Small Optimality Gaps Research Articles

A WaveGAN Approach for mmWave-Based FANET Topology Optimization.

The integration of dynamic Flying Ad hoc Networks (FANETs) and millimeter Wave (mmWave) technology can offer a promising solution for numerous data-intensive applications, as it enables the establishment of a robust flying infrastructure with significant data transmission capabilities. However, to enable effective mmWave communication within this dynamic network, it is essential to precisely align the steerable antennas mounted on Unmanned Aerial Vehicles (UAVs) with their corresponding peer units. Therefore, it is important to design a novel approach that can quickly determine an optimized alignment and network topology. In this paper, we propose a Generative Adversarial Network (GAN)-based approach, called WaveGAN, for FANET topology optimization aiming to maximize the network throughput by selecting the communication paths with the best channel conditions. The proposed approach consists of a WaveGAN model followed by a beam search. The former learns how to generate optimized network topologies from a supervised dataset, while the latter adjusts the generated topologies to meet the structure requirements of the mmWave-based FANET. The simulation results show that the proposed approach is able to quickly find FANET topologies with a very small optimality gap for different network sizes.

A Tractable Linearization-Based Approximated Solution Methodology to Stochastic Multi-Period AC Security-Constrained Optimal Power Flow

The growing presence of renewable energy sources (RES), energy storage systems (ESSs) and flexible loads (FLs) in power systems necessitates a new approach to N-1 security in day-ahead operation planning considering stochasticity and time coupling. To this end, a comprehensive approach based on stochastic multi-period AC security-constrained optimal power flow (S-MP-AC-SCOPF) was proposed recently in a form of a large-scale non-linear programming problem, which is not scalable. This paper proposes the first-time new tractable solution methodology for the most complete AC-SCOPF problem to date: S-MP-AC-SCOPF. As other main novelty, the proposed methodology achieves tractability by solving sequentially a limited number of different linear approximations of the full S-MP-AC-SCOPF problem. These linear approximations differ in terms of: state dependency regarding losses approximation, carefully reduced sets of constraints or tightening of critical constraints. The performance of the methodology is demonstrated in power system models of Sweden and Portugal. Extensive numerical experiments have shown that the methodology is able to reduce progressively until complete removal the number of violated thermal or voltage constraints. It leads to reasonably accurate solution with small optimality gap within few iterations and is at least 85% faster than the competitor IPOPT solver.

Integrating train service route design with passenger flow allocation for an urban rail transit line

Train service design problem considers many operating strategies, i.e., multiple service routes, multiple train compositions, and express/local modes. Incorporating multiple operating strategies, this study first formulates an integer linear programming model for integrating train service route design with the passenger flow allocation problem. Consistency between the two components is enforced by a set of linking constraints that consider the relationship between the number of transit trips assigned to a route and the capacity of a single train. To solve the proposed model on real-life instances, we develop an approach that utilizes the Alternating Direction Method of Multipliers (ADMM). This dualizes the linking constraints and decomposes the problem into two subproblems: a train service route design subproblem and a passenger flow allocation subproblem. The latter can be subdivided into a set of passenger group-specific subproblems and is solved by a label correcting algorithm. Through Lagrangian multipliers, the interplay between the train service route design and passenger flow allocation subproblems is explored. To address the nonlinearities that arise in ADMM, we describe a new linearization technique for the quadratic penalty terms in the two subproblems by exploiting the rolling update mechanism of ADMM. The proposed approach is tested on synthetic and real-life instances from an urban rail company in China. The numerical results show that the proposed ADMM approach provides objective values that are on average 7.63% better than the conventional sequential approach. We also demonstrate that ADMM provides smaller optimality gaps in general when compared to a Lagrangian relaxation approach.

A column generation-based framework for ATFM incorporating a user-driven prioritization process

A column generation-based framework is proposed for air traffic flow management (ATFM) incorporating a user-driven prioritization process (UDPP). Airspace Users' (AUs') preferences and priorities can be explicitly reflected in the framework. AUs are allowed to propose their preferred routes as initial options and AUs' delay cost structure can be customized and not revealed to other AUs. Major UDPP features, including Fleet Delay Reordering, Selective Flight Protection, and Margins, are fully incorporated in the proposed framework. In the master problem, the model selects 4D-routes for flights subject to sector capacity constraints. In the subproblems, new 4D-routes are generated. Numerical experiments demonstrate that the proposed framework can efficiently solve the ATFM problem considering UDPP features with often zero or small optimality gaps. The effectiveness of UDPP is also verified using case studies. Parallelism of subproblems considerably further reduces the runtime and the large-scale scenario computational time can be reduced to around 30 min.

