掌握branch and cut算法原理附带C++求解TSP问题代码

branch and cut其实还是和branch and bound脱离不了干系的。所以,在开始本节的学习之前,请大家还是要务必掌握branch and bound算法的原理。

01 应用背景

Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values.

Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations.

Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called cut and branch.[1]

02 总体描述

前面说过,branch and cut其实还是和branch and bound脱离不了干系。其实是有很大干系的。在应用branch and bound求解整数规划问题的时候,如下图(好好复习一下该过程):

假如,我们现在求一个整数规划最大化问题,在分支定界过程中,求解整数规划模型的LP松弛模型得到的非整数解作为上界,而此前找到的整数解作为下界。如果出现某个节点upper bound低于现有的lower bound,则可以剪掉该节点。否则,如果不是整数解继续分支。

此外,在求解整数规划模型的LP松弛时,If cutting planes are used to tighten LP relaxations。那么这时候的branch and bound就变成了branch and cut。

那么,什么是cutting planes呢?如下图:

红色部分是整数规划的可行解空间。

蓝色部分是整数规划的LP松弛可行解空间。

在求解LP松弛时,加入橙色的cut,缩小解空间,同时又不影响整数解的解空间,可使解收敛得更快。

这就是branch and cut的过程了。比branch and bound高明之处就在于多了一个cutting planes,可能使branch and bound的效率变得更高。至于cutting planes是什么,等下一篇推文吧~

03 举个例子

为了让大家更好了解到branch and cut的精髓,必须得举一个简单的例子。对于同一个问题:

branch and cut(左)和branch and bound(右)求解过程如下:

可以看到,两者的不同之处就在子问题P2的处理上。

  • 对于branch and bound来说,求解线性松弛得到的Z = -29.5 < Z = -28。可知该支是可能隐含有更优解的,于是二话不说分支。无奈,分了两支以后发现居然没更优解,这种付出了却没有回报的感觉就像是受到了欺骗一样。
  • 对于branch and cut来说,在求解线性松弛得到的Z = -29.5 < Z = -28时,并没有被兴奋冲昏头脑,它尝试着在线性松弛的解空间上砍下一块,但又不能影响到整数解的解空间范围。琢磨半天终于找到一块能砍的,于是Add cut: 2x1 + x2 <= 7。砍下来以后,形成新的子问题P3,赶紧看看P3的最优解是多少。在P3中,Z=-27.8 > -28,这一支果然不可取。

从上面的算法过程我们可以看到,求解同一个问题,branch and cut只用了3步,而branch and bound却用了4步。

There are many methods to solve the mixed-integer linear programming. Gomory Cutting Planes is fast, but unreliable. Branch and Bound is reliable but slow. The Branch and cut combine the advantages from these two methods and improve the defects.

It has proven to be a very successful approach for solving a wide variety of integer programming problems.

We can solve the MILP by taking some cutting planes before apply whole system to the branch and bound, Branch and cut is not only reliable, but faster than branch and bound alone. Finally, we understand that using branch and cut is more efficient than using branch and bound.[2]

04 算法过程

关于branch and cut的过程,可以总结如下:[1]

相比branch and bound,其多了一个Cutting Planes的过程,先用Cutting Planes tighten LP relaxations,然后求解LP relaxations再判断是否有分支的必要。

其伪代码如下:

