Reformulation-linearization Technique Research Articles

Partial reformulation-linearization based optimization models for the Golomb ruler problem

In this paper, we provide a straightforward proof of a conjecture proposed in \cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in \cite{Kocuk2019}, utilizing the reformulation-linearization technique. \modif{Finally, we provide the correct outputs of the greedy heuristic proposed in \cite{Duxbury2021} and correct some false conclusions stated or implied therein.}

Omnichannel product selection and shelf space planning optimization

As the retail industry is evolving toward an omnichannel paradigm, we study a product selection and shelf space planning problem faced by omnichannel retailers. We aim to maximize aggregated profits across channels by determining the optimal product offerings for the offline and online channels while optimizing the allocation of limited shelf space to these products in brick-and-mortar stores. Furthermore, the showroom effect of physical stores on online sales is also taken into account. We propose a corresponding model that jointly optimizes these decisions. To solve the nonlinear nonconvex model of practical scale, we first reformulate it into a mixed integer quadratic programming by exploring the structural properties. Next, we use the reformulation linearization technique to further improve the computational speed. Our numerical studies validate the efficiency of the proposed solution approach. In addition, we derive some managerial insights, including that retailers should adopt joint optimization and consider how the involved effects and parameters may influence their profits and decisions.

Characterizing linearizable QAPs by the level-1 reformulation-linearization technique

The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are linearizable; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form.

Game-theoretic analysis of the impact of crew overnight hotel cost on airlines’ fleet assignment and crew pairing

Crew costs account for a high proportion of the total operational costs of an airline. The cost of overnight stay for the crew at a hotel is an important additional cost. To address this problem, this study establishes an integrated model to depict the interaction between the fleet assignment problem and the crew pairing problem. Because this cost is largely determined by the price agreed by the agreement hotel group, we further present a Stackelberg price game model to capture the interdependence between the integrated fleet assignment and crew pairing of airlines and the price game among hotel groups that provide accommodation services for airlines. We combine the idle time re-allocation with a spatio-temporal graph convolutional network to ensure the robustness of crew pairing. The solvability of the model is improved using the Reformulation-Linearization Technique. To obtain the overall Nash equilibrium of the Stackelberg price game model, we design a heuristic algorithm based on a decomposition algorithm and a proximity search algorithm. A case study is made to demonstrate that our robust model is cost-saving for the airline.

An efficient adaptive large neighborhood search algorithm based on heuristics and reformulations for the generalized quadratic assignment problem

Operations Research (OR) analytics play a crucial role in optimizing decision-making processes for real-world problems. The assignment problem, with applications in supply chains, healthcare logistics, and production scheduling, represents a prominent optimization challenge. This paper focuses on addressing the Generalized Quadratic Assignment Problem (GQAP), a well-known NP-hard combinatorial optimization problem. To tackle the GQAP, we propose an OR analytical approach that incorporates efficient relaxations, reformulations, heuristics, and a metaheuristic algorithm. Initially, we employ the Reformulation Linearization Technique (RLT) to generate various linear relaxation models, carefully selecting the most efficient ones. Building upon this foundation, we introduce a robust reformulation based on Benders Decomposition (BD), which serves as the basis for an iterative optimization algorithm applied to the GQAP. Furthermore, we develop a constructive heuristic algorithm to identify near-optimal solutions, followed by an enhancement utilizing an Adaptive Large Neighborhood Search (ALNS) metaheuristic algorithm. This ALNS algorithm is enhanced through the integration of a tabu list derived from Tabu Search (TS) and a decision rule inspired by Simulated Annealing (SA). To validate our approach and evaluate its performance, we conduct a comparative analysis against state-of-the-art algorithms documented in the literature. This comparison confirms the significant improvements achieved in terms of solution quality and computational efficiency through our proposed methodology. These advancements contribute to the state of the art in solving the GQAP and hold the potential to enhance decision-making processes in a wide range of domains. Our methodology demonstrates remarkable improvements in solution quality and computational efficiency when compared to existing approaches, as evidenced by the comparative results with state-of-the-art algorithms. The potential implications of our research extend to optimizing decision-making processes in diverse fields, rendering it highly relevant and impactful.