Efficient Algorithms for Stochastic Ride-Pooling Assignment with Mixed Fleets

Ride-pooling, which accommodates multiple passenger requests in a single trip, has the potential to substantially enhance the throughput of mobility-on-demand (MoD) systems. This paper investigates MoD systems that operate mixed fleets composed of “basic supply” and “augmented supply” vehicles. When the basic supply is insufficient to satisfy demand, augmented supply vehicles can be repositioned to serve rides at a higher operational cost. We formulate the joint vehicle repositioning and ride-pooling assignment problem as a two-stage stochastic integer program, where repositioning augmented supply vehicles precedes the realization of ride requests. Sequential ride-pooling assignments aim to maximize total utility or profit on a shareability graph: a hypergraph representing the matching compatibility between available vehicles and pending requests. Two approximation algorithms for midcapacity and high-capacity vehicles are proposed in this paper; the respective approximation ratios are [Formula: see text] and [Formula: see text], where p is the maximum vehicle capacity plus one. Our study evaluates the performance of these approximation algorithms using an MoD simulator, demonstrating that these algorithms can parallelize computations and achieve solutions with small optimality gaps (typically within 1%). These efficient algorithms pave the way for various multimodal and multiclass MoD applications. History: This paper has been accepted for the Transportation Science Special Issue on Emerging Topics in Transportation Science and Logistics. Funding: This work was supported by the National Science Foundation [Grants CCF-2006778 and FW-HTF-P 2222806], the Ford Motor Company, and the Division of Civil, Mechanical, and Manufacturing Innovation [Grants CMMI-1854684, CMMI-1904575, and CMMI-1940766]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2021.0349 .

The promise of EV-aware multi-period optimal power flow problem: Cost and emission benefits

Increased electric vehicle (EV) penetration brings considerable challenges to the daily planning of the power grid operations. A careful coordination of the grid operations and charging schedules is needed to alleviate these challenges, and turn them into opportunities. For this purpose, we study the Multi-Period Optimal Power Flow problem (MOPF) with electric vehicles under emission considerations. We integrate three different real-world datasets: household electricity consumption, marginal emission factors, and EV driving profiles. We present a systematic solution approach based on second-order cone programming to find globally optimal solutions for the resulting nonconvex optimization problem. To the best of our knowledge, our paper is the first to propose such a comprehensive model integrating multiple real datasets and a promising solution method for the EV-aware MOPF Problem. Our computational experiments on various instances with up to 2000 buses demonstrate that our solution approach leads to high-quality feasible solutions with provably small optimality gaps. In addition, we show the importance of coordinated EV charging to achieve significant emission savings and reductions in cost. In turn, our findings can provide quantitative insights to decision-makers on how to incentivize EV drivers depending on the trade-off between cost and emission.

Joint VNF Placement, CPU Allocation, and Flow Routing for Traffic Changes

The emerging network-softwarization technologies, such as software-defined networking and network function virtualization play important roles in 5G communication and future networks. One of the critical challenges of the practical application of the softwarized networks is to appropriately place virtual network functions (VNFs). The underlying resources and traffic requirements are often factored in the previous works of VNF placement. However, VNFs’ dynamic abilities of changing traffic are usually ignored. Resources allocated to VNFs can vary their traffic-change ratios, and the adjustment of resource volumes should be provisioned to support traffic changes. In this work, we pay attention to the joint optimization problem of VNF Placement, CPU Allocation, and flow Routing (VNFPAR) in the scenarios consisting of VNFs that can dynamically change traffic. We employ the logarithmic functions to approximate VNFs’ traffic change relations and formulate VNFPAR as a mixed-integer nonlinear programming (MINLP) problem. We demonstrate that this problem is highly nonconvex and involves highly coupled variables. For small-scale VNFPAR problems, we propose an optimal algorithm based on relaxation and programming to consume the minimum bandwidth resources. Because VNFPAR is NP-hard, to quickly find near-optimal solutions for large-scale VNFPAR problems, we present heuristic algorithms based on multistage greedy and simulated annealing, respectively. Besides, to achieve a tradeoff between solution quality and execution time, we decompose VNFPAR into subproblems and design an alternating optimization-based method. We evaluate our algorithms on real-work topologies and traffic patterns. Extensive simulations show that our proposed heuristic algorithms are convergent, stable, and effective in terms of solving VNFPARs. The proposed algorithms have small optimality gaps within 7.4%–26.3%. Meanwhile, they save 39.3%–48.4% bandwidth resources compared with relevant baseline technologies.

