Emiliano Traversi

Automatic Dantzig---Wolfe reformulation of mixed integer programs

Dantzig–Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig–Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig–Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

Partial convexification of general MIPs by Dantzig-Wolfe reformulation

Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered block-diagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proof-of-concept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 0-1 dynamic knapsack problem (without block-diagonal structure) and on general MIPLIB2003 instances. Our results support that Dantzig-Wolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.

Optimal linear arrangements using betweenness variables

We solve for the first time to proven optimality the small instances in the classical literature benchmark of Minimum Linear Arrangement. This is achieved by formulating the problem as an ILP in a somehow unintuitive way, using variables expressing the fact that a vertex is between two other adjacent vertices in the arrangement. Using (only) these variables appears to be the key idea of the approach. Indeed, with these variables already the use of very simple constraints leads to good results, which can however be improved with a more detailed study of the underlying polytope.

Separable Non-convex Underestimators for Binary Quadratic Programming

We present a new approach to constrained quadratic binary programming. Dual bounds are computed by choosing appropriate global underestimators of the objective function that are separable but not necessarily convex. Using the binary constraint on the variables, the minimization of this separable underestimator can be reduced to a linear minimization problem over the same set of feasible vectors. For most combinatorial optimization problems, the linear version is considerably easier than the quadratic version. We explain how to embed this approach into a branch-and-bound algorithm and present experimental results.

Hybrid SDP Bounding Procedure

The principal idea of this paper is to exploit Semidefinite Programming (SDP) relaxation within the framework provided by Mixed Integer Nonlinear Programming (MINLP) solvers when tackling Binary Quadratic Problems. We included the SDP relaxation in a state-of-the-art MINLP solver as an additional bounding technique and demonstrated that this idea could be computationally useful. The Quadratic Stable Set Problem is adopted as the case study. The tests indicate that the Hybrid SDP Bounding Procedure allows an average 50% cut of the overall computing time and a cut of more than one order of magnitude for the branching nodes.

Solving the Temporal Knapsack Problem via Recursive Dantzig-Wolfe Reformulation

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig-Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursively applying DWR, i.e., by using DWR not only for solving the original MIP but also for recursively solving the pricing sub-problems. In a binary case (like the TKP), the Recursive DWR method can be performed in such a way that the only two components needed during the optimization are a Linear Programming solver and an algorithm for solving Knapsack Problems. The Recursive DWR allows us to solve Temporal Knapsack Problem instances through computation of strong dual bounds, which could not be obtained by exploiting the best-known previous approach based on DWR. We solve the Temporal Knapsack Problem using what we call a Multilevel Dantzig-Wolfe Reformulation (DWR) method.The idea of Multilevel DWR is to solve a Mixed Integer Program by recursively applying DWR.The method allows fast computation of strong bounds.

Exact approaches for the knapsack problem with setups

Abstract We consider a generalization of the knapsack problem in which items are partitioned into classes, each characterized by a fixed cost and capacity. We study three alternative Integer Linear Programming formulations. For each formulation, we design an efficient algorithm to compute the linear programming relaxation (one of which is based on Column Generation techniques). We theoretically compare the strength of the relaxations and derive specific results for a relevant case arising in benchmark instances from the literature. Finally, we embed the algorithms above into a unified implicit enumeration scheme which is run in parallel with an improved Dynamic Programming algorithm to effectively solve the problem to proven optimality. An extensive computational analysis shows that our new exact algorithm is capable of efficiently solving all the instances of the literature and turns out to be the best algorithm for instances with a low number of classes.

On the separation of split inequalities for non-convex quadratic integer programming

We investigate the computational potential of split inequalities for non-convex quadratic integer programming, first introduced by Letchford (2010) and further examined by Burer and Letchford (2012). These inequalities can be separated by solving convex quadratic integer minimization problems. For small instances with box-constraints, we show that the resulting dual bounds are very tight; they can close a large percentage of the gap left open by both the RLT- and the SDP-relaxations of the problem. The gap can be further decreased by separating the so-called non-standard split inequalities, which we examine in the case of ternary variables.

Quadratic Combinatorial Optimization Using Separable Underestimators

Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems ...

Optimum Vehicle Flows in a Fully Automated Vehicle Network

This paper provides a novel assignment method and a solution algorithm that allows to determine the optimum vehicle flows in a fully automated vehicle network. This assignment method incorporates the following specific features: (1) optimal redistribution of occupied and unoccupied vehicles; (2) inter-vehicle spacing is adapted to meet the minimum safe distance criteria on congested link, (no collision in the worst failure case); (3) trip-time minimization of all traffic participants by a centralized vehicle routing. The latter feature allows the realization of a so called system optimum solution, which minimizes the total time of all trips. This assignment method is applied to two, topologically different, test networks at different travel demand levels, in order to determine: the share of unoccupied vehicle, the minimum number of required vehicles, the share of congested links, the lost trip-time of occupied vehicles due to the presents of unoccupied vehicles. Furthermore, the advantage of a centralized vehicle routing is quantified by comparing the total trip-times of a scenario using a system optimum solution with a scenario applying the user equilibrium solution, without considering unoccupied vehicle flows. Regarding the investigated scenarios, the share of unoccupied vehicle flows with centralized vehicle routing in a uniform, random demand scenario is approximately 11%−14%.