The advantage of this approach is twofold. First, our C++ interface is more comfortable than the typical native interfaces of the solvers. Second, changing the underlying solver in a certain application using LEMON's LP interface needs no effort. So, for example, one may try her idea using an open source solver, demonstrate its usability for a customer and if it works well, but the performance should be improved, then the customer may decide to purchase and use a better commercial solver.
Currently, the following linear and mixed integer programming packages are supported: GLPK, Clp, Cbc, ILOG CPLEX and SoPlex. However, additional wrapper classes for new solvers can also be implemented quite easily.
// Create an instance of the default LP solver class // (it will represent an "empty" problem at first) Lp lp; // Add two columns (variables) to the problem Lp::Col x1 = lp.addCol(); Lp::Col x2 = lp.addCol(); // Add rows (constraints) to the problem lp.addRow(x1 - 5 <= x2); lp.addRow(0 <= 2 * x1 + x2 <= 25); // Set lower and upper bounds for the columns (variables) lp.colLowerBound(x1, 0); lp.colUpperBound(x2, 10); // Specify the objective function lp.max(); lp.obj(5 * x1 + 3 * x2); // Solve the problem using the underlying LP solver lp.solve(); // Print the results if (lp.primalType() == Lp::OPTIMAL) { std::cout << "Objective function value: " << lp.primal() << std::endl; std::cout << "x1 = " << lp.primal(x1) << std::endl; std::cout << "x2 = " << lp.primal(x2) << std::endl; } else { std::cout << "Optimal solution not found." << std::endl; }
Lp::Col type represents the columns, i.e. the (primal) variables of the LP problem. The numerical operators can be used to form expressions from columns, which are required for specifying rows (constraints) and the objective function. Due to the suitable operator overloads, a problem can be described in C++ conveniently, directly as it is expressed in mathematics. After specifying all the parameters of the problem, we can solve it using the solve() function. The status of the solution (namely OPTIMAL, UNBOUNDED, INFEASIBLE etc.) can be obtained using primalType() and the results can be obtained using primal().
The above problem has an optimal solution. If you execute this example, it will print out the following results.
Objective function value: 67.5 x1 = 7.5 x2 = 10
However, if you, for example, removed the lines that specify the rows, you would obtain an LP problem, for which the objective function value is unbounded over the feasible solutions. Thus the program would print out this line.
Optimal solution found.
If you would like to build linear expressions or constraints in more steps, then you can use the classes Lp::Expr and Lp::Constr. For example, this line
lp.addRow(x1 - 5 <= x2);
Lp::Expr e1, e2; e1 = x1; e1 -= 5; e2 = x2; Lp::Constr c = e1 <= e2; lp.addRow(c);
These classes make it easy to build more complex expressions and constraints, e.g. using loops. For example, we can sum all the variables (columns) in an expression using the Lp::ColIt iterator, which can be used the same as node and arc iterators for graphs.
Lp::Expr sum; for (Lp::ColIt col(lp); col != INVALID; ++col) { sum += col; }
After solving the problem, you can query the value of any primal expression, not just the individual columns and the objective function.
std::cout << lp.primal(sum) << std::endl;
Of course, working with the dual problem is also supported by the LP interface. Lp::Row represents the rows, i.e. the dual variables of the LP problem.
Lp::Row r = lp.addRow(x2 >= 3);
The dual solutions of an LP problem can be obtained using the functions dualType() and dual() (after solving the problem).
std::cout << lp.dual(r) << std::endl;
Lp::DualExpr can be used to build dual expressions from rows and Lp::RowIt can be used to list the rows.
Lp::DualExpr dual_sum; for (Lp::RowIt row(lp); row != INVALID; ++row) { dual_sum += row; }
The LP solver interface provides several other features, which are documented in the classes LpBase and LpSolver. For detailed documentation, see these classes in the reference manual.
If you would like to use a specific solver instead of the default one, you can use GlpkLp, ClpLp, CplexLp etc. instead of the class Lp.
The following example demonstrates the usage of the MIP solver interface. The whole program can be found in mip_demo.cc.
// Create an instance of the default MIP solver class // (it will represent an "empty" problem at first) Mip mip; // Add two columns (variables) to the problem Mip::Col x1 = mip.addCol(); Mip::Col x2 = mip.addCol(); // Add rows (constraints) to the problem mip.addRow(x1 - 5 <= x2); mip.addRow(0 <= 2 * x1 + x2 <= 25); // Set lower and upper bounds for the columns (variables) mip.colLowerBound(x1, 0); mip.colUpperBound(x2, 10); // Set the type of the columns mip.colType(x1, Mip::INTEGER); mip.colType(x2, Mip::REAL); // Specify the objective function mip.max(); mip.obj(5 * x1 + 3 * x2); // Solve the problem using the underlying MIP solver mip.solve(); // Print the results if (mip.type() == Mip::OPTIMAL) { std::cout << "Objective function value: " << mip.solValue() << std::endl; std::cout << "x1 = " << mip.sol(x1) << std::endl; std::cout << "x2 = " << mip.sol(x2) << std::endl; } else { std::cout << "Optimal solution not found." << std::endl; }
mip.sol() instead of mip.solValue()? Or could we write primalValue() for Lp in lp_demo.cc?x1 is specified to be INTEGER valued. x2 is set to REAL, which is the default option.
// Set the type of the columns mip.colType(x1, Mip::INTEGER); mip.colType(x2, Mip::REAL);
Because of this integrality requirement, the results will be different from the LP solution.
Objective function value: 67 x1 = 8 x2 = 9
The documentation of the MIP solver interface can be found in the reference manual at the class MipSolver. The common parts of the LP and MIP interfaces are documented in their common ancestor class LpBase.
TEMPLATE_DIGRAPH_TYPEDEFS(GR); // Create an instance of the default LP solver Lp lp; // Add a column to the problem for each arc typename GR::template ArcMap<Lp::Col> f(g); lp.addColSet(f); // Capacity constraints for (ArcIt a(g); a != INVALID; ++a) { lp.colLowerBound(f[a], 0); lp.colUpperBound(f[a], capacity[a]); } // Flow conservation constraints for (NodeIt n(g); n != INVALID; ++n) { if (n == source || n == target) continue; Lp::Expr e; for (OutArcIt a(g, n); a != INVALID; ++a) e += f[a]; for (InArcIt a(g, n); a != INVALID; ++a) e -= f[a]; lp.addRow(e == 0); } // Objective function Lp::Expr o; for (OutArcIt a(g, source); a != INVALID; ++a) o += f[a]; for (InArcIt a(g, source); a != INVALID; ++a) o -= f[a]; lp.max(); lp.obj(o); // Solve the LP problem lp.solve();