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The multi-depot vehicle scheduling problem (MDVSP) is a critical planning challenge for transit agencies. We introduce a novel approach to MDVSP by incorporating service reliability through chance-constrained programming (CCP), targeting the pivotal issue of travel time uncertainty and its impact on transit service quality. Our model guarantees service reliability measured by on-time performance (OTP), a primary metric for transit agencies, and fairness across different service areas.We propose an exact branch-and-cut (B&C) scheme to solve our CCP model. We present several cut-generation procedures that exploit the underlying problem structure and analyze the relationship between the obtained cut families. Additionally, we design a Lagrangian-based heuristic to handle large-scale instances reflective of real-world transit operations. Our approach partitions the set of trips, each subset leading to a subproblem that can be efficiently solved with our B&C algorithm, and then employs a procedure to combine the subproblem solutions to create a vehicle schedule that satisfies all the planning constraints of the MDVSP. Our empirical evaluation demonstrates the superiority of our stochastic variant in achieving cost-effective schedules with reliable OTP guarantees compared to alternatives commonly used by practitioners, as well as the computational benefits of our methodologies.

Adaptive robust optimization problems have received significant attention in recent years, but remain notoriously difficult to solve when recourse decisions are discrete in nature. In this paper, we propose new reformulation techniques for adaptive robust binary optimization (ARBO) problems with objective uncertainty. Without loss of generality, we focus on ARBO problems with "selective adaptability", a term we coin to describe a common class of linking constraints between first-stage and second-stage solutions. Our main contribution revolves around a collection of exact and approximate network flow reformulations for the ARBO problem, which we develop by building upon ideas from the decision diagram literature. Our proposed models can generate feasible solutions, primal bounds and dual bounds, while their size and approximation quality can be precisely controlled through user-specified parameters. Furthermore, and in contrast with existing solution methods, these models are easy to implement and can be solved directly with standard off-the-shelf solvers. Through an extensive set of computational experiments, we show that our models can generate high-quality solutions and dual bounds in significantly less time than popular benchmark methods, often by orders of magnitude.

We investigate the stochastic transfer synchronization problem, which seeks to synchronize the timetables of different routes in a transit network to reduce transfer waiting times, delay times, and unnecessary in-vehicle times. We present a sophisticated two-stage stochastic mixed-integer programming model that takes into account variability in passenger walking times between bus stops, bus running times, dwell times, and demand uncertainty. Our model incorporates new features related to dwell time determination by considering passenger arrival patterns at bus stops which have been neglected in the literature on transfer synchronization and timetabling. We solve a sample average approximation of our model using a problem-based scenario reduction approach, and the progressive hedging algorithm. As a proof of concept, our computational experiments on two single transfer nodes in the City of Toronto, with a mixture of low- and high-frequency routes, demonstrate the potential advantages of the proposed model. Our findings highlight the necessity and value of incorporating stochasticity in transfer-based timetabling models.

This paper explores the integration of Automated Guided Vehicles (AGVs) in warehouse order picking, a crucial and cost-intensive aspect of warehouse operations. The booming AGV industry, accelerated by the COVID-19 pandemic, is witnessing widespread adoption due to its efficiency, reliability, and cost-effectiveness in automating warehouse tasks. This paper focuses on enhancing the picker-to-parts system, prevalent in small to medium-sized warehouses, through the strategic use of AGVs. We discuss the benefits and applications of AGVs in various warehouse tasks, highlighting their transformative potential in improving operational efficiency. We examine the deployment of AGVs by leading companies in the industry, showcasing their varied functionalities in warehouse management. Addressing the gap in research on optimizing operational performance in hybrid environments where humans and AGVs coexist, our study delves into a dynamic picker-to-parts warehouse scenario. We propose a novel approach Neural Approximate Dynamic Programming approach for coordinating a mixed team of human and AGV workers, aiming to maximize order throughput and operational efficiency. This involves innovative solutions for non-myopic decision making, order batching, and battery management. We also discuss the integration of advanced robotics technology in automating the complete order-picking process. Through a comprehensive numerical study, our work offers valuable insights for managing a heterogeneous workforce in a hybrid warehouse setting, contributing significantly to the field of warehouse automation and logistics.

