Integer Quadratic Programming Research Articles

Optimizing a linear function over the efficient set of a multiple objective integer quadratic program

In this paper, we investigate the optimization of a function over the efficient set of a Multiple Objective Integer Quadratic Program (MOIQP). The method described employs a branch-and-cut approach to address the problem. At every node within the search tree, a profound cut is derived by considering the ascending trends of the criteria. This cut enables the removal of branches where the associated sub-problems don’t contain efficient solutions. The proposed methodology is illustrated in an example. Extensive numerical experiments are conducted to demonstrate the method’s effectiveness against state-of-the-art methods.

A two-layer strategy for sustainable energy management of microgrid clusters with embedded energy storage system and demand-side flexibility provision

The intermittent nature of renewable energy generation and the fluctuating demands pose persistent challenges in microgrid operations. In response, stakeholders and operators have turned to clustering the geographically adjacent microgrids as a solution. In this context, this paper introduces a novel two-layer energy management strategy for microgrid clusters, utilizing demand-side flexibility and the capabilities of shared battery energy storage (SBES) to minimize operational costs and emissions, while ensuring a spinning reserve within individual microgrids to prevent load-shedding. In the lower layer, the proposed approach devises optimal day-ahead operation policies, while the upper layer employs a cooperative strategy to further optimize the operational efficiency across the entire cluster. The energy management problem is accurately formulated as a mixed integer quadratic programming (MIQP) optimization, which incorporates linear terms in the problem's constraints. The formulation accounts for operational costs associated with SBES including expenses of charging/discharging and changes in operating states (CiOS). Real-world case studies with a cluster of three microgrids in Australia validate the effectiveness of this approach. Results show a reduction in operational costs for the base case scenario by 6.96 % compared to conventional microgrid management strategies. Sensitivity analyses further demonstrate the economic benefits of varying SBES capacity and flexibility pricing, with savings ranging from 6.5 % to 8.1 %. The proposed strategy also reduces CO2 emissions by up to 11.6 %, while improving system reliability. This strategy holds promise for integration into distributed energy systems with high renewable penetration and clustered local grids, offering significant advantages for utility operators and end-users through improved energy efficiency and reduced emissions.

Robust pricing and inventory decisions in ship‐from‐store omnichannel operations

AbstractThis work studies the deployment of the ship‐from‐store omnichannel strategy and the pricing and inventory decisions for an online retailer. Robust optimization models are constructed for the online‐only and the ship‐from‐store modes under a budgeted uncertainty set. The ARIMA model is used to predict the parameter values of the budgeted uncertainty set using historical demand data. The closed‐form optimal solution for the online‐only mode is obtained. The robust counterpart model for the ship‐from‐store mode is converted to a mixed integer quadratic programming model. Numerical studies are conducted to validate the theoretical results and to verify the effectiveness and practicality of the developed robust optimization solution approach. The results show that adopting a ship‐from‐store strategy may hurt the retailer's profit if a significant proportion of consumers are time‐sensitive with high travel cost. The ship‐from‐store strategy is optimal if it significantly boosts market growth.

Integer‐Sheet‐Pump Quantization for Hexahedral Meshing

AbstractSeveral state‐of‐the‐art algorithms for semi‐structured hexahedral meshing involve a so called quantization step to decide on the integer DoFs of the meshing problem, corresponding to the number of hexahedral elements to embed into certain regions of the domain. Existing reliable methods for quantization are based on solving a sequence of integer quadratic programs (IQP). Solving these in a timely and predictable manner with general‐purpose solvers is a challenge, even more so in the open‐source field. We present here an alternative robust and efficient quantization scheme that is instead based on solving a series of continuous linear programs (LP), for which solver availability and efficiency are not an issue. In our formulation, such LPs are used to determine where inflation or deflation of virtual hexahedral sheets are favorable. We compare our method to two implementations of the former IQP formulation (using a commercial and an open‐source MIP solver, respectively), finding that (a) the solutions found by our method are near‐optimal or optimal in most cases, (b) these solutions are found within a much more predictable time frame, and (c) the state of the art run time is outperformed, in the case of using the open‐source solver by orders of magnitude.