QoS-Aware Fog Node Placement for Intensive IoT Applications in SDN-Fog Scenarios

The advent of the Internet of Things (IoT) paradigm to intensive domains, such as industry, is a key enabler for the automation of critical, real-world processes. The strict Quality-of-Service (QoS) requirements of these domains make low-latency computing paradigms, such as fog computing, very attractive for meeting these requirements. Moreover, the requirements of scalability and flexibility in the underlying network communications motivate the use of software-defined networking (SDN) in the infrastructure. To enable these fog-SDN environments, fog nodes (FNs) that have both computing and SDN capabilities can be deployed, thus easing the deployment of fog in SDN networks. However, the exact placement of these FNs is key to the latency of the hosts that make use of them, and thus, must be carefully assessed to meet the stringent QoS requirements of critical, time-strict IoT applications. This article focuses on this FN placement problem by formalizing it and solving it through both optimal and approximated methods, including comparisons with state-of-the-art benchmarks. In particular, we analyze the performance of each of these methods in terms of latency and execution time in both SDN Internet topologies and Industrial IoT infrastructures. Our proposed heuristic provides placements with near-optimal latencies, with smaller optimality gaps than the benchmark, and computes them in tractable times.

Dynamic pricing and inventory management in the presence of online reviews

We study the joint pricing and inventory management problem in the presence of online customer reviews. Customers who purchase the product may post reviews that would influence future customers' purchasing behaviors. We develop a stochastic joint pricing and inventory management model to characterize the optimal policy in the presence of online reviews. We show that a rating‐dependent base‐stock/list‐price policy is optimal. Interestingly, we can reduce the dynamic program that characterizes the optimal policy to one with a single‐dimensional state space (the aggregate net rating). The presence of online reviews gives rise to the trade‐off between generating current profits and inducing future demands, thus having several important implications for the firm's operations decisions. First, online reviews drive the firm to deliver a better service and attract more customers to post a review. Hence, the safety‐stock and base‐stock levels are higher in the presence of online reviews. Second, the evolution of the aggregate net rating process follows a mean‐reverting pattern: When the current rating is low (respectively, high), it has an increasing (respectively, decreasing) trend in expectation. Third, although myopic profit optimization leads to significant optimality losses in the presence of online reviews, balancing current profits, and near‐future demands suffices to exploit the network effect induced by online reviews. We propose a dynamic look‐ahead heuristic policy that leverages this idea well and achieves small optimality gaps that decay exponentially in the length of the look‐ahead time window. Finally, we develop a general paid‐review strategy, which provides monetary incentives for customers to leave reviews. This strategy facilitates the retailer to (partially) separately generate current profits and induce future demands via the network effect of online reviews.

Progressive Slice Recovery With Guaranteed Slice Connectivity After Massive Failures

In presence of multiple failures affecting their network infrastructure, operators are faced with the Progressive Network Recovery (PNR) problem, i.e., deciding the best sequence of repairs during recovery. With incoming deployments of 5G networks, PNR must evolve to incorporate new recovery opportunities offered by network slicing. In this study, we introduce the new problem of Progressive Slice Recovery (PSR), which is addressed with eight different strategies, i.e., allowing or not to change slice embedding during the recovery, and/or by enforcing different versions of slice connectivity (i.e., network vs. content connectivity). We propose a comprehensive PSR scheme, which can be applied to all recovery strategies and achieves fast recovery of slices. We first prove the PSR’s NP-hardness and design an integer linear programming (ILP) model, which can obtain the best recovery sequence and is extensible for all the recovery strategies. Then, to address scalability issues of the ILP model, we devise an efficient two-phases progressive slice recovery (2-phase PSR) meta-heuristic algorithm, small optimality gap, consisting of two main steps: i) determination of recovery sequence, achieved through a linear-programming relaxation that works in polynomial time; and ii) slice-embedding recovery, for which we design an auxiliary-graph-based column generation to re-embed failed slice nodes/links to working substrate elements within a given number of actions. Numerical results compare the different strategies and validate that amount of recovered slices can be improved up to 50% if operators decide to reconfigure only few slice nodes and guarantee content connectivity.

