Mixed-integer Programming Problem Research Articles

Direct Search Techniques for Mixed Stochastic Nonlinear Programming Model

Stochastic programming is a methodology utilized for the purpose of achieving optimal planning and decision-making outcomes when faced with uncertain data. The subject of investigation pertains to a stochastic optimization problem wherein the results of stochastic data are not disclosed during runtime, and the optimization of the decision does not necessitate foresight into forthcoming outcomes. This establishes a strong correlation with the imperative need for immediate optimization in uncertain data settings, enabling effective decision-making in the present moment. The present study introduces a novel methodology for achieving global optimization of the model for nonlinear mixed-stochastic programming problem. The present study centers on stochastic problems that are two-staged and entail non-linearities in both the objective function and constraints. The first stage variables are discrete in nature, whereas the second stage variables are a combination of continuous and mixed types. Scenario-based representations are utilized for formulating problems. The fundamental approach to address the non-linear mixed-stochastic programming problem involves converting the model into a deterministic non-linear mixed-count program that is equivalent in form. The feasibility of this proposition stems from the discrete distribution assumption of uncertainty, which can be represented by a limited set of scenarios. The magnitude of the model size will increase significantly due to the quantity of scenarios and time horizons involved. The utilization of filtered probability space in conjunction with data mining techniques will be employed for the purpose of scenario generation. The methodology employed for addressing nonlinear mixed-integer programming problems of significant scale involves elevating the value of a non-basic variable beyond its boundaries in order to compel a basis variable to attain a cumulative value. Subsequently, the problem is simplified by maintaining a constant count variable and modifying it incrementally in discrete intervals to achieve an optimal solution at a global level.

Energy Criticality Avoidance-Based Delay Minimization Ant Colony Algorithm for Task Assignment in Mobile-Server-Assisted Mobile Edge Computing.

Mobile edge computing has been an important computing paradigm for providing delay-sensitive and computation-intensive services to mobile users. In this paper, we study the problem of the joint optimization of task assignment and energy management in a mobile-server-assisted edge computing network, where mobile servers can provide assisted task offloading services on behalf of the fixed servers at the network edge. The design objective is to minimize the system delay. As far as we know, our paper presents the first work that improves the quality of service of the whole system from a long-term aspect by prolonging the operational time of assisted mobile servers. We formulate the system delay minimization problem as a mixed-integer programming (MIP) problem. Due to the NP-hardness of this problem, we propose a dynamic energy criticality avoidance-based delay minimization ant colony algorithm (EACO), which strives for a balance between delay minimization for offloaded tasks and operational time maximization for mobile servers. We present a detailed algorithm design and deduce its computational complexity. We conduct extensive simulations, and the results demonstrate the high performance of the proposed algorithm compared to the benchmark algorithms.

Self-Paced Learning Based Graph Convolutional Neural Network for Mixed Integer Programming (Student Abstract)

Graph convolutional neural network (GCN) based methods have achieved noticeable performance in solving mixed integer programming problems (MIPs). However, the generalization of existing work is limited due to the problem structure. This paper proposes a self-paced learning (SPL) based GCN network (SPGCN) with curriculum learning (CL) to make the utmost of samples. SPGCN employs a GCN model to imitate the branching variable selection during the branch and bound process, while the training process is conducted in a self-paced fashion. Specifically, SPGCN contains a loss-based automatic difficulty measurer, where the training loss of the sample represents the difficulty level. In each iteration, a dynamic training dataset is constructed according to the difficulty level for GCN model training. Experiments on four NP-hard datasets verify that CL can lead to generalization improvement and convergence speedup in solving MIPs, where SPL performs better than predefined CL methods.

部材の密集度を考慮した骨組構造物の大域的トポロジー最適化

It is known that the topology optimization (TopOpt) problem of frame structures can be formulated as a mixed-integer second-order cone programming problem (MISOCP), when the compliance is minimized and the cross-section of each member is selected from a set of predetermined candidates. However, when considering the constructability of real-world building structures, it is insufficient to only minimize compliance. In this paper, in order to control the density of members, the constraints of the number of members connected to each node are additionally considered into TopOpt by MISOCP treated in the previous study.

