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In this paper, we consider a class of mixed integer programming problems(MIPs) whose objective functions are DC functions, that is, functionsrepresentable in terms of the difference of two convex functions. These MIPscontain a very wide class of computationally difficult nonconvex MIPs since theDC functions have powerful expressability. Recently, Maehara, Marumo, andMurota provided a continuous reformulation without integrality gaps fordiscrete DC programs having only integral variables. They also presented a newalgorithm to solve the reformulated problem. Our aim is to extend their resultsto MIPs and give two specific algorithms to solve them. First, we propose analgorithm based on DCA originally proposed by Pham Dinh and Le Thi, whereconvex MIPs are solved iteratively. Next, to handle nonsmooth functionsefficiently, we incorporate a smoothing technique into the first method. Weshow that sequences generated by the two methods converge to stationary pointsunder some mild assumptions.

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