# A new approach for solving mixed integer DC programs using a continuous relaxation with no integrality gap and smoothing techniques

## TL;DR

This paper introduces a novel approach for solving mixed integer DC programming problems by extending continuous reformulation techniques, employing DCA and smoothing methods to achieve convergence to stationary points.

## Contribution

It extends continuous reformulation without integrality gaps to mixed integer problems and proposes two algorithms, including a smoothing-enhanced variant, with proven convergence.

## Key findings

- Algorithms converge to stationary points under mild assumptions
- Smoothing technique improves handling of nonsmooth functions
- Extension of continuous reformulation to mixed integer DC programs

## Abstract

In this paper, we consider a class of mixed integer programming problems (MIPs) whose objective functions are DC functions, that is, functions representable in terms of the difference of two convex functions. These MIPs contain a very wide class of computationally difficult nonconvex MIPs since the DC functions have powerful expressability. Recently, Maehara, Marumo, and Murota provided a continuous reformulation without integrality gaps for discrete DC programs having only integral variables. They also presented a new algorithm to solve the reformulated problem. Our aim is to extend their results to MIPs and give two specific algorithms to solve them. First, we propose an algorithm based on DCA originally proposed by Pham Dinh and Le Thi, where convex MIPs are solved iteratively. Next, to handle nonsmooth functions efficiently, we incorporate a smoothing technique into the first method. We show that sequences generated by the two methods converge to stationary points under some mild assumptions.