Two stage beamforming and combining scheme for FDD massive MIMO systems with multi‐antenna users

AbstractThis paper proposes a two‐stage beamforming and combining scheme in frequency division duplex (FDD) massive multiple‐input multiple‐output (MIMO) systems with multi‐antenna users. Specifically, the proposed scheme is a two‐stage scheme, where the pre‐beamforming matrix and pre‐combining matrix are designed using the channel covariance matrix (CCM) in the first stage. The problem of the pre‐beamforming matrix and pre‐combining matrix design are formulated as an 0–1 quadratic integer programming problem. To solve this problem, it is further transformed into an 0–1 mixed linear integer programming problem. In the second stage, the sparse code multiple access and multi‐user precoding and combining are adopted to mitigate the inter‐user interference. Different from the previous transmission scheme using CCM, the proposed scheme consider the multi‐antenna user system, and uses the CCM to design both the pre‐beamforming and pre‐combining matrices to sparsify the effective channel matrix, such that the overhead of pilot and feedback can be reduced. Moreover, the sparse code multiple access can help to better reduce the inter‐user interference. Simulation results validate the good performance of the proposed scheme.

A mathematical programming approach to export pathway planning of distributed hydrogen production in offshore wind farm

Hydrogen production is a vital development trend to improve the economics and penetration rate of offshore wind farms (OWFs), and distributed hydrogen production (DHP) is a preferred solution for deep and far sea OWFs without expensive submarine cables and offshore substations. The export pathway of OWF-DHP consists of collection pipelines, transmission pipelines, and offshore compressor stations. In this paper, mathematical programming is introduced to explore the export pathway planning problem of OWF-DHP. Firstly, the area of OWF is discretized into multiple grids with the center of the grids being the candidate locations of the offshore compressor station. Then, a binary integer quadratic programming problem is established to optimize both the offshore compressor station location and pipeline construction with different topologies. Further, the mathematical model is solved by the branch and cut algorithm integrated with the proposed dimension reduction methods. Finally, the effectiveness of the proposed export pathway planning approach is verified by the actual data of Baltic Eagle OWF in Germany, which would support the construction of the OWF-DHP project.

Hybrid model predictive control for vehicle trajectory tracking with integrated attitude adjustment

This paper presents a vehicle trajectory tracking method that integrates attitude adjustment. The method utilizes the asymmetric adjustment of the damping of the adjustable damping shock absorber to lift the left and right vehicle body and carry out the attitude adjustment of the vehicle. Additionally, it actively tilts toward the inside of the curve during turning. In order to address the nonlinear constraints of damping asymmetry regulation and the mixed logic relationship between suspension asymmetry damping switching and left and right body lifting during body attitude regulation, a mixed logic dynamic system (MLD) for vehicle attitude adjustment is established. Subsequently, the vehicle trajectory tracking controller is designed using the hybrid model predictive control theory (HMPC), which integrates active steering control and attitude adjustment. The optimal control problem of hybrid systems is transformed into a hybrid integer quadratic programming problem and solved through finite time domain rolling optimization. The effectiveness of the proposed integrated controller is confirmed through Simulink simulation. Compared with pure tracking control and roll tracking control with zero roll angle as the control target, this controller demonstrates its ability to effectively track the reference trajectory and enhance the vehicle’s dynamic performance.

Optimised operation of integrated community energy system considering integrated energy pricing strategy: A two-layer Stackelberg game approach

Aiming at the problems of new energy absorption and inter-subject interest conflict in the integrated community energy system, this study proposed an integrated energy pricing strategy that considered the interconversion of various energy sources and built a two-layer optimisation model of the electric-gas-heat‑hydrogen interconnection integrated community energy system. The upper-layer operator model maximises the revenue by setting energy price and selling energy to users. The lower-layer users model minimises the energy cost by actively adjusting energy usage strategy according to energy price. The two sides rationally pursue the maximisation of their respective interests, and the process of bargaining can be described by Stackelberg game. The model is also considered for combining the characteristics of elastic load and vehicle-to-grid on the demand side and operates jointly with the bus battery swapping station and two-stage power-to-gas for optimizing system operation. Through theoretical derivation, it is proved that there is a unique Stackelberg equilibrium solution for the proposed game model. Further, the two-layer optimisation model is transformed into a mixed integer quadratic programming problem by applying the Karush-Kuhn-Tucher optimal condition, linear relaxation technique, and duality theorem. Finally, the global optimal energy price is obtained by solving the converted model. The optimisation results in different scenarios show that the proposed model and method not only protect the interests of the operator and users but also improve the wind power absorption capacity of the community and reduce the load fluctuation.

