We propose improvements to some of the best heuristic algorithms for optimal product pricing problem originally designed by Dobson and Kalish in the late 1980’s and in the early 1990’s. Our improvements are based on a detailed study of a fundamental decoupling structure of the underlying mixed integer programming (MIP) problem and on incorporating more … Read more
For stochastic mixed-integer programs, we revisit the dual decomposition algorithm of Car\o{}e and Schultz from a computational perspective with the aim of its parallelization. We address an important bottleneck of parallel execution by identifying a formulation that permits the parallel solution of the \textit{master} program by using structure-exploiting interior-point solvers. Our results demonstrate the potential … Read more
We consider semi-continuous network flow problems, that is, a class of network flow problems where some of the variables are restricted to be semi-continuous. We introduce the semi-continuous inflow set with variable upper bounds as a relaxation of general semi-continuous network flow problems. Two particular cases of this set are considered, for which we present … Read more
In this paper we study the t-branch split cuts introduced by Li and Richard (2008). They presented a family of mixed-integer programs with n integer variables and a single continuous variable and conjectured that the convex hull of integer solutions for any n has unbounded rank with respect to (n-1)-branch split cuts. It was shown … Read more
Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, … Read more
Gunluk and Pochet [O. Gunluk, Y. Pochet: Mixing mixed integer inequalities. Mathematical Programming 90(2001) 429-457] proposed a procedure to mix mixed integer rounding (MIR) inequalities. The mixed MIR inequalities define the convex hull of the mixing set $\{(y^1,\ldots,y^m,v) \in Z^m \times R_+:\alpha_1 y^i + v \geq \b_i,i=1,\ldots,m\}$ and can also be used to generate valid … Read more
In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (2008). By analyzing $n$-dimensional lattice-free sets, we prove that for every integer $n$ there exists a positive integer $t$ such that every facet-defining inequality of the … Read more
We study a mixed integer linear program with $m$ integer variables and $k$ non-negative continuous variables in the form of the relaxation of the corner polyhedron that was introduced by Andersen, Louveaux, Weismantel and Wolsey [\emph{Inequalities from two rows of a simplex tableau}, Proc.\ IPCO 2007, LNCS, vol.~4513, Springer, pp.~1–15]. We describe the facets of … Read more
In the paper we prove that any nonconvex quadratic problem over some set $K\subset \mathbb{R}^n$ with additional linear and binary constraints can be rewritten as linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and … Read more
In a recent paper, Dash, Dey and Gunluk (2010) showed that many families of inequalities for the two-row continuous group relaxation and variants of this relaxation are cross cuts or crooked cross cuts, both of which generalize split cuts. Li and Richard (2008) recently studied t-branch split cuts for mixed-integer programs for integers t >= … Read more