Interior Point Algorithms for Integer Programming

An algorithm for solving linear programming problems can be used as a subroutine in methods for more complicated problems. Such methods usually involve solving a sequence of related linear programming problems, where the solution to one linear program is close to the solution to the next, that is, it provides a warm start for the next linear program. The branch and cut method for solving integer programming problems is of this form: linear programming relaxations are solved until eventually the integer programming problem has been solved. Within the last ten years, interior point methods have become accepted as powerful tools for solving linear programming problems. It appears that interior point methods may well solve large linear programs substantially faster than the simplex method. A natural question, therefore, is whether interior point methods can be successfully used to solve integer programming problems. This requires the ability to exploit a warm start. As discussed in Chapter 5 of this book, branch and cut methods for integer programming problems are a combination of cutting plane methods and branch and bound methods. In a cutting plane method for solving an integer programming problem, the linear programming relaxation of the integer program is solved. If the optimal solution to the linear program is feasible in the integer program, it also solves the integer program; otherwise, a constraint is added to the linear program which separates the optimal solution from the set of feasible solutions to the integer program, the new linear program is solved, and the process is repeated. In a branch and bound method for solving an integer programming problem, the rst step is also to solve the linear programming relaxation. If the optimal solution is feasible in the integer program, it solves the integer program. Otherwise, the relaxation is split into two subproblems, usually by xing a particular variable at zero or one. This is the start of a tree of subproblems. A subproblem is selected and the linear programming relaxation of that subproblem is solved. Four outcomes are possible: the linear programming relaxation is infeasible, in which case the integer subproblem is also infeasible, and the tree