Same-Day Delivery (SDD) services aim to maximize the fulfillment of online orders while minimizing delivery delays but are beset by operational uncertainties such as those in order volumes and courier planning. Our work aims to enhance the operational efficiency of SDD by focusing on the ultra-fast Order Dispatching Problem (ODP), which involves matching and dispatching orders to couriers within a centralized warehouse setting, and completing the delivery within a strict timeline (e.g., within minutes). We introduce important extensions to ultra-fast ODP such as order batching and explicit courier assignments to provide a more realistic representation of dispatching operations and improve delivery efficiency. As a solution method, we primarily focus on NeurADP, a methodology that combines Approximate Dynamic Programming (ADP) and Deep Reinforcement Learning (DRL), and our work constitutes the first application of NeurADP outside of the ride-pool matching problem. NeurADP is particularly suitable for ultra-fast ODP as it addresses complex one-to-many matching and routing intricacies through a neural network-based VFA that captures high-dimensional problem dynamics without requiring manual feature engineering as in generic ADP methods. We test our proposed approach using four distinct realistic datasets tailored for ODP and compare the performance of NeurADP against myopic and DRL baselines by also making use of non-trivial bounds to assess the quality of the policies. Our numerical results indicate that the inclusion of order batching and courier queues enhances the efficiency of delivery operations and that NeurADP significantly outperforms other methods. Detailed sensitivity analysis with important parameters confirms the robustness of NeurADP under different scenarios, including variations in courier numbers, spatial setup, vehicle capacity, and permitted delay time.

Ride-pooling services have been growing in popularity, increasing the need for efficient and effective operations. The main goal of ride-pooling services is to maximize the number of passengers served while minimizing wait and delay times. However, factors such as the timing and volume of passenger requests, pick-up and drop-off locations, available vehicle capacity, and the trajectory to fulfill multiple requests introduce high degrees of uncertainty, creating challenges for ride-pooling operators. This study aims to expand the current state-of-the-art Approximate Dynamic Programming (ADP) approach for ride-pooling services, introduce key extensions, and perform a comparative analysis with the Neural Approximate Dynamic Programming (NeurADP) approach to optimize the efficiency and effectiveness of these services. Specifically, we develop an ADP approach that incorporates three important problem specifications: (i) pick-up and drop-off deadlines, (ii) vehicle rebalancing, and (iii) allowing more than two passengers in a vehicle. We conduct a detailed numerical study with the New York City taxi-cab dataset and a novel dataset of taxi-cab requests collected in the city of Chicago. We also provide a sensitivity analysis on key model parameters such as wait and delay times, passenger group sizes, and vehicle capacity, along with the investigation of the effects of vehicle rebalancing. Our comparative analysis highlights the strengths and limitations of both ADP and NeurADP methodologies. Network density and road directionality are found to significantly impact the performance. NeurADP is found to be more efficient in learning value function approximations for larger and more complex problem settings than the ADP approach. However, for smaller settings, ADP is shown to outperform NeurADP.

In this work, we design primal and dual bounding methods for multistage adaptive robust optimization (MSARO) problems motivated by two decision rules rooted in the stochastic programming literature. From the primal perspective, this is achieved by applying decision rules that restrict the functional forms of only a certain subset of decision variables resulting in an approximation of MSARO as a two-stage adjustable robust optimization problem. We leverage the two-stage robust optimization literature in the solution of this approximation. From the dual perspective, decision rules are applied to the Lagrangian multipliers of a Lagrangian dual of MSARO, resulting in a two-stage stochastic optimization problem. As the quality of the resulting dual bound depends on the distribution chosen when developing the dual formulation, we define a distribution optimization problem with the aim of optimizing the obtained bound and develop solution methods tailored to the nature of the recourse variables. Our framework is general-purpose and does not require strong assumptions such as a stage-wise independent uncertainty set, and can consider integer recourse variables. Computational experiments on newsvendor, location-transportation, and capital budgeting problems show that our bounds yield considerably smaller optimality gaps compared to the existing methods.