Learning for Spatial Branching: An Algorithm Selection Approach

The use of machine learning techniques to improve the performance of branch-and-bound optimization algorithms is a very active area in the context of mixed integer linear problems, but little has been done for nonlinear optimization. To bridge this gap, we develop a learning framework for spatial branching and show its efficacy in the context of the Reformulation-Linearization Technique for polynomial optimization problems. The proposed learning is performed offline, based on instance-specific features and with no computational overhead when solving new instances. Novel graph-based features are introduced, which turn out to play an important role for the learning. Experiments on different benchmark instances from the literature show that the learning-based branching rule significantly outperforms the standard rules. History: Accepted by Andrea Lodi, Area Editor/Design & Analysis of Algorithms – Discrete. Funding: This work was supported by Ivey Business School (David G. Burgoyne Faculty Fellowship); FEDER [MTM2014-60191-JIN]; Spanish Ministry of Education [FPU Grant 17/02643, FPU Grant 20/01555]; Conselleria de Cultura, Educacion e Universidade [ED431C 2021/24]; Natural Sciences and Engineering Research Council of Canada [Discovery Grant 2017-04185]; Spanish Ministry of Science and Technology [MTM2017-87197-C3] and Spanish Ministry of Science and Innovation [PID2021-124030NB-C32].

Multi-time scale stochastic production simulation under VHVDC long-term contract trading electricity quantity constraint

A two-layer multi-time scale stochastic production simulation framework is constructed to account for the long-term contract electricity quantity of ultra-high voltage direct current (UHVDC) transmission. On the upper layer, based on the characteristics of load demand and renewable energy output extracted from the historical operating data, monthly and daily production simulation models are carried out considering the seasonal characteristics of hydropower during a high-water period and low-water period to optimize the distribution of contract electric quantity sending through UHVDC transmission in the target year or month. According to the DC transmission electric quantity optimized by the daily production simulation in the upper layer, together with the forecast scenario, the lower layer of the framework provides the optimization of day-ahead scheduling and intra-day rolling dispatch in the implementation process. The day-ahead dispatch optimization makes full use of the adjustment capability of transmission and optimizes the DC transmission electric quantity correction. Its compensation is based on the result of the daily production simulation, then the correction will be returned to the upper layer to restart the optimization of the remaining UHVDC contract electric quantity of the subsequent period and its distribution plan. Combined with the day-ahead DC transmission plan, the intra-day rolling optimization is carried out to adjust the output of the unit using more accurate forecasting scenarios. The distributionally robust optimization model is used in the lower layer to convert an uncertain problem into a deterministic quadratically constrained quadratic programming (QCQP) problem according to the form of an uncertain distribution set. Then the QCQP problem is further converted into a linear programming (LP) problem by using the reformulation linearization technique (RLT). A test system with the energy composition and distribution referring to a real provincial power grid in northwest China is established for verification. The results show that the proposed method can effectively improve the economics of system operation and the accommodation of renewable energy based on ensuring security.

The Value of Randomized Strategies in Distributionally Robust Risk-Averse Network Interdiction Problems

Conditional value at risk (CVaR) is widely used to account for the preferences of a risk-averse agent in extreme loss scenarios. To study the effectiveness of randomization in interdiction problems with an interdictor that is both risk- and ambiguity-averse, we introduce a distributionally robust maximum flow network interdiction problem in which the interdictor randomizes over the feasible interdiction plans in order to minimize the worst case CVaR of the maximum flow with respect to both the unknown distribution of the capacity of the arcs and the interdictor’s own randomized strategy. Using the size of the uncertainty set, we control the degree of conservatism in the model and reformulate the interdictor’s distributionally robust optimization problem as a bilinear optimization problem. For solving this problem to any given optimality level, we devise a spatial branch-and-bound algorithm that uses the McCormick inequalities and reduced reformulation linearization technique to obtain a convex relaxation of the problem. We also develop a column-generation algorithm to identify the optimal support of the convex relaxation, which is then used in the coordinate descent algorithm to determine the upper bounds. The efficiency and convergence of the spatial branch-and-bound algorithm is established in numerical experiments. Further, our numerical experiments show that randomized strategies can have significantly better performance than optimal deterministic ones.History: Accepted by David Alderson, Area Editor for Network Optimization: Algorithms & Applications.Funding: The first author’s research is supported by a Group for Research in Decision Analysis postdoctoral fellowship and Fonds de Recherche du Québec–Nature et Technologies postdoctoral research scholarship [Grants 275296 and 301065]. The second author acknowledges support from the Canadian Natural Sciences and Engineering Research Council and the Canada Research Chair program [Grants RGPIN-2016-05208, 492997-2016, and 950-230057].Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.1257 . The data file and codes are posted on GitHub ( https://github.com/Utsav19/Value-of-Randomization ).

