# # 绘制优化后的骨架（蓝色）# for x, y in optimized_nodes:#     cv2.circle(img, (int(x), int(y)), 2, (255, 0, 0), -1)# for i, j in links:#     cv2.line(img, tuple(optimized_nodes[i].astype(int)), tuple(optimized_nodes[j].astype(int)), (255, 0, 0), 1)cv2.imwrite("res0.jpg",img)def out_gpd(links_gpdf,nodes_gpdf,links,pts,nodes):links_gpdf = links_gpdf.set_geometry([G.LineString(pts[b:e]).simplify(0.5, True)for (b, e) in [links[k]['rng'] for k in links_gpdf['linkid']]])nodes_gpdf = nodes_gpdf.set_geometry([G.Point(pts[nodes[k]['vecs'][0][0]])for k in nodes_gpdf['nodeid']])gpd.GeoDataFrame(links_gpdf).to_file('./link2_out.geojson')gpd.GeoDataFrame(nodes_gpdf).to_file('./node2_out.geojson')return def main():"""主函数"""# 读取二值图像binary_img = read_binary_image('./1.jpg')link_path = "./link.geojson"node_path = "./node.geojson"link_gpd = gpd.read_file(link_path) node_gpd = gpd.read_file(node_path) # 提取节点和连边# 初始化节点字典，保存节点及其连出的向量信息nodes = {}for r in node_gpd.itertuples():fixed = (not np.isnan(r.node_type))nodes[r.nodeid] = {'vecs': []}# 初始化点集列表和线段字典pts = []e = 0 #记录pts的idxlinks = {} for r in link_gpd.itertuples(): #按照固定点类别区分模式mode = (0 if (r.snode_type in [250]) else1 if (r.snode_type in [1001, 1002, 1003]) else 2)xy = np.array(r.geometry.coords, np.double)if mode == 1:if len(xy) == 2:#如果固定点挂接的link是两个形点，那么添加一个靠近重点的四等分点，组成三个形点的link。xy = np.array([xy[0], (xy[0] + xy[1]*3)/4, xy[1]])#在插值的时候保留第一个点xy = np.concatenate([xy[:1], resample(xy[1:])])else:xy = resample(xy)pts.append(xy)b = ee += len(xy) #当前link对应的pts点序从b到e-1links[r.linkid] = {'rng': (b, e), 'mode': mode}nodes[r.snodeid]['vecs'].append((b + 0, b + 1))nodes[r.enodeid]['vecs'].append((e - 1, e - 2))#三维展开成二维，即将所有的line坐标平铺，取消line维度  pts = np.concatenate(pts)# 用于记录同一个点的信息，对于每个节点，如果它连出多条边，将这些边的起点合并成一个。follow = {} for n in nodes.values():l = n['vecs']if len(l) > 2:#计算node节点连的所有边的角度,并按照顺/逆时针调整顺序v = np.array([pts[i2] - pts[i1] for (i1, i2) in l]).Tl[:] = [l[i] for i in np.argsort(np.arctan2(v[1], v[0]))]#follow 字典，代表同一个点if len(l) > 1:for (i, _) in l[1:]:follow[i] = l[0][0]def fi(i): return follow.get(i, i)print(pts.shape)print(links["7cba8b52-3e8e-11ef-bae9-7993d366a9e1"])# 生成距离变换图dist_transform = generate_distance_transform(binary_img)# lambda1  #保持原有长度损失
# lambda2  #grid损失
# lambda3  #角度平滑损失
# lambda4 #固定点不动约束
# lambda5 #固定点射出角约束
# lambda6  #多度挂接角度均匀 # 优化整个骨架optimized_nodes = smooth_skeleton(pts, links, nodes, dist_transform, fi, follow, lambda1=1, lambda2=0.01, lambda3 = 2000000, lambda4 = 10000000,lambda5 = 10000000,lambda6=1400000, max_iterations=5000)# 可视化优化前后的骨架 visualize_skeleton(binary_img, pts.astype(np.int32), links, optimized_nodes,fi)# out gpdout_gpd(link_gpd,node_gpd,links,optimized_nodes,nodes)if __name__ == '__main__': main()

优化后的结果见（蓝色）

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