Convex mixed integer nonlinear programming problems and an outer approximation algorithm

TL;DR

This paper extends the outer approximation algorithm to convex mixed-integer nonlinear programming problems with non-differentiable data by using subgradients, providing a convergent method to find optimal solutions.

Contribution

It generalizes the outer approximation method to handle non-differentiable convex MINLPs using subgradients and establishes convergence of the algorithm.

Findings

Developed an outer approximation algorithm for non-differentiable convex MINLPs.

Proved convergence of the proposed algorithm.

Extended the applicability of outer approximation methods to a broader class of problems.

Abstract

In this paper, we mainly study one class of convex mixed-integer nonlinear programming problems (MINLPs) with non-differentiable data. By dropping the differentiability assumption, we substitute gradients with subgradients obtained from KKT conditions, and use the outer approximation method to reformulate convex MINLP as one equivalent MILP master program. By solving a finite sequence of subproblems and relaxed MILP problems, we establish an outer approximation algorithm to find the optimal solution of this convex MINLP. The convergence of this algorithm is also presented. The work of this paper generalizes and extends the outer approximation method in the sense of dealing with convex MINLPs from differentiable case to non-differentiable one.