Advances in MINLP to Identify Energy-Efficient Distillation Configurations

Separation of mixtures of chemicals, ubiquitous in chemical and petrochemical industries, by distillation is energy intensive. Nearly 3% of the overall energy is used for distillation in the United States. Improving the distillation process is crucial for making chemical industries more sustainable. However, designing distillation sequences is challenging because the choice set is vast, and the equations governing the physical process are highly nonconvex. Traditional design practices rely on heuristics and often result in suboptimal solutions. Tumbalam Gooty et al. present the first approach that reliably identifies the distillation sequence that requires the least energy for a given separation. By embedding convex hulls of substructures and adapting the reformulation-linearization technique to fractions of polynomials, they demonstrated that their approach outperforms the state-of-the-art. Their work will help the chemical industry reduce greenhouse gas emissions associated with distillation.

Nested vehicle routing problem: Optimizing drone-truck surveillance operations

Unmanned aerial vehicles or drones are becoming increasingly popular due to their low cost and high mobility. In this paper we address the routing and coordination of a drone-truck pairing where the drone travels to multiple locations to perform specified observation tasks and rendezvous periodically with the truck to swap its batteries. We refer to this as the Nested-Vehicle Routing Problem (Nested-VRP) and develop a Mixed Integer Quadratically Constrained Programming (MIQCP) formulation with critical operational constraints, including drone battery capacity and synchronization of both vehicles during scheduled rendezvous. An enhancement of the MIQCP model for the Nested-VRP is achieved by deriving the equivalent Mixed Integer Linear Programming (MILP) formulation as well as leveraging lifting and Reformulation-Linearization techniques to strengthen the subtour elimination constraints of the drone. Given the NP-hard nature of the Nested-VRP, we further propose an efficient neighborhood search (NS) heuristic where we generate and improve on a good initial solution by iteratively solving the Nested-VRP on a local scale. We provide comparisons of both the exact approaches based on MIQCP or its enhanced formulations and NS heuristic methods with a relaxation lower bound in the cases of small and large problem sizes, and present the results of a computational study to show the effectiveness of the MIQCP model and its variants as well as the efficiency of the NS heuristic, including for a real-life instance with 631 locations. We envision that this framework will facilitate the planning and operations of combined drone-truck missions.

Variable Bound Tightening and Valid Constraints for Multiperiod Blending

Multiperiod blending has a number of important applications in a range of industrial sectors. It is typically formulated as a nonconvex mixed integer nonlinear program (MINLP), which involves binary variables and bilinear terms. In this study, we first propose a reformulation of the constraints involving bilinear terms using lifting. We introduce a method for calculating tight bounds on the lifted variables calculated by aggregating multiple constraints. We propose valid constraints derived from the reformulation-linearization technique (RLT) that use the bounds on the lifted variables to further tighten the formulation. Computational results indicate our method can substantially reduce the solution time and optimality gap.Summary of Contribution: In this paper, we study the multiperiod blending problem, which has a number of important applications in a range of industrial sectors, such as refining, chemical production, mining, and wastewater management. Solving this problem efficiently leads to significant economic and environmental benefits. However, solving even medium-scale instances to global optimality remains challenging. To address this challenge, we propose a variable bound tightening algorithm and tightening constraints for multiperiod blending. Computational results show that our methods can substantially reduce the solution time and optimality gap.

Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere

We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optimization. Using the reformulation-linearization technique (RLT) and Lasserre-type hierarchies, relaxations of the problem are introduced and analyzed. For our analysis we assume throughout the existence of a dual optimal solution as well as strong duality. For the GMP over the simplex we prove a convergence rate of O(1/r) for a linear programming, RLT-type hierarchy, where r is the level of the hierarchy, using a quantitative version of Pólya’s Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020. https://doi.org/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65–92, 2008. https://doi.org/10.1007/s10107-006-0085-1) over the sphere has a convergence rate of O(1/r^2). Moreover, we show the introduced linear RLT-relaxation is a generalization of a hierarchy for minimizing forms of degree d over the simplex, introduced by De Klerk et al. (J Theor Comput Sci 361(2–3):210–225, 2006).

Optimization Approach to Data-Driven Air Traffic Flow Management

An integer programming model for data-driven air traffic flow management that extends the state-of-the-art approach of Bertsimas and Patterson was developed. The major contribution of our model is that the structure of the route set is modified to improve the flexibility of rerouting operations. To improve the computational performance, we employed reformulation-linearization techniques to turn the model into a linear integer programming model, and we propose two sets of valid inequalities to speed up the computation process. The computational results indicated that the optimal strategies for flights can be efficiently obtained for large-scale instances.

The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore–Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. We also present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem.

Distributionally robust day-ahead scheduling of park-level integrated energy system considering generalized energy storages

The optimal scheduling of park-level integrated energy system can improve the efficiency of energy utilization and promote the consumption level of renewable energy. However, the uncertainty of renewable energy sources’ output power may lead to negative impacts on the scheduling of park-level integrated energy system. Therefore, a distributionally robust day-ahead scheduling model of PIES considering generalized energy storages is proposed in this paper, aiming to reduce the operating cost, renewable energy curtailment, and carbon emission of park-level integrated energy system. In the proposed model, the actual multi-energy storage devices, integrated demand response and pipeline energy storages are synergistically modeled as generalized energy storages to improve the operating flexibility of park-level integrated energy system; the Wasserstein metric-based distributionally robust optimization method is utilized to handle the uncertainty problems in the scheduling of park-level integrated energy system, which can obtain the expected operating costs of park-level integrated energy system under the worst-case probability distribution restricted in an ambiguity set; the strong duality theory and reformulation–linearization technique are utilized to linearize the proposed non-convex model and make it easier to be solved by the commercial solver. Case studies are performed on a park-level integrated energy system that consists of an IEEE 33-bus distribution network, a 44-node district heating network and a 20-node natural gas network for verifying the effectiveness and advantages of the proposed model.

Bi-level non-convex joint optimization model of energy storage in energy and primary frequency regulation markets

Large-scale energy storage as a new type of flexible market player can arbitrage in the energy market and provide primary frequency regulation (PFR) service to make profits. The operation of the battery energy storage system and the market clearing process interact with each other. Therefore, this paper proposes a bi-level optimization joint model of energy storage in energy and primary frequency regulation markets, where the upper-level maximizes the storage profit considering the battery degradation and the lower level simulates the joint market clearing process. Different from the existing research, dynamic frequency nonlinear constraints and the non-convex constraints of binary decision variables are considered in the lower level problem. The reformulation–linearization technique (RLT) is used to transform them into linear constraints. The relaxation and primary-dual reformulation of the original, and penalization of the associated duality gap are used to transform the non-convex model into a single-level mixed integer linear programming problem (MILP) which can be solved directly. The results reflect the influence of energy storage on system frequency and market clearing price. The interactive decision-making relationship between energy storage operation and clearing price is balanced.

Integer programming approach and application of reformulation-linearization technique to liver exchange problem

Organ transplants are essential for many end-stage organic disease patients. Unfortunately, because of medical or biological incompatibilities, not all donors can donate to their intended recipients. These incompatibilities can be overcome by organ exchange programs, which find new compatible donor–patient pairs by exchanging donors between patients. Organ exchange programs have become prevalent in the last decade for kidneys, and liver exchanges have also been increasing steadily. However, despite the growing number of liver exchanges, since the procedure is relatively new, there is a lack of studies attempting to optimize exchange plans through mathematical programming. This paper develops a new integer programming model for liver exchange programs that takes into account the unique characteristics of liver transplantation. In addition, a new enhanced model is obtained by applying the reformulation-linearization technique (RLT), which provides tight linear programming (LP) relaxation bounds and is computationally efficient.