// ILP branch and cut solution pseudocode, assuming objective is to be maximizedILP_solution branch_and_cut_ILP(IntegerLinearProgram initial_problem) {    queue active_list; // L, above    active_list.enqueue(initial_problem); // step 1    // step 2    ILP_solution optimal_solution; // this will hold x* above    double best_objective = -std::numeric_limits<double>::infinity; // will hold v* above    while (!active_list.empty()) { // step 3 above        LinearProgram& curr_prob = active_list.dequeue(); // step 3.1        do { // steps 3.2-3.7            RelaxedLinearProgram& relaxed_prob = LP_relax(curr_prob); // step 3.2            LP_solution curr_relaxed_soln = LP_solve(relaxed_prob); // this is x above            bool cutting_planes_found = false;            if (!curr_relaxed_soln.is_feasible()) { // step 3.3                continue; // try another solution; continues at step 3            }            double current_objective_value = curr_relaxed_soln.value(); // v above            if (current_objective_value <= best_objective) { // step 3.4                continue; // try another solution; continues at step 3            }            if (curr_relaxed_soln.is_integer()) { // step 3.5                best_objective = current_objective_value;                optimal_solution = cast_as_ILP_solution(curr_relaxed_soln);                continue; // continues at step 3            }            // current relaxed solution isn't integral            if (hunting_for_cutting_planes) { // step 3.6                violated_cutting_planes = search_for_violated_cutting_planes(curr_relaxed_soln);                if (!violated_cutting_planes.empty()) { // step 3.6                    cutting_planes_found = true; // will continue at step 3.2                    for (auto&& cutting_plane : violated_cutting_planes) {                        active_list.enqueue(LP_relax(curr_prob, cutting_plane));                    }                    continue; // continues at step 3.2                }            }            // step 3.7: either violated cutting planes not found, or we weren't looking for them            auto&& branched_problems = branch_partition(curr_prob);            for (auto&& branch : branched_problems) {                active_list.enqueue(branch);            }            continue; // continues at step 3        } while (hunting_for_cutting_planes /* parameter of the algorithm; see 3.6 */               && cutting_planes_found);        // end step 3.2 do-while loop    } // end step 3 while loop    return optimal_solution; // step 4}
// tsp.cc - traveling salesman code based on Gurobi using branch and cut
#include <string>#include <iostream>#include <iomanip>#include <vector>#include <cmath>#include <chrono>#include <vector>#include <deque>#include <limits>// Gurobi#include "gurobi_c++.h"
// Data structure to represent an edge of the input graphstruct Edge{  int end1;  int end2;  double weight;};
// If memory is limited, a soft limit for the maximum number of LPs in the// queue can be set for the branch and cut algorithm.// This is effectively disabled by default.const int lp_soft_limit = 1000000;
// Tolerance to determine whether or not a number is integralconst double tol = 1.0e-9;
// Auxiliary function to decide whether or not a number is integralinline bool is_integral(double d){  return std::fabs(d - std::round(d)) < tol;}
void find_components(int                       num_nodes,                     int                       num_edges,                     std::vector<Edge>         edges,                     const std::vector<double> &x,                     int                       &num_components,                     std::vector<int>          &components){  // Mark all unassigned nodes with -1.  components.resize(num_nodes);  for (int n = 0; n < num_nodes; n++) {    components[n] = -1;  }
  // Component index  int c = 0;  // Node indices  int n1, n2;  // Find all connected components.  while (true) {    // Find an unassigned node.    n1 = -1;    for (int n = 0; n < num_nodes; n++) {      if (components[n] == -1) {        n1 = n;        break;      }    }    if (n1 == -1) {      // All nodes have been assigned.      break;    }
    // Assign node to current component.    components[n1] = c;    // Mark the entire connected component.    while (true) {      // Find an unassigned connected node.      n2 = -1;      for (int e = 0; e < num_edges; e++) {        // Skip edges that are not used in the current solution.        if (x[e] < tol) continue;        if (edges[e].end1 == n1 && components[edges[e].end2] == -1) {          n2 = edges[e].end2;          break;        }        if (edges[e].end2 == n1 && components[edges[e].end1] == -1) {          n2 = edges[e].end1;          break;        }      }      // No connected node found. Continue with next component.      if (n2 == -1) break;