A Greedy heuristic for solving scheduling problems with bounded rejection cost

We study single machine scheduling problems with the option of job rejection. The scheduling measures considered are: makespan, total completion time, total tardiness and number of tardy jobs. Following many real-life settings, the total permitted rejection cost is assumed to be bounded. All the problems are known to be NP-hard. The paper focuses on the introduction of an efficient greedy-type heuristic, and performing an extensive numerical study. For makespan minimization, the heuristic is proved to be asymptotically optimal under very general conditions. For each of the problems, a pseudo-polynomial dynamic programming algorithm is introduced, and consequently, the results obtained by the greedy heuristic are compared to the optimum. We show that the greedy heuristic leads to very small optimality gaps in all our tests, and that the dynamic programming algorithms can solve medium and large size instances in reasonable times.

Constrained Clustering for the Capacitated Vehicle Routing Problem (CC-CVRP)

ABSTRACT eCommerce, postal and logistics’ planners require to solve large-scale capacitated vehicle routing problems (CVRPs) on a daily basis. CVRP problems are NP-Hard and cannot be easily solved for large problem instances. Given their complexity, we propose a methodology to reduce the size of CVRP problems that can be later solved with state-of-the-art optimization solvers. Our method is an efficient version of clustering that considers the constraints of the original problem to transform it into a more tractable version. We call this approach Constrained Clustering Capacitated Vehicle Routing Solver (CC-CVRS) because it produces a soft-clustered vehicle routing problem with reduced decision variables. We demonstrate how this method reduces the computational complexity associated with the solution of the original CVRP and how the computed solution can be transformed back into the original space. Extensive numerical experiments show that our method allows to solve very large CVRP instances within seconds with optimality gaps of less than 16%. Therefore, our method has the following benefits: it can compute improved solutions with small optimality gaps in near real-time, and it can be used as a warm-up solver to compute an improved solution that can be used as an initial solution guess by an exact solver.

A Novel Tractable Methodology to Stochastic Multi-Period AC OPF in Active Distribution Systems Using Sequential Linearization Algorithm

The ongoing transition of passive distribution networks to active distribution systems (ADSs) calls for the development of sophisticated and tractable tools to address the needs and challenges of future ADSs. In this regard, this work proposes a novel tractable methodology for flexibility procurement via stochastic multi-period AC optimal power flow (S-MP-OPF). The first novelty of the proposed methodology is the formulation of a new mixed-integer linear programming (MILP) model, which develops novel linear approximations for all nonlinear constraints (i.e., active and reactive power flows and branch currents) that ensure tractability. The proposed linearization approach relies on the second-order Taylor series expansion of trigonometric terms, does not hinge on the flat-voltage, near-voltage and small angle assumptions, and formulates the linear approximations using square of voltage magnitude and voltage angle difference variables. The second novelty is a sophisticated heuristic sequential linearization algorithm (SLA) which incorporates and improves the accuracy of the MILP model in an iterative manner. The bench-marking of proposed SLA is done on a 34-bus ADS for a comprehensive set of flexible options, whereas its versatility and scalability is demonstrated on real-world 31-bus weakly-meshed and 191-bus radial ADSs. Extensive numerical analyses show that the proposed SLA leads to a highly accurate and feasible solution with small optimality gap within few iterations, under normal and stressed operating conditions, outperforms alternative methods in terms of solution accuracy and computational efficiency, and is scalable to large S-MP-OPF problems.