We consider a multi-stage stochastic lot-sizing problem with service level constraints and supplier-driven product substitution. A firm has multiple products and it has the option to meet demand from substitutable products at a cost. Considering the uncertainty in future demands, the firm wishes to make ordering decisions in every period such that the probability that all demands can be met in the next period meets or exceeds a minimum service level. We propose a rolling-horizon policy in which a two-stage joint chance-constrained stochastic program is solved to make decisions in each time period. We demonstrate how to effectively solve this formulation. In addition, we propose two policies based on deterministic approximations. We demonstrate that the proposed chance-constraint policy can achieve the service levels more reliably and at a lower cost. We also explore the value of product substitution in this model, demonstrating that the substitution option allows achieving service levels while reducing costs by 7% to 25% in our experiments, and that the majority of the benefit can be obtained with limited levels of substitution allowed.

The Alternating Current Optimal Transmission Switching (ACOTS) problem incorporates line switching decisions into the fundamental AC optimal power flow (ACOPF) problem. The advantages of the ACOTS problem are well-known in terms of reducing the operational cost and improving system reliability. ACOTS optimization models contain discrete variables and nonlinear, non-convex structures, which make it difficult to solve. We derive strengthened quadratic convex (QC) relaxations for ACOTS by combining several methodologies recently developed in the ACOPF literature. First, we relax the ACOTS model with the on/off QC relaxation, which has been empirically observed to be both tight and computationally efficient in approximating the ACOPF problem. Further, we tighten this relaxation by using strong linearization with extreme-point representation, and by adding several types of new valid inequalities. In particular, we derive a novel kind of "on/off cycle-based polynomial constraints", by taking advantage of the network structure. Those constraints are linearized using convex-hull representations and implemented in an efficient "branch-and-cut" framework. We also tighten the relaxation using the optimization-based bound tightening algorithm. Our extensive numerical experiments on medium-scale PGLib instances show that, compared with the state-of-the-art formulations, our strengthening techniques are able to improve the quality of ACOTS relaxations on many of the PGLib instances, with some being substantial improvements.

Kidney paired donation programs (KPDPs) match patients with willing but incompatible donors to compatible donors with an assurance that when they donate, their intended recipient receives a kidney in return from a different donor. A patient and donor join a KPDP as a pair, represented as a vertex in a compatibility graph, where arcs represent compatible kidneys flowing from a donor in one pair to a patient in another. A challenge faced in real-world KPDPs is the possibility of a planned match being cancelled, e.g., due to late detection of organ incompatibility or patient-donor dropout. We therefore develop a two-stage robust optimization approach to the kidney exchange problem wherein (1) the first stage determines a kidney matching solution according to the original compatibility graph, and then (2) the second stage repairs the solution after observing transplant cancellations. In addition to considering homogeneous failure, we present the first approach that considers non-homogeneous failure between vertices and arcs. To this end, we develop solution algorithms with a feasibility-seeking master problem and evaluate two types of recourse policies. Our framework outperforms the state-of-the-art kidney exchange algorithm under homogeneous failure on publicly available instances. Moreover, we provide insights on the scalability of our solution algorithms under non-homogeneous failure for two recourse policies and analyze their impact on highly-sensitized patients, patients for whom few kidney donors are available and whose associated exchanges tend to fail at a higher rate than non-sensitized patients., Comment: Preprint

Two-stage stochastic programs with binary recourse are challenging to solve and efficient solution methods for such problems have been limited. In this work, we generalize an existing binary decision diagram-based (BDD-based) approach of Lozano and Smith (Math. Program., 2018) to solve a special class of two-stage stochastic programs with binary recourse. In this setting, the first-stage decisions impact the second-stage constraints. Our modified problem extends the second-stage problem to a more general setting where logical expressions of the first-stage solutions enforce constraints in the second stage. We also propose a complementary problem and solution method which can be used for many of the same applications. In the complementary problem we have second-stage costs impacted by expressions of the first-stage decisions. In both settings, we convexify the second-stage problems using BDDs and parametrize either the arc costs or capacities of these BDDs with first-stage solutions depending on the problem. We further extend this work by incorporating conditional value-at-risk and we propose, to our knowledge, the first decomposition method for two-stage stochastic programs with binary recourse and a risk measure. We apply these methods to a novel stochastic dominating set problem and present numerical results to demonstrate the effectiveness of the proposed methods.