Multi-View Randomized Kernel Classification via Nonconvex Optimization

Multi kernel learning (MKL) is a representative supervised multi-view learning method widely applied in multi-modal and multi-view applications. MKL aims to classify data by integrating complementary information from predefined kernels. Although existing MKL methods achieve promising performance, they fail to consider the tradeoff between diversity and classification accuracy of kernels, preventing further improvement of classification performance. In this paper, we tackle this problem by generating a number of high-quality base learning kernels and selecting a kernel subset with maximum pairwise diversity and minimum generalization errors. We first formulate this idea as a nonconvex quadratic integer programming problem. Then we transform this nonconvex problem into a convex optimization problem and prove it is equivalent to a semidefinite relaxation problem, which a semidefinite-based branch-and-bound algorithm can quickly solve. Experimental results on the real-world datasets demonstrate the superiority of the proposed method. The results also show that our method works for the support vector machine (SVM) classifier and other state-of-the-art kernel classifiers.

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.

Optimal household appliances scheduling for smart energy management considering inclining block rate tariff and net-metering system

Smart grids that integrate household renewable energy sources and share information with households can help create and maintain a smarter data-driven environment. Within this environment, flexible home energy management policies that minimize household energy costs can be adopted. This paper considers a smart home with a renewable energy source that favors satisfying its energy needs at minimum cost. This is achievable by smartly scheduling the use of its domestic appliances to match a given energy grid tariff. Focusing on the case of Egypt in which an inclining block rate (IBR) tariff is imposed, this paper fills a gap in the literature regarding the load scheduling models aiming to minimize energy cost at the household level whenever such a tariff exists. A new mixed integer quadratic programming (MIQP) model is formulated for this scheduling problem, considering the adopted net metering system with installed domestic photovoltaic (PV) systems in Egypt. The model generates the optimal household load schedule and the optimal amounts of energy to exchange with the grid while considering all the system and consumer utility constraints. To assess the applicability of the proposed model, a survey is conducted to identify the diversity and characteristics of using the electrical appliances by the Egyptian households. Based on the collected survey results, the effectiveness of the proposed MIQP model is investigated. Results confirm the effectiveness of the proposed model to minimize energy cost for different categories of the Egyptian households.

A Unified Framework Integrating Trajectory Planning and Motion Optimization Based on Spatio-Temporal Safety Corridor for Multiple AGVs

Planning safe and smooth trajectories for multiple autonomous ground vehicles (MAGVs) in a complex dynamic unstructured environment is a fundamental and challenging task. In this paper, a novel unified framework integrating trajectory planning and motion optimization (TPMO) is proposed based on spatio-temporal safety corridor (STSC), which guarantees collision avoidance and trajectory smoothness. The proposed TPMO framework consists of two parts. The first part is to establish the STSC for each AGV based on the mixed integer quadratic programming (MIQP) algorithm. The proposed STSC method ensures collision avoidance in the environment of static and dynamic obstacles, and provides a longitudinal and lateral coupled trajectory (LLCT) for trajectory planning. The second part is to design a motion optimization methodology, which considers the constraints of AGV geometry as well as longitudinal and lateral coupled motion characteristics. Moreover, our formulation provides a theoretical guarantee that the entire trajectory is optimal under collision avoidance. Finally, the proposed TPMO framework is applied to solve the optimal cooperative trajectory and motion planning problem of MAGVs in a near-natural simulation and real vehicle environments, validating the proposed framework's effectiveness and practicality.

Research on Indoor Digital Design and Experience Based on Perception and Behavioral Association

Abstract This study examines interior digital design and experience based on perceptual-behavioral associations, focusing on how users’ perceptual cognition affects the internal design experience. This study collects data on users’ physiological and psychological responses when observing interior design by combining eye movement and electroencephalography techniques. Morphological analysis was used to deconstruct and reorganize the interior design elements, while the Mixed Integer Quadratic Programming (MIQP) method was used to design the interior space layout. The experimental results show that subjects’ average perceptual imagery ratings of interior schemes in the internal design perception and behavioral experiments differed significantly, and the EEG data and eye movement data exhibited specific behavioral characteristics. For example, there were statistically significant differences in perceptual imagery ratings and physiological data across interior design schemes within a particular time window. Combining eye-movement and EEG data can capture users’ perceptual perceptions and behavioral responses.