An MISOCP-Based Decomposition Approach for the Unit Commitment Problem with AC Power Flows

Unit Commitment (UC) and Optimal Power Flow (OPF) are two fundamental problems in short-term electric power systems planning that are traditionally solved sequentially. The state-of-the-art mostly uses a direct current (DC) approximation of the power flow equations. However, utilizing the DC approach in the UC-level may lead to infeasible or suboptimal generator commitment schedules for the OPF problem. In this paper, we aim to simultaneously solve the UC Problem with alternating current (AC) power flow equations, which combines the challenging nature of both UC and OPF Problems. Due to the highly nonconvex nature of the AC flow equations, we utilize the mixed-integer second-order cone programming (MISOCP) relaxation of the UC Problem as the basis of our solution approach. The MISOCP relaxation is utilized for finding both a lower bound and a candidate generator commitment schedule. Once this schedule is obtained, we solve a multi-period OPF problem to obtain feasible solutions for the UC problem with AC power flows. For smaller instances, we develop two different algorithms that exploit the recent advances in the OPF literature and obtain high-quality feasible solutions with provably small optimality gaps. For solving larger instances, we develop a Lagrangian decomposition based approach that yields promising results.

Improving Column Generation for Vehicle Routing Problems via Random Coloring and Parallelization

We consider a variant of the vehicle routing problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation-based linear-programming relaxation for this VRP. Specifically, we focus on algorithms for the “pricing problem,” which corresponds to the resource-constrained elementary shortest path problem. The first algorithm extends the pulse algorithm for which we derive a new bounding scheme on the maximum load of any route. The second algorithm is based on random coloring from parameterized complexity which can be also combined with other techniques in the literature for improving VRPs, including cutting planes and column enumeration. We conduct numerical studies using VRP benchmarks (with 50–957 nodes) and instances of a medical home care delivery problem using census data in Wayne County, Michigan. Using parallel computing, both pulse and random coloring can significantly improve column generation for solving the linear programming relaxations and we can obtain heuristic integer solutions with small optimality gaps. Combining random coloring with column enumeration, we can obtain improved integer solutions having less than 2% optimality gaps for most VRP benchmark instances and less than 1% optimality gaps for the medical home care delivery instances, both under a 30-minute computational time limit. The use of cutting planes (e.g., robust cuts) can further reduce optimality gaps on some hard instances, without much increase in the run time. Summary of Contribution: The vehicle routing problem (VRP) is a fundamental combinatorial problem, and its variants have been studied extensively in the literature of operations research and computer science. In this paper, we consider general-purpose algorithms for solving VRPs, including the column-generation approach for the linear programming relaxations of the integer programs of VRPs and the column-enumeration approach for seeking improved integer solutions. We revise the pulse algorithm and also propose a random-coloring algorithm that can be used for solving the elementary shortest path problem that formulates the pricing problem in the column-generation approach. We show that the parallel implementation of both algorithms can significantly improve the performance of column generation and the random coloring algorithm can improve the solution time and quality of the VRP integer solutions produced by the column-enumeration approach. We focus on algorithmic design for VRPs and conduct extensive computational tests to demonstrate the performance of various approaches.

An Efficient Simulation-Based Approach to Ambulance Fleet Allocation and Dynamic Redeployment

We present an efficient approach to ambulance fleet allocation and dynamic redeployment, where the goal is to position an entire fleet of ambulances to base locations to maximize the service level (or utility) of the Emergency Medical Services (EMS) system. We take a simulation-based approach, where the utility of an allocation is measured by directly simulating emergency requests. In both the static and dynamic settings, this modeling approach leads to an exponentially large action space (with respect to the number of ambulances). Futhermore, the utility of any particular allocation can only be measured via a seemingly “black box” simulator. Despite this complexity, we show that embedding our simulator within a simple and efficient greedy allocation algorithm produces good solutions. We derive data-driven performance guarantees which yield small optimality gap. Given its efficiency, we can repeatedly employ this approach in real-time for dynamic repositioning. We conduct simulation experiments based on real usage data of an EMS system from a large Asian city, and demonstrate significant improvement in the system’s service levels using static allocations and redeployment policies discovered by our approach.

Mixed integer programming approaches to partial disassembly line balancing and sequencing problem