We introduce an aggregation framework to address multi-stage stochastic programs with mixed-integer state variables and continuous local variables (MSILPs). Our aggregation framework imposes additional structure to the integer state variables by leveraging the information of the underlying stochastic process, which is modeled as a Markov chain (MC). We demonstrate that the aggregated MSILP can be solved exactly via a branch-and-cut algorithm integrated with a variant of stochastic dual dynamic programming. To improve tractability, we propose to use this approach to obtain dual bounds. Moreover, we apply two-stage linear decision rule (2SLDR) approximations, in particular a new MC-based variant that we propose, to obtain high-quality decision policies with significantly reduced computational effort. We test the proposed methodologies in an MSILP model for hurricane disaster relief logistics planning. Our empirical evaluation compares the effectiveness of the various proposed approaches and analyzes the trade-offs between policy flexibility, solution quality, and computational effort. Specifically, the 2SLDR approximation yields provable high-quality solutions for our test instances supported by the proposed bounding procedure. We also extract valuable managerial insights from the solution behaviors exhibited by the underlying decision policies.

Stochastic Programming is a powerful modeling framework for decision-making under uncertainty. In this work, we tackle two-stage stochastic programs (2SPs), the most widely used class of stochastic programming models. Solving 2SPs exactly requires optimizing over an expected value function that is computationally intractable. Having a mixed-integer linear program (MIP) or a nonlinear program (NLP) in the second stage further aggravates the intractability, even when specialized algorithms that exploit problem structure are employed. Finding high-quality (first-stage) solutions -- without leveraging problem structure -- can be crucial in such settings. We develop Neur2SP, a new method that approximates the expected value function via a neural network to obtain a surrogate model that can be solved more efficiently than the traditional extensive formulation approach. Neur2SP makes no assumptions about the problem structure, in particular about the second-stage problem, and can be implemented using an off-the-shelf MIP solver. Our extensive computational experiments on four benchmark 2SP problem classes with different structures (containing MIP and NLP second-stage problems) demonstrate the efficiency (time) and efficacy (solution quality) of Neur2SP. In under 1.66 seconds, Neur2SP finds high-quality solutions across all problems even as the number of scenarios increases, an ideal property that is difficult to have for traditional 2SP solution techniques. Namely, the most generic baseline method typically requires minutes to hours to find solutions of comparable quality., Comment: To appear in the proceedings of NeurIPS 2022

Sector duration optimization (SDO) is a problem arising in treatment planning for stereotactic radiosurgery on Gamma Knife. Given a set of isocenter locations, SDO aims to select collimator size configurations and irradiation times thereof such that target tissues receive prescribed doses in a reasonable amount of treatment time, while healthy tissues nearby are spared. We present a multiobjective linear programming model for SDO to generate a diverse collection of solutions so that clinicians can select the most appropriate treatment. We develop a generic two-phase solution strategy based on the epsilon-constraint method for solving multiobjective optimization models, which aims to systematically increase the number of high-quality solutions obtained, instead of conducting a traditional uniform search. To improve solution quality further and to accelerate the procedure, we incorporate some general and problem-specific enhancements. Moreover, we propose an alternative version of our two-phase strategy, which makes use of machine learning tools to reduce the computational effort. In our computational study on eight previously treated real test cases, a significant portion of obtained solutions outperformed clinical results and those from a single-objective model from the literature. In addition to significant benefits of the algorithmic enhancements, our experiments illustrate the usefulness of machine learning strategies to reduce the overall run times nearly by half while maintaining or besting the clinical practice.

The City of Toronto Long Term Care Homes & Services (LTCH&S) division is one of the largest providers of long-term care in the Canadian province of Ontario, providing care to 2,640 residents at 10 homes across Toronto. Our collaboration with LTCH&S was initiated to facilitate the increasingly challenging task of scheduling nursing staff and reduce high absenteeism rate observed among the part-time nurses. We developed a spreadsheet-based scheduling tool to automate the generation of schedules and incorporate nurses' preferences for different shifts into the schedules. At the core of the scheduling tool is a hierarchical optimization model that generates a feasible schedule with the highest total preference score while satisfying the maximum possible demand. Feasible schedules had to abide by a set of complex seniority requirements which prioritized more senior nurses when allocating the available shifts. Our scheduling tool was implemented in a 391-bed home in Toronto. The tool allowed nursing managers to generate feasible schedules within a fraction of an hour, in contrast to the status-quo manual approach which could took up to tens of hours. In addition, the schedules successfully accounted for preferences with on average above 94% of the allocated shifts ranked as most preferred.

