Using Interior-point Methods within an Outer Approximation Framework for Mixed Integer Nonlinear Programming

Interior-point methods for nonlinear programming have been demonstrated to be quite efficient, especially for large scale problems, and, as such, they are ideal candidates for solving the nonlinear subproblems that arise in the solution of mixed-integer nonlinear programming problems via outer approximation. However, traditionally, infeasible primal-dual interior-point methods have had two main perceived deficiencies: (1) lack of infeasibility detection capabilities, and (2) poor performance after a warmstart. In this paper, we propose the exact primal-dual penalty approach as a means to overcome these deficiencies. The generality of this approach to handle any change to the problem makes it suitable for the outer approximation framework, where each nonlinear subproblem can differ from the others in the sequence in a variety of ways. Additionally, we examine cases where the nonlinear subproblems take on special forms, namely those of second-order cone programming problems and semidefinite programming problems. Encouraging numerical results are provided.