/***********************************************************************
Xpress Optimizer Examples
=========================
file TSP.cs
````````````
Solve a MIP using cuts/constraints that are lazily separated.
We take a random instance of the symmetric TSP and solve that using
lazily separated constraints.
(c) 2021-2023 Fair Isaac Corporation
***********************************************************************/
using System;
using System.Linq;
using Optimizer;
namespace XPRSexamples
{
/// <summary>Example for solving a MIP using lazily separated cuts/constraints.</summary>
/// <remarks>
/// We solve a random instance of the symmetric TSP using lazily separated
/// cuts/constraints.
///
/// <p>
/// The model is based on a graph G = (V,E).
/// We have one binary variable x[e] for each edge e in E. That variable
/// is set to 1 if edge e is selected in the optimal tour and 0 otherwise.
/// </p>
/// <p>
/// The model contains only one explicit constraint:
/// <code>
/// for each v in V: sum(u in V : u != v) x[uv] == 2
/// </code>
/// This states that from all the edges incident to a node u, exactly two
/// must be selected in the optimal tour.
/// </p>
/// <p>
/// The above constraints ensures that the selected edges form tours. However,
/// it allows multiple tours, also known as subtours. So we need a constraint
/// that requires that there is only one tour (which then necessarily hits
/// all the nodes). This constraint is known as subtour elimination constraint
/// and is
/// <code>
/// sum(e in S) x[e] <= |S|-1 for each subtour S
/// </code>
/// Since there are exponentially many subtours in a graph, this constraint
/// is not stated explicitly. Instead we check for any solution that the
/// optimizer finds, whether it satisfies the subtour elimination constraint.
/// If it does then we accept the solution. Otherwise we reject the solution
/// and augment the model by the violated subtour eliminiation constraint.
/// </p>
/// <p>
/// This lazy addition of constraints is implemented using two callbacks:
/// - a preintsol callback that rejects any solution that violates a
/// subtour elimination constraint,
/// - an optnode callback that injects any violated subtour elimination
/// constraints.
/// </p>
/// <p>
/// An important thing to note about this strategy is that dual reductions
/// have to be disabled. Since the optimizer does not see the whole model
/// (subtour elimination constraints are only generated on the fly), dual
/// reductions may cut off the optimal solution.
/// </p>
/// </remarks>
public class TSP
{
/// <summary>
/// A point in the plane.
/// </summary>
public class Point
{
public Point(double x, double y)
{
X = x;
Y = y;
}
public double X { get; set; }
public double Y { get; set; }
}
/** Number of nodes in the instance. */
private readonly int nodes;
/** Number of edges in the instance. */
private readonly int edges;
private readonly Point[] coordinates;
/** Variable indices for the edges. */
private readonly int[,] x;
/// <summary>Construct a new random instance with random seed 0 and 10 nodes.</summary>
public TSP() : this(10, 0) {}
/// <summary>Construct a new random instance with random seed 0.</summary>
/// <param name="nodes">The number of nodes in the instance.</param>
public TSP(int nodes) : this(nodes, 0) {}
/// <summary>Construct a new random instance.</summary>
/// <param name="nodes">The number of nodes in the instance.</param>
/// <param name="seed">Random number seed.</param>
public TSP(int nodes, int seed)
{
this.nodes = nodes;
edges = (nodes * (nodes - 1)) / 2;
Random rand = new Random(seed);
coordinates = Enumerable
.Range(0, nodes)
.Select((i) => new Point(4.0 * rand.NextDouble(), 4.0 * rand.NextDouble()))
.ToArray();
x = new int[nodes, nodes];
}
/// <summary>Get the distance between two nodes.</summary>
/// <param name="u">First node.</param>
/// <param name="v">Second node.</param>
/// <returns>The distance between <c>u</c> and <c>v</c>. The distance is symmetric.</returns>
public double Distance(int u, int v)
{
double deltaX = coordinates[u].X - coordinates[v].X;
double deltaY = coordinates[u].Y - coordinates[v].Y;
return Math.Sqrt(deltaX * deltaX + deltaY * deltaY);
}
/// <summary>Find the tour rooted at 0 in a solution.</summary>
/// <remarks>As a side effect, the tour is printed to the console.</remarks>
/// <param name="sol">The current solution.</param>
/// <param name="from">Stores the tour. <c>from[u]</c> yields the predecessor of <c>u</c> in the tour.
/// If <c>from[u]</c> is negative then <c>u</c> is not in the tour. This parameter can be <c>null</c>.</param>
/// <returns>The length of the tour.</returns>
private int FindTour(double[] sol, int[] from)
{
if (from == null)
from = new int[nodes];
for (int i = 0; i < from.Length; ++i)
from[i] = -1;
bool[] inTour = new bool[edges]; // Marks edges on the subtour
int u = 0;
int used = 0;
Console.Write("0");
do
{
for (int v = 0; v < nodes; ++v)
{
if (u == v) // no self-loops
continue;
else if (from[v] != -1) // node already on tour
continue;
else if (sol[x[u, v]] < 0.5) // edge not selected in solution
continue;
else if (inTour[x[u, v]]) // edge already on tour
continue;
else
{
Console.Write(" -> " + v);
inTour[x[u, v]] = true;
from[v] = u;
used += 1;
u = v;
break;
}
}
} while (u != 0);
Console.WriteLine();
return used;
}
/// <summary>Integer solution check callback.</summary>
private void PreIntsolCallback(XPRSprob prob, object cbdata, bool soltype, ref int p_reject, ref double p_cutoff) {
Console.WriteLine("Checking feasible solution ...");
// Get current solution and check whether it is feasible
double[] sol = prob.GetLpSolX();
int used = FindTour(sol, null);
Console.Write("Solution is ");
if (used<nodes)
{
Console.WriteLine("infeasible (" + used + " edges)");
p_reject = 1;
}
else
{
Console.WriteLine("feasible");
}
}
/// <summary>Optimal node callback.</summary>
/// <remarks>
/// This callback is invoked after the LP relaxation of a node is solved.
