Stabilized Benders methods for large-scale combinatorial optimization, with appllication to data privacy

The Cell Suppression Problem (CSP) is a challenging Mixed-Integer Linear Problem arising in statistical tabular data protection. Medium sized instances of CSP involve thousands of binary variables and million of continuous variables and constraints. However, CSP has the typical structure that allows application of the renowned Benders’ decomposition method: once the “complicating” binary variables are fixed, the problem decomposes into a large set of linear subproblems on the “easy” continuous ones. This allows to project away the easy variables, reducing to a master problem in the complicating ones where the value functions of the subproblems are approximated with the standard cutting-plane approach. Hence, Benders’ decomposition suffers from the same drawbacks of the cutting-plane method, i.e., oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with randomly generated and real-world CSP instances with up to 3600 binary variables, 90M continuous variables and 15M inequality constraints show that our approach is competitive with both the current state-of-the-art (cutting-plane-based) code for cell suppression, and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders is able to quickly provide a very good solution in less than one minute, while the other approaches were not able to find any feasible solution in one hour.Peer ReviewedPreprin

One Benders cut to rule all schedules in the neighbourhood

Logic-Based Benders Decomposition (LBBD) and its Branch-and-Cut variant, namely Branch-and-Check, enjoy an extensive applicability on a broad variety of problems, including scheduling. Although LBBD offers problem-specific cuts to impose tighter dual bounds, its application to resource-constrained scheduling remains less explored. Given a position-based Mixed-Integer Linear Programming (MILP) formulation for scheduling on unrelated parallel machines, we notice that certain $k-$OPT neighbourhoods could implicitly be explored by regular local search operators, thus allowing us to integrate Local Branching into Branch-and-Check schemes. After enumerating such neighbourhoods and obtaining their local optima - hence, proving that they are suboptimal - a local branching cut (applied as a Benders cut) eliminates all their solutions at once, thus avoiding an overload of the master problem with thousands of Benders cuts. However, to guarantee convergence to optimality, the constructed neighbourhood should be exhaustively explored, hence this time-consuming procedure must be accelerated by domination rules or selectively implemented on nodes which are more likely to reduce the optimality gap. In this study, the realisation of this idea is limited on the common 'internal (job) swaps' to construct formulation-specific $4$-OPT neighbourhoods. Nonetheless, the experimentation on two challenging scheduling problems (i.e., the minimisation of total completion times and the minimisation of total tardiness on unrelated machines with sequence-dependent and resource-constrained setups) shows that the proposed methodology offers considerable reductions of optimality gaps or faster convergence to optimality. The simplicity of our approach allows its transferability to other neighbourhoods and different sequencing optimisation problems, hence providing a promising prospect to improve Branch-and-Check methods

Accelerated Benders Decomposition and Local Branching for Dynamic Maximum Covering Location Problems

The maximum covering location problem (MCLP) is a key problem in facility location, with many applications and variants. One such variant is the dynamic (or multi-period) MCLP, which considers the installation of facilities across multiple time periods. To the best of our knowledge, no exact solution method has been proposed to tackle large-scale instances of this problem. To that end, in this work, we expand upon the current state-of-the-art branch-and-Benders-cut solution method in the static case, by exploring several acceleration techniques. Additionally, we propose a specialised local branching scheme, that uses a novel distance metric in its definition of subproblems and features a new method for efficient and exact solving of the subproblems. These methods are then compared through extensive computational experiments, highlighting the strengths of the proposed methodologies

NIHBA : A network interdiction approach for metabolic engineering design

Funding Information: This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) for funding project ‘Synthetic Portabolomics: Leading the way at the crossroads of the Digital and the Bio Economies (EP/N031962/1)’. N.K. was funded by a Royal Academy of Engineering Chair in Emerging Technology award.Peer reviewedPublisher PD

The Recoverable Robust Tail Assignment Problem

This is the author accepted manuscript. The final version is available from Institute for Operations Research and the Management Sciences (INFORMS) via the DOI in this record Schedule disruptions are commonplace in the airline industry with many flight-delaying events occurring each day. Recently there has been a focus on introducing robustness into airline planning stages to reduce the effect of these disruptions. We propose a recoverable robustness technique as an alternative to robust optimisation to reduce the effect of disruptions and the cost of recovery. We formulate the recoverable robust tail assignment problem (RRTAP) as a stochastic program, solved using column generation in the master and subproblems of the Benders decomposition. We implement a two-phase algorithm for the Benders decomposition incorporating the Magnanti-Wong [21] enhancement techniques. The RRTAP includes costs due to flight delays, cancellation, and passenger rerouting, and the recovery stage includes cancellation, delay, and swapping options. To highlight the benefits of simultaneously solving planning and recovery problems in the RRTAP we compare our tail assignment solution with the tail assignment generated using a connection cost function presented in Gr¨onkvist [15]. Using airline data we demonstrate that by developing a better tail assignment plan via the RRTAP framework, one can reduce recovery costs in the event of a disruption.Australian Research Council Centre of Excellence for MathematicsMASCOS

A Benders Based Rolling Horizon Algorithm for a Dynamic Facility Location Problem

This study presents a well-known capacitated dynamic facility location problem (DFLP) that satisfies the customer demand at a minimum cost by determining the time period for opening, closing, or retaining an existing facility in a given location. To solve this challenging NP-hard problem, this paper develops a unique hybrid solution algorithm that combines a rolling horizon algorithm with an accelerated Benders decomposition algorithm. Extensive computational experiments are performed on benchmark test instances to evaluate the hybrid algorithm’s efficiency and robustness in solving the DFLP problem. Computational results indicate that the hybrid Benders based rolling horizon algorithm consistently offers high quality feasible solutions in a much shorter computational time period than the stand-alone rolling horizon and accelerated Benders decomposition algorithms in the experimental range

Quadratic stabilization of Benders decomposition

The foundational Benders decomposition, or variable decomposition, is known to have the inherent instability of cutting plane-based methods. Several techniques have been proposed to improve this method, which has become the state of the art for important problems in operations research. This paper presents a complementary improvement featuring quadratic stabilization of the Benders cutting-plane model. Inspired by the level-bundle methods of nonsmooth optimization, this algorithmic improvement is designed to reduce the number of iterations of the method. We illustrate the interest of the stabilization on two classical problems: network design problems and hub location problems. We also prove that the stabilized Benders method has the same theoretical convergence properties as the usual Benders method

Solving the optimum communication spanning tree problem

This paper presents an algorithm based on Benders decomposition to solve the optimum communication spanning tree problem. The algorithm integrates within a branch-and-cut framework a stronger reformulation of the problem, combinatorial lower bounds, in-tree heuristics, fast separation algorithms, and a tailored branching rule. Computational experiments show solution time savings of up to three orders of magnitude compared to state-of-the-art exact algorithms. In addition, our algorithm is able to prove optimality for five unsolved instances in the literature and four from a new set of larger instances.Peer ReviewedPostprint (author's final draft