Optimization problems with discrete decisions are nonconvex and thus lack strong duality, which limits the usefulness of tools such as shadow prices and the KKT conditions. It was shown in Burer(2009) that mixed-binary quadratic programs can be written as completely positive programs, which are convex. Completely positive reformulations of discrete optimization problems therefore have strong duality if a constraint qualification is satisfied. We apply this perspective in two ways. First, we write unit commitment in power systems as a completely positive program, and use the dual copositive program to design a new pricing mechanism. Second, we reformulate integer programming games in terms of completely positive programming, and use the KKT conditions to solve for pure strategy Nash equilibria. To facilitate implementation, we also design a cutting plane algorithm for solving copositive programs exactly.

Digital Annealer (DA) is a computer architecture designed for tackling combinatorial optimization problems formulated as quadratic unconstrained binary optimization (QUBO) models. In this paper, we present the results of an extensive computational study to evaluate the performance of DA in a systematic way in comparison to multiple state-of-the-art solvers for different problem classes. We examine pure QUBO models, as well as QUBO reformulations of three constrained problems, namely quadratic assignment, quadratic cycle partition, and selective graph coloring, with the last two being new applications for DA. For the selective graph coloring problem, we also present a size reduction heuristic that significantly increases the number of eligible instances for DA. Our experimental results show that despite being in its development stage, DA can provide high-quality solutions quickly and in that regard rivals the state of the art, particularly for large instances. Moreover, as opposed to established solvers, within its limit on the number of decision variables, DA's solution times are not affected by the increase in instance size. These findings illustrate that DA has the potential to become a successful technology in tackling combinatorial optimization problems.

In a telecommunication network, Routing and Wavelength Assignment (RWA) is the problem of finding lightpaths for incoming connection requests. When facing a dynamic traffic, greedy assignment of lightpaths to incoming requests based on predefined deterministic policies leads to a fragmented network that cannot make use of its full capacity due to stranded bandwidth. At this point service providers try to recover the capacity via a defragmentation process. We study this setting from two perspectives: (i) while granting the connection requests via the RWA problem and (ii) during the defragmentation process by lightpath rerouting. For both problems, we present the first two-stage stochastic integer programming model incorporating incoming request uncertainty to maximize the expected grade of service. We develop a decomposition-based solution approach, which uses various relaxations of the problem and a newly developed problem-specific cut family. Simulation of two-stage policies for a variety of instances in a rolling-horizon framework of 52 stages shows that our stochastic models provide high-quality solutions compared to traditionally used deterministic ones. Specifically, the proposed provisioning policies yield improvements of up to 19% in overall grade of service and 20% in spectrum saving, while the stochastic lightpath rerouting policies grant up to 36% more requests using up to just 4% more bandwidth spectrum.

Kidney paired donation programs allow patients registered with an incompatible donor to receive a suitable kidney from another donor, as long as the latter's co-registered patient, if any, also receives a kidney from a different donor. The kidney exchange problem (KEP) aims to find an optimal collection of kidney exchanges taking the form of cycles and chains. Existing exact solution methods for KEP either are designed for the case where only cyclic exchanges are considered, or can handle long chains but are scalable as long as cycles are short. We develop the first decomposition method that is able to deal with long cycles and long chains for large realistic instances. More specifically, we propose a branch-and-price framework, in which the pricing problems are solved (for the first time in packing problems in a digraph) through multi-valued decision diagrams. Also, we present a new upper bound on the optimal value of KEP, stronger than the one proposed in the literature, which is obtained via our master problem. Computational experiments show superior performance of our method over the state of the art by optimally solving almost all instances in the PrefLib library for multiple cycle and chain lengths.