/// This is where we can inject additional constraints as cuts.
/// </remarks>
private void OptnodeCallback(XPRSprob prob, object cbdata, out int p_infeasible)
{
// We don't consider this node infeasible. Instead we add cuts to solve
// the LP relaxation again.
p_infeasible = 0;
// Only separate constraints on nodes that are integer feasible.
if (prob.MIPInfeas != 0)
return;
// Get the current solution.
double[] sol = prob.GetLpSolX();
// Get the tour starting at 0 and check whether it covers all
// nodes. If it does not then it is infeasible and we must
// generate a subtour elimination constraint.
int[] from = new int[nodes];
int used = FindTour(sol, from);
if (used < nodes)
{
// The tour is too short. Get the edges on the tour and add a
// subtour elimination constraint
int[] ind = new int[used];
double[] val = Enumerable.Repeat(1.0, used).ToArray();
for (int u = 0, next = 0; u < nodes; ++u)
{
if (from[u] >= 0)
ind[next++] = x[u, from[u]];
}
// Since we created the constraint in the original space, we must
// crush it to the presolved space before we can add it.
XPRSprob.RowInfo r = prob.PresolveRow(ind, val, 'L', used - 1);
if (r != null)
prob.AddCut(1, r);
}
// Here we would set p_infeasible=1 if the current node is considered
// infeasible.
}
/// <summary>Create a feasible tour and add this as initial MIP solution.</summary>
private void CreateInitialTour(XPRSprob prob)
{
int[] ind = new int[nodes];
double[] val = new double[nodes];
// Create a tour that just visits each node in order, i.e., goes from u to u+1.
prob.AddMipSol(Enumerable.Repeat(1.0, nodes).ToArray(),
Enumerable.Range(0, nodes).Select((u) => x[u, (u + 1) % nodes]).ToArray());
}
/// <summary>Solve the TSP represented by this instance.</summary>
public void Solve()
{
using (XPRSprob prob = new XPRSprob("")) {
// Create variables. We create one variable for each edge in
// the (complete) graph. x[u][v] gives the index of the variable
// that represents edge uv. Since edges are undirected, x[v][u]
// gives the same variable.
// All variables are binary.
for (int i = 0; i < nodes; ++i)
{
for (int j = i + 1; j < nodes; ++j)
{
x[j, i] = x[i, j] = prob.BinVar("x_" + i + "_" + j);
}
}
// Objective. All variables are in the objective and their
// respective coefficient is the distance between the two nodes.
int[] objInd = new int[edges];
double[] objVal = new double[edges];
for (int u = 0, nz = 0; u < nodes; ++u)
{
for (int v = u + 1; v < nodes; ++v)
{
objInd[nz] = x[u, v];
objVal[nz] = Distance(u, v);
++nz;
}
}
prob.SetObjective(objInd, objVal, ObjSense.Minimize);
// Constraint: In the graph that is induced by the selected
// edges, each node should have degree 2.
// This is the only constraint we add explicitly.
// Subtour elimination constraints are added
// dynamically via a callback.
for (int u = 0; u < nodes; ++u)
{
int[] ind = Enumerable.Range(0, nodes)
.Where((v) => v != u)
.Select((v) => x[u,v])
.ToArray();
double[] val = Enumerable.Repeat(1.0, ind.Length).ToArray();
prob.AddRow(ind, val, 'E', 2);
}
// Create a starting solution.
// This is optional but having a feasible solution available right
// from the beginning can improve optimizer performance.
CreateInitialTour(prob);
// Write out the model in case we want to look at it.
prob.WriteProb("tsp.lp", "l");
// We don't have all constraints explicitly in the matrix, hence
// we must disable dual reductions. Otherwise MIP presolve may
// cut off the optimal solution.
prob.MIPDualReductions = 0;
// Add a callback that rejects solutions that do not satisfy
// the subtour constraints.
prob.AddPreIntsolCallback(PreIntsolCallback);
// Add a callback that separates subtour elimination constraints
prob.AddOptnodeCallback(OptnodeCallback);
// Add a message listener to display log information.
prob.AddMessageCallback((p, d, m, l, t) => { if (l > 0) Console.WriteLine(m); });
prob.MipOptimize();
double[] sol = prob.GetMipSolX();
// Print the optimal tour.
Console.WriteLine("Tour with length " + prob.MIPBestObjVal);
FindTour(sol, null);
}
}
public static void Main(String[] args)
{
new TSP(10).Solve();
}
}
}
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