Product recovery has received greater attention in recent years mainly due to increased environmental awareness of consumers and stricter environmental regulations imposed by governments. In product recovery, disassembly of the product into its constituent parts is the most significant activity and generally performed on a disassembly line. During disassembly, a complete or partial disassembly of the product may be preferred. In complete disassembly, all parts must be disassembled, while partial disassembly allows to disassemble a subset of parts (e.g., the ones with relatively high revenues). This study deals with a partial disassembly line balancing and sequencing (PDLBS) problem considering revenues of parts to be disassembled, general workstation cost, additional cost of workstation(s) with hazardous parts, and cost of direction changes. For the PDLBS problem, a generic mixed integer programming (MIP) model, with the aim of maximizing total profit, is developed. To strengthen the MIP formulation, two sets of valid inequalities are proposed. The computational results show that the MIP model with valid inequalities is able to provide optimal solutions for the PDLBS problems with up to 30 tasks. To obtain near-optimal solutions for large-sized problems, a MIP-based solution approach is proposed. The proposed approach decomposes the entire MIP model into selection and assignment (SA) and sequencing (SEQ) models. The SA model is an exact relaxation of the MIP model (with valid inequalities) obtained by removing all the sequencing variables and constraints. Hence, SA model also produces an efficient upper bound for the PDLBS problem. The SEQ model, accordingly, aims to find an optimal sequence of tasks subject to the fixed selection and assignment of tasks provided by the SA model. The computational results show that the proposed MIP-based solution approach provides efficient solutions with small optimality gaps for large-sized problems.

Incremental Space-Filling Design Based on Coverings and Spacings: Improving Upon Low Discrepancy Sequences

The paper addresses the problem of defining families of ordered sequences \(\{\mathbf {x}_i\}_{i\in \mathbb {N}}\) of elements of a compact subset \({\mathscr {X}}\) of \(\mathbb {R}^d\) whose prefixes \(\mathbf {X}_n=\{\mathbf {x}_i\}_{i=1}^{n}\), for all orders n, have good space-filling properties as measured by the dispersion (covering radius) criterion. Our ultimate aim is the definition of incremental algorithms that generate sequences \(\mathbf {X}_n\) with small optimality gap, i.e., with a small increase in the maximum distance between points of \({\mathscr {X}}\) and the elements of \(\mathbf {X}_n\) with respect to the optimal solution \(\mathbf {X}_n^{\star} \). The paper is a first step in this direction, presenting incremental design algorithms with proven optimality bound for one-parameter families of criteria based on coverings and spacings that both converge to dispersion for large values of their parameter. The examples presented show that the covering-based method outperforms state-of-the-art competitors, including coffee-house, suggesting that it inherits from its guaranteed 50% optimality gap.

Formulations and a Lagrangian relaxation approach for the prize collecting traveling salesman problem

AbstractThe prize collecting traveling salesman problem (PCTSP) is a generalization of the traveling salesman problem (TSP) in which a tour starting at a root node must visit a subset of nodes to collect a prescribed amount of total prize minimizing the summation of travel costs and penalty costs associated with nonvisited nodes. We propose new formulations and compare them with adaptations to the PCTSP of strong formulations from the literature for the asymmetric TSP. One of the formulations proposed in this paper provided the tighter linear relaxation bounds on computational experiments based on benchmark instances for the asymmetric TSP. We then develop a Lagrangian relaxation approach embedding heuristic schemes to obtain both lower and upper bounds. The proposed heuristic has shown to be successful in obtaining optimal or near‐optimal solutions with information from the Lagrangian subproblems. Our Lagrangian approach can thereupon provide small optimality gaps, being an efficient alternative to bound optimal values of large instances.

Least Squares Monte Carlo and Pathwise Optimization for Merchant Energy Production

We study merchant energy production modeled as a compound switching and timing option. The resulting Markov decision process is intractable. Least squares Monte Carlo combined with information relaxation and duality is a state-of-the-art reinforcement learning methodology to obtain operating policies and optimality gaps for related models. Pathwise optimization is a competing technique developed for optimal stopping settings, in which it typically provides superior results compared to this approach, albeit with a larger computational effort. We apply these procedures to merchant energy production. Employing pathwise optimization requires methodological extensions. We use principal component analysis and block coordinate descent in novel ways to respectively precondition and solve the ensuing ill-conditioned and large scale linear program, which even a cutting-edge commercial solver is unable to handle directly. Both techniques yield near optimal operating policies on realistic ethanol production instances. However, at the cost of both considerably longer run times and greater memory usage, pathwise optimization leads to substantially tighter dual bounds compared to least squares Monte Carlo, even when specified in a simple fashion, complementing it in this case. Thus, it plays a critical role in obtaining small optimality gaps. Our numerical observations on the magnitudes of these bound improvements differ from what is currently known. This research has potential relevance for other commodity merchant operations contexts and motivates additional algorithmic work in the area of pathwise optimization.