A Hierarchical Multi-Vehicle Coordinated Motion Planning Method Based on Interactive Spatio-Temporal Corridors

Multi-vehicle coordinated motion planning has always been challenged to safely and efficiently resolve conflicts under non-holonomic dynamic constraints. Constructing spatial-temporal corridors for multi-vehicle can decouple the high-dimensional conflicts and further reduce the difficulty of obtaining feasible trajectories. Therefore, this paper proposes a novel hierarchical multi-vehicle coordinated motion planning method based on interactive spatio-temporal corridors (ISTCs). In the first layer, based on the initial guidance trajectories, Mixed Integer Quadratic Programming is designed to construct ISTCs capable of resolving conflicts in generic multi-vehicle scenarios. And then in the second layer, Non-Linear Programming is settled to generate in-corridor trajectories that satisfy the vehicle dynamics. By introducing ISTCs, the multi-vehicle coordinated motion planning problem is able to be decoupled into single-vehicle trajectory optimization problems, which greatly decentralizes the computational pressure and has great potential for real-world applications. Besides, the proposed method searches for feasible solutions in the 3-D <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(x,y,t)$</tex-math></inline-formula> configuration space, preserving more possibilities than the traditional velocity-path decoupling method. Simulated experiments in unsignalized intersection and challenging dense scenarios have been conducted to verify the feasibility and adaptability of the proposed framework.

Optimal Solution for the Integrated Model of the Problems in Capacitated Vehicle Routing and Vehicle Scheduling at the Multi-door Depot

To have an efficient Supply Chain (SC), the coordination and integration of the activities in the SC are mandatory. Routing vehicles is an optimization problem in the SC. When the routed vehicles return to the depot and the doors at the depot are busy, the returned vehicles must wait to unload the accumulated shipments. Therefore, properly scheduling these vehicles to those doors at the depot to minimize the waiting time is an optimization problem in the SC. Therefore, in this study, routing vehicles to collect the shipments from suppliers and scheduling vehicles to doors at the depot, based on a first-come-first-serve basis, are simultaneously solved. Hence, the objective of this integrated vehicle routing and scheduling problem (VR&amp;SP) is to minimize the total cost which contains the following components: vehicle travelling cost between suppliers, loading cost at the suppliers, vehicle waiting cost, unloading cost at the depot and vehicle operations cost. A Mixed Integer Quadratic Programming (MIQP) model is developed to solve the integrated VR&amp;SP. The Branch and Bound algorithm method is employed to obtain the exact optimal solution to this MIQP using LINGO optimization software. The compatibility of the developed MIQP model is verified by the randomly generated small-scale instances of VR&amp;SP. Therefore, it can be concluded that, this model solves the vehicle routing to suppliers and vehicle scheduling to doors at the depot simultaneously. Since VR&amp;SP is a NP-hard problem, heuristics or meta-heuristic methods are recommended to solve the large-scale instances of VR&amp;SP. Moreover, it can be recommended for further studies to amend this model by incorporating additional constraints to make it more applicable to real-world scenarios.

Optimal design of hydrogen-based storage with a hybrid renewable energy system considering economic and environmental uncertainties

Hydrogen and electricity derived from renewable sources present feasible alternative energy options for the decarbonisation of the transportation and power sectors. This study presents the utilisation of hydrogen generated from solar and wind energy resources as a clean fuel for mobility and backup storage for stationary applications under economic and environmental uncertainties. This is achieved by developing a detailed techno-economic model of an integrated system consisting of a hydrogen refuelling station and an electric power generation system using Mixed Integer Quadratic Constrained Programming (MIQCP), which is further relaxed to Mixed Integer Linear Programming (MILP). The model is implemented in the Advanced Interactive Multidimensional Modelling Software (AIMMS) and considering the inherent uncertainties in the wind resource, solar resource, costs and discount rate, the total cost of the three configurations (Hybrid PV-Wind, Standalone PV, and Standalone wind energy system) was minimised using robust optimisation technique, and the corresponding optimal sizes of the components, levelised cost of energy (LCOE), excess energy, greenhouse emission avoided, and carbon tax were evaluated. The levelised cost of the deterministic optimisation solution for all the configuration ranges between 0.0702 $/kWh to 0.0786 $/kWh, while the levelised cost of the robust optimisation solution ranges between 0.07188 $/kWh to 0.1125 $/kWh. The proposed integration has the advantages of affordable hydrogen and electricity prices, minimisation of carbon emissions and grid export of excess energy.