Multistage stochastic programming provides a modeling framework for sequential decision-making problems that involve uncertainty. One typically overlooked aspect of this methodology is how uncertainty is incorporated into modeling. Traditionally, statistical forecasting techniques with simple forms, e.g., (first-order) autoregressive time-series models, are used to extract scenarios to be added to optimization models to represent the uncertain future. However, often times, the performance of these forecasting models are not thoroughly assessed. Motivated by the advances in probabilistic forecasting, we incorporate a deep learning-based time-series forecasting method into multistage stochastic programming framework, and compare it with the cases where a traditional forecasting method is employed to model the uncertainty. We assess the impact of more accurate forecasts on the quality of two commonly used look-ahead policies, a deterministic one and a two-stage one, in a rolling\red{-}horizon framework on a practical problem. Our results illustrate that more accurate forecasts contribute substantially to the model performance, and enable obtaining high-quality solutions even from computationally cheap heuristics. They also show that the probabilistic forecasting capabilities of deep learning-based methods can be especially beneficial when used as a (conditional) sampling tool for scenario-based models, and to predict the worst-case scenario for risk-averse models.

The routing and wavelength assignment with protection is an important problem in telecommunications. Given an optical network and incoming connection requests, a commonly studied variant of the problem aims to grant maximum number of requests by assigning lightpaths at minimum network resource usage level, while ensuring the provided services remain functional in case of a single-link failure through dedicated path protection. We consider a practically relevant version where alternative lightpaths for requests are assumed to be given as a precomputed set, and show that it is NP-hard. We formulate the problem as an integer programming (IP) model, and also use it as a foundation to develop a novel quadratic unconstrained binary optimization (QUBO) model, which can be both directly solved by a state-of-the-art solver like GUROBI. We present necessary and sufficient conditions on objective function parameters to prioritize request granting objective over wavelength-link usage for both models, and a sufficient condition to ensure the exactness of the QUBO model. Moreover, we implement a problem-specific branch-and-cut algorithm for the IP model, and employ a new quantum-inspired technology, Digital Annealer (DA), for the QUBO model. We conduct computational experiments on a large suite of instances that are hard to optimally solve in order to assess the efficiency and efficacy of all of these approaches as well as a problem-specific heuristic. The results show that the emerging technology DA outperforms the considered established techniques coupled with GUROBI, in finding mostly significantly better or as good solutions in only two minutes compared to two hours of run time, whereas the problem-specific heuristic fails to be competitive.

Inverse optimization, determining parameters of an optimization problem that render a given solution optimal, has received increasing attention in recent years. While significant inverse optimization literature exists for convex optimization problems, there have been few advances for discrete problems, despite the ubiquity of applications that fundamentally rely on discrete decision-making. In this paper, we present a new set of theoretical insights and algorithms for the general class of inverse mixed integer linear optimization problems. Specifically, a general characterization of optimality conditions is established and leveraged to design new cutting plane solution algorithms. Through an extensive set of computational experiments, we show that our methods provide substantial improvements over existing methods in solving the largest and most difficult instances to date.

Multistage stochastic programs can be approximated by restricting policies to follow decision rules. Directly applying this idea to problems with integer decisions is difficult because of the need for decision rules that lead to integral decisions. In this work, we introduce Lagrangian dual decision rules (LDDRs) for multistage stochastic mixed-integer programming (MSMIP) which overcome this difficulty by applying decision rules in a Lagrangian dual of the MSMIP. We propose two new bounding techniques based on stagewise (SW) and nonanticipative (NA) Lagrangian duals where the Lagrangian multiplier policies are restricted by LDDRs. We demonstrate how the solutions from these duals can be used to drive primal policies. Our proposal requires fewer assumptions than most existing MSMIP methods. We compare the theoretical strength of the restricted duals and show that the restricted NA dual can provide relaxation bounds at least as good as the ones obtained by the restricted SW dual. In our numerical study on two problem classes, one traditional and one novel, we observe that the proposed LDDR approaches yield significant optimality gap reductions compared to existing general-purpose bounding methods for MSMIP problems.

The Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so-called discretization assumptions allow to discretize the search space of DGP instances. In this paper, we study the Discretizable Molecular Distance Geometry Problem whose feasible solutions provide a discretization scheme for the DGP. We propose the first constraint programming formulations as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints and valid inequalities. Our computational results indicate that our formulations outperform the state-of-the-art integer programming formulations, both for feasible and infeasible instances.

The distributed operating room (OR) scheduling problem aims to find an assignment of surgeries to ORs across collaborating hospitals that share their waiting lists and ORs. We propose a stochastic extension of this problem where surgery durations are considered to be uncertain. In order to obtain solutions for the challenging stochastic model, we use sample average approximation, and develop two enhanced decomposition frameworks that use logic-based Benders (LBBD) optimality cuts and binary decision diagram based Benders cuts. Specifically, to the best of our knowledge, deriving LBBD optimality cuts in a stochastic programming context is new to the literature. Our computational experiments on a hospital dataset illustrate that the stochastic formulation generates robust schedules, and that our algorithms improve the computational efficiency.

Given an integer dimension K and a simple, undirected graph G with positive edge weights, the Distance Geometry Problem (DGP) aims to find a realization function mapping each vertex to a coordinate in K-dimensional space such that the distance between pairs of vertex coordinates is equal to the corresponding edge weights in G. The so-called discretization assumptions reduce the search space of the realization to a finite discrete one which can be explored via the branch-and-prune (BP) algorithm. Given a discretization vertex order in G, the BP algorithm constructs a binary tree where the nodes at a layer provide all possible coordinates of the vertex corresponding to that layer. The focus of this paper is finding optimal BP trees for a class of Discretizable DGPs. More specifically, we aim to find a discretization vertex order in G that yields a BP tree with the least number of branches. We propose an integer programming formulation and three constraint programming formulations that all significantly outperform the state-of-the-art cutting plane algorithm for this problem. Moreover, motivated by the difficulty in solving instances with a large and low density input graph, we develop two hybrid decomposition algorithms, strengthened by a set of valid inequalities, which further improve the solvability of the problem.

This paper provides a novel framework for solving multiobjective discrete optimization problems with an arbitrary number of objectives. Our framework formulates these problems as network models, in that enumerating the Pareto frontier amounts to solving a multicriteria shortest path problem in an auxiliary network. We design techniques for exploiting the network model in order to accelerate the identification of the Pareto frontier, most notably a number of operations to simplify the network by removing nodes and arcs while preserving the set of nondominated solutions. We show that the proposed framework yields orders-of-magnitude performance improvements over existing state-of-the-art algorithms on five problem classes containing both linear and nonlinear objective functions., Comment: 32 pages, 15 figures, 5 tables

In this paper, we present lower bounds on the rank of the split closure, the multi-branch closure and the lattice-free closure for packing sets as a function of the integrality gap. We also provide a similar lower bound on the split rank of covering polyhedra. These results indicate that whenever the integrality gap is high, these classes of cutting planes must necessarily be applied for many rounds in order to obtain the integer hull.

Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn, Wiesemann, and Georghiou (Math. Program., 130, 177-209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. A computational study on two example problems demonstrates that using a two-stage LDR can yield significantly better primal policies and modestly better dual policies than using policies based on a static LDR.

In this paper, we study the strength of Chvatal-Gomory (CG) cuts and more generally aggregation cuts for packing and covering integer programs (IPs). Aggregation cuts are obtained as follows: Given an IP formulation, we first generate a single implied inequality using aggregation of the original constraints, then obtain the integer hull of the set defined by this single inequality with variable bounds, and finally use the inequalities describing the integer hull as cutting-planes. Our first main result is to show that for packing and covering IPs, the CG and aggregation closures can be 2-approximated by simply generating the respective closures for each of the original formulation constraints, without using any aggregations. On the other hand, we use computational experiments to show that aggregation cuts can be arbitrarily stronger than cuts from individual constraints for general IPs. The proof of the above stated results for the case of covering IPs with bounds require the development of some new structural results, which may be of independent interest. Finally, we examine the strength of cuts based on k different aggregation inequalities simultaneously, the so-called multi-row cuts, and show that every packing or covering IP with a large integrality gap also has a large k-aggregation closure rank. In particular, this rank is always at least of the order of the logarithm of the integrality gap.