Matrix Non-Structural Model and Its Application in Heat Exchanger Network without Stream Split

Heat integration by a heat exchanger network (HEN) is an important topic in chemical process system synthesis. From the perspective of optimization, the simultaneous synthesis of HEN belongs to a mixed-integer and nonlinear programming problem. Both the stage-wise superstructure (SWS) model and the chessboard model are the most widely adopted and belong to structural models, in which a framework is assumed for stream matching, and the global optimal solution outside its feasible domain may be defined by the framework. A node-wise non-structural model (NW-NSM) is proposed to find more universal stream matching options, but it requires a mass of structural variables and extra multiple correction strategies. The aim of this paper is to develop a novel matrix non-structural model (M-NSM) for HEN without stream splits from the perspectives of global optimization methods and superstructure models. In the proposed M-NSM, the heat exchanger position order is quantized by matrix elements at each stream, and a HEN structure is initialized by the random generation of matrix elements. An approach for solving HEN problems based on a matrix real-coded genetic algorithm is employed in this model. The results show that M-NSM provides more flexibility to expand the search region for feasible solutions with higher efficiency than previous models.

A multi-depot vehicle routing problem with time windows, split pickup and split delivery for surplus food recovery and redistribution

The problem faced by the food banks operating on the front-end model is the collection of surplus food from donation points and re-distribution of it to the demand points using a fleet of vehicles available at different volunteer locations/depots within specified time windows. Further, due to the limited vehicle capacity, demand splitting is allowed at the pickup as well and delivery points. Therefore, this problem can be seen as a rich variant of the vehicle routing problem (VRP) and named as a multi-depot VRP with time windows, split pickup and split delivery (MDVRP-TW-SP-SD). The problem is to determine the optimal number of vehicles from each depot to be hired and their optimal routes (sequence for visiting pickup and demand nodes) such that the total cost of hiring the vehicles, routing cost and penalty for unmet demand are minimized and it is formulated as a mixed-integer programming problem. To the best of our knowledge, this is the first attempt to formulate the MDVRP-TW-SP-SD. To efficiently and effectively solve the problem, an elitist GA and a GA hybridized with the local search are proposed. The performance of the proposed algorithms is tested through extensive computational experiments carried out on 180 modified benchmark problem instances from the literature. The results of computational experiments show the superiority of the GA hybridized with 3-opt local search. A case of the Robin Hood Army operating in the Lucknow city of India is also presented to demonstrate the applicability of the proposed model to practice.

Daily peak shaving operation of mixed pumped-storage hydro plants considering cascade hydraulic coupling

Growing peaking regulation pressure of the thermal-dominant power grid in China caused by increasing peak–valley differences is of concern in recent years. As the second largest power supply in China, hydropower is an important supporting resource for daily peak shaving operations. It is effective to install new pumped-storage hydro turbines (PSHT) based on the constructed cascade reservoirs to make full use of water resources, which is become a popular measure recently. This paper investigates the peak shaving of cascade hydropower with mixed pumped-storage (CHMPS) to reduce the variance of the residual load of the external grid. The hydraulic constraints, unit operation constraints, and impact of water head on unit generation are considered to formulate a precise model. The nonlinear and nonconvex problem is linearized and converted into a mixed-integer quadratic programming (MIQP) problem to enhance computational efficiency. Simulation results based on a hydropower system in China verify the effect of the proposed model on daily peak shaving.

Joint-optimized feature selection and classifier hyperparameters by salp swarm algorithm in piano score difficulty measurement problem

This study proposes the semantic-explicit features that characterize difficulty, and jointly optimizes feature selection and classifier hyperparameters by the salp swarm algorithm (SSA) to deal with the corresponding mixed-integer programming problem with constructing large-scale piano score difficulty level datasets. The difficulty level of piano scores is essential for piano learners to choose the appropriate piece, especially for beginners and amateurs. However, the previous studies lack an open-access baseline dataset and sufficient difficulty-related features, as well as the separate optimization of and feature and model hyperparameter. To address such problems, this study constructs large-scale difficulty-level datasets, proposes novel difficulty-related features, and jointly optimizes feature selection and classifier hyperparameters due to the coupled effect of feature selection and model optimization. The search space of the joint optimization is complex due to the strong mutual constraint relationship of difficulty levels in