Trade-off between optimal design and operation in district cooling networks

Especially in densely populated areas, district cooling represents an opportunity to reduce energy consumption and emissions. Nevertheless, this technology is characterised by large capital costs which impede its diffusion. As a consequence, optimization tools can significantly help to unleash their potential. In this paper, a methodology is proposed to combinedly optimize the design and operation of a district cooling system based on a Mixed Integer Quadratic Programming. The model is compared to the design only optimization, based on a properly tailored heuristic approach. The models, when applied to a case study characterized by seasonal demand, provide similar solutions, which differ by 0.5 % in terms of objective value for a standard scenario. The simultaneous design and operation optimization does not provide sensible savings with respect to optimizing solely the design. A sensitivity analysis is performed to prove the robustness of the results. The results showed that the simultaneous operation and design optimization would be limited to 1 % of total costs in the case of seasonal cooling demand. On the other hand, if the cooling demand persists throughout the year, as in tropical climates, the combined optimization provides significant benefits, since these savings reach 4.7 % of total costs.

Optimization of a quadratic programming problem over an integer efficient set

Multi-objective programming problem often contains numerous efficient solutions, which confuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms.

Cooperative Beamforming for RIS-Aided Cell-Free Massive MIMO Networks

The combination of cell-free massive multiple-input multiple-output (CF-mMIMO) and reconfigurable intelligent surface (RIS) is envisioned as a promising paradigm to improve network capacity and enhance coverage capability. However, to reap full benefits of RIS-aided CF-mMIMO, the main challenge is to efficiently design cooperative beamforming (CBF) at base stations (BSs), RISs, and users. Firstly, we investigate the fractional programing to convert the weighted sum-rate (WSR) maximization problem into a tractable optimization problem. Then, the alternating optimization framework is employed to decompose the transformed problem into a sequence of subproblems, i.e., hybrid BF (HBF) at BSs, passive BF at RISs, and combining at users. In particular, the alternating direction method of multipliers algorithm is utilized to solve the HBF subproblem at BSs. Concretely, the analog BF design with unit-modulus constraints is solved by the manifold optimization (MO) while we obtain a closed-form solution to the digital BF design that is essentially a convex least-square problem. Additionally, the passive BF at RISs and the analog combining at users are designed by primal-dual subgradient and MO methods. Moreover, considering heavy communication costs in conventional CF-mMIMO systems, we propose a partially-connected CF-mMIMO (P-CF-mMIMO) framework to decrease the number of connections among BSs and users. To better compromise WSR performance and network costs, we formulate the BS selection problem in the P-CF-mMIMO system as a binary integer quadratic programming (BIQP) problem, and develop a relaxed linear approximation algorithm to handle this BIQP problem. Finally, numerical results demonstrate superiorities of our proposed algorithms over baseline counterparts.

Risk-based distributionally robust optimal dispatch for multiple cascading failures in regional integrated energy system using surrogate modeling

Regional integrated energy system (RIES) with the close coupling of multiple energies faces various cascading failure risks in its operation, and formulating the risk-based optimal dispatch scheme is important to ensure the secure and economic operation of the RIES. Considering that the occurrence probability of each subsequent failure path caused by the initial failure in the cascading failure chain varies with the change of dispatch scheme, the cooling/heating and gas pipeline dynamics and the uncertainty of renewable energy, a risk-based distributionally robust optimal dispatch model for multiple cascading failures in a RIES is proposed. A kernel ambiguity set is constructed by using the maximum mean discrepancy between the real and reference probability distributions and the kernel mean embedding technology, which can obtain the more accurate worst-case probability distribution. Moreover, a surrogate modeling method based on piecewise convex quadratic functions is proposed to learn the mapping between the cascading failure risks and the decision variables, and the original large-scale mixed integer nonlinear programming model is transformed into a small-scale mixed integer quadratic constrained programming model for efficient solution. Finally, case study on an actual RIES demonstrates the effectiveness and advantages of the proposed method. Compared to optimal dispatch without considering the cascading failure risks, the proposed method can reduce the total failure risk cost by nearly 60%.