Convex hull representations for bounded products of variables

It is well known that the convex hull of \(\{{(x,y,xy)}\}\), where (x, y) is constrained to lie in a box, is given by the reformulation-linearization technique (RLT) constraints. Belotti et al. (Electron Notes Discrete Math 36:805–812, 2010) and Miller et al. (SIAG/OPT Views News 22(1):1–8, 2011) showed that if there are additional upper and/or lower bounds on the product \(z=xy\), then the convex hull can be represented by adding an infinite family of inequalities, requiring a separation algorithm to implement. Nguyen et al. (Math Progr 169(2):377–415, 2018) derived convex hulls for \(\{(x,y,z)\}\) with bounds on \(z=xy^b\), \(b\ge 1\). We focus on the case where \(b=1\) and show that the convex hull with either an upper bound or lower bound on the product is given by RLT constraints, the bound on z and a single second-order cone (SOC) constraint. With both upper and lower bounds on the product, the convex hull can be represented using no more than three SOC constraints, each applicable on a subset of (x, y) values. In addition to the convex hull characterizations, volumes of the convex hulls with either an upper or lower bound on z are calculated and compared to the relaxation that imposes only the RLT constraints. As an application of these volume results, we show how spatial branching can be applied to the product variable so as to minimize the sum of the volumes for the two resulting subproblems.

Joint Offloading and Energy Harvesting Design in Multiple Time Blocks for FDMA Based Wireless Powered MEC

The combination of mobile edge computing (MEC) and wireless power transfer (WPT) is recognized as a promising technology to solve the problem of limited battery capacities and insufficient computation capabilities of mobile devices. This technology can transfer energy to users by radio frequency (RF) in wireless powered mobile edge computing. The user converts the harvested energy, stores it in the battery, and utilizes the harvested energy to execute corresponding local computing and offloading tasks. This paper adopts the Frequency Division Multiple Access (FDMA) technique to achieve task offloading from multiple mobile devices to the MEC server simultaneously. Our objective is to study multiuser dynamic joint optimization of computation and wireless resource allocation under multiple time blocks to solve the problem of maximizing residual energy. To this end, we formalize it as a nonconvex problem that jointly optimizes the number of offloaded bits, energy harvesting time, and transmission bandwidth. We adopt convex optimization technology, combine with Karush–Kuhn–Tucker (KKT) conditions, and finally transform the problem into a univariate constrained convex optimization problem. Furthermore, to solve the problem, we propose a combined method of Bisection method and sequential unconstrained minimization based on Reformulation-Linearization Technique (RLT). Numerical results demonstrate that the performance of our joint optimization method outperforms other benchmark schemes for the residual energy maximization problem. Besides, the algorithm can maximize the residual energy, reduce the computation complexity, and improve computation efficiency.

Convexification techniques for linear complementarity constraints

We develop convexification techniques for mathematical programs with complementarity constraints. Specifically, we adapt the reformulation-linearization technique of Sherali and Adams (SIAM J Discrete Math 3, 411–430, 1990) to problems with linear complementarity constraints and discuss how this procedure reduces to its standard specification for binary mixed-integer programs. Then, we consider specially structured complementarity sets that appear in KKT systems with linear constraints. For sets with a single complementarity constraint, we develop a convexification procedure that generates all nontrivial facet-defining inequalities and has an appealing “cancel-and-relax” interpretation. This procedure is used to describe the convex hull of problems with few side constraints in closed-form. As a consequence, we delineate cases when the factorable relaxation techniques yield the convex hull from those for which they do not. We then discuss how these results extend to sets with multiple complementarity constraints. In particular, we show that a suitable sequential application of the cancel-and-relax procedure produces all nontrivial inequalities of their convex hull. We conclude by illustrating, on a set of randomly generated problems, that the relaxations we propose can be significantly stronger than those available in the literature.