      // Assign the connected node to the current component.      components[n2] = c;      // Merge the two nodes.      for (int e = 0; e < num_edges; e++) {        if (edges[e].end1 == n2) {          edges[e].end1 = n1;        }        if (edges[e].end2 == n2) {          edges[e].end2 = n1;        }      }    }    c++;  }  num_components = c;}
void solve_TSP(int                     num_nodes,               int                     num_edges,               const std::vector<Edge> &edges,               bool                    integral_weights,               std::vector<double>     &x_opt,               int                     &lp_solves,               int                     &subtour_constraints){  // Allocate memory for the solution.  std::vector<double> x;  x.resize(num_edges);
  // Set up environment.  GRBEnv env;  // Create initial model.  GRBModel initial_model(env);  // Add variables.  initial_model.addVars(num_edges, GRB_CONTINUOUS);  initial_model.update();  GRBVar *vars = initial_model.getVars();  // Set up objective function.  GRBLinExpr obj;  for (int e = 0; e < num_edges; e++) {    obj += GRBLinExpr(vars[e], edges[e].weight);  }  initial_model.setObjective(obj, GRB_MINIMIZE);  // Add initial constraints.  for (int n = 0; n < num_nodes; n++) {    GRBLinExpr lhs;    // Add all edges that are adjacent to the current node.    for (int e = 0; e < num_edges; e++) {      if (edges[e].end1 == n || edges[e].end2 == n) {        lhs += GRBLinExpr(vars[e], 1.0);      }    }    initial_model.addConstr(lhs, GRB_EQUAL, GRBLinExpr(2.0));  }  for (int e = 0; e < num_edges; e++) {    initial_model.addConstr(GRBLinExpr(vars[e], 1.0),                            GRB_LESS_EQUAL,                            GRBLinExpr(1.0));  }  initial_model.update();
  lp_solves = 0;  subtour_constraints = 0;
  // Branch and cut.  double cost;  double cost_opt = std::numeric_limits<double>::infinity();  int num_components;  std::vector<int> components;  std::deque<GRBModel> problems;  problems.push_back(initial_model);  while (problems.size() > 0) {    // Get next problem in queue.    GRBModel model = problems.front();    problems.pop_front();
    // In this loop, the LP is solved repeatedly until a solution without    // subtours is found.    bool skipped = false;    while (true) {      // Solve current model.      model.optimize();      lp_solves++;
      if (model.get(GRB_IntAttr_Status) != GRB_OPTIMAL) {        // Do not continue branch if problem is infeasible.        skipped = true;        break;      }
      // Check cost. If it is too high, stop following this branch.      cost = model.get(GRB_DoubleAttr_ObjVal);      // We can cut off branches more aggressively when the weights,      // and thus the optimal cost, are integral.      if (integral_weights && cost > cost_opt - 1.0 + tol) {          skipped = true;          break;      }      if (cost > cost_opt) {        skipped = true;        break;      }
      // Get current solution.      delete[] vars;      vars = model.getVars();      for (int e = 0; e < num_edges; e++) {        x[e] = vars[e].get(GRB_DoubleAttr_X);      }
      // Find connected components of the solution and eliminate subtours.      find_components(num_nodes,                      num_edges,                      edges,                      x,                      num_components,                      components);
      if (num_components == 1) {        // There are no more subtours that could be eliminated.        break;      }
      // We will add one constraint per connected component.      std::vector<GRBLinExpr> lhs;      lhs.resize(num_components);      for (int e = 0; e < num_edges; e++) {        // Identify the component this edge belongs to.        const int c1 = components[edges[e].end1];        const int c2 = components[edges[e].end2];        // Skip edges that connect two different components.        if (c1 != c2) continue;        // Add edge to the subtour elimination constraint.        lhs[c1] += GRBLinExpr(vars[e], 1.0);      }      // Compute the size of each component.      // This is required for the right-hand side of the constraints.      std::vector<int> component_sizes;      component_sizes.resize(num_components);      for (int c = 0; c < num_components; c++) {        component_sizes[c] = 0;      }      for (int n = 0; n < num_nodes; n++) {        component_sizes[components[n]]++;      }      // Add constraints to model.      for (int c = 0; c < num_components; c++) {        model.addConstr(lhs[c],                        GRB_LESS_EQUAL,                        GRBLinExpr(component_sizes[c] - 1.0));        subtour_constraints++;      }      std::cout << "Added " << num_components                << " subtour elimination constraints." << std::endl;      model.update();    }
    if (skipped) continue;
    // Branch using a fractional variable.    bool integral_sol = true;    for (int e = 0; e < num_edges; e++) {      if (!is_integral(x[e])) {        integral_sol = false;        // Add model with <= constraint for fractional variable.        GRBModel model_le(model);        model_le.addConstr(GRBLinExpr(vars[e], 1.0),                           GRB_LESS_EQUAL,                           GRBLinExpr(std::floor(x[e])));        model_le.update();        // Add model with >= constraint for fractional variable.        GRBModel model_ge(model);        model_ge.addConstr(GRBLinExpr(vars[e], 1.0),                           GRB_GREATER_EQUAL,                           GRBLinExpr(std::ceil(x[e])));        model_ge.update();        // Check if the soft limit for the number of LPs is hit.        if (problems.size() < lp_soft_limit) {          // Add new problems at the end of the queue.          // This corresponds to breadth-first search.          problems.push_back(model_le);          problems.push_back(model_ge);        }        else {          // Add new problems at the beginning of the queue.          // This corresponds to depth-first search.          problems.push_front(model_le);          problems.push_front(model_ge);        }        // Print information about the queue.        std::cout << "Branching; there are now "                  << problems.size()                  << " models in the queue."                  << std::endl << std::endl;        // Stop after creating one branch!        break;      }    }
    // Update optimal cost and optimal solution if integral solution was found.    if (integral_sol) {      cost_opt = cost;      x_opt = x;    }  }}
int main(int argc, char **argv){  // Read problem from stdin.  std::string line;  // Read problem size.  std::getline(std::cin, line);  // Remove leading spaces.  while (line[0] == ' ') line = line.substr(1);  const int num_nodes = std::stoi(line.substr(0, line.find(" ")));  const int num_edges = std::stoi(line.substr(line.find(" ") + 1));  // Read graph.  std::vector<Edge> edges;  edges.resize(num_edges);  for (int e = 0; e < num_edges; e++) {    std::getline(std::cin, line);    // Remove leading spaces.    while (line[0] == ' ') line = line.substr(1);    edges[e].end1 = std::stoi(line.substr(0, line.find(" ")));    line = line.substr(line.find(" ") + 1);    edges[e].end2 = std::stoi(line.substr(0, line.find(" ")));    line = line.substr(line.find(" ") + 1);    edges[e].weight = std::stod(line);  }  std::cout << "Loaded TSP with " << num_nodes << " nodes and "                                  << num_edges << " edges.\n";
  // Check if the edge weights are integral.  // If so, we can optimize a bit more aggressively in some places.  bool integral_weights = true;  for (int e = 0; e < num_edges; e++) {    if (!is_integral(edges[e].weight)) {      integral_weights = false;      break;    }  }
  std::cout << "Computation begins.\n";  // Start timer.  const auto t_start = std::chrono::high_resolution_clock::now();  // Solve TSP using Gurobi (for the LPs).  std::vector<double> x_opt;  int lp_solves;  int subtour_constraints;  try {    solve_TSP(num_nodes,              num_edges,              edges,              integral_weights,              x_opt,              lp_solves,              subtour_constraints);  }  catch (const GRBException &e) {    std::cerr << "Gurobi exception: " << e.getMessage() << std::endl;    std::exit(1);  }  // Stop timer.  const auto t_end = std::chrono::high_resolution_clock::now();  const std::chrono::duration<double> dtime = t_end - t_start;
  std::cout << "Computation finished ("             << std::fixed << std::setprecision(3)            << dtime.count() << "s).\n";
  // Print additional information.  std::cout << "Solved a total of " << lp_solves << " LPs." << std::endl;  std::cout << "Added a total of " << subtour_constraints            << " subtour elimination constraints." << std::endl;
  // Print optimal solution.  std::cout << "The best tour is:\n";  double c_optimal = 0.0;  // Set output format.  if (integral_weights) {    std::cout << std::setprecision(0);  }  else {    std::cout << std::setprecision(1);  }  for (int e = 0; e < num_edges; e++) {    // See if the edge is used.    if (x_opt[e] > 0.0) {      std::cout << edges[e].end1 << " "                << edges[e].end2 << " "                << edges[e].weight << std::endl;      c_optimal += x_opt[e]*edges[e].weight;    }  }  std::cout << "The cost of the best tour is: " << c_optimal << std::endl;
  return 0;}

reference

  • [1] (https://en.wikipedia.org/wiki/Branch_and_cut)
  • [2] (https://optimization.mccormick.northwestern.edu/index.php/Branch_and_cut)

原文发布于微信公众号 - 程序猿声(ProgramDream)

原文发表时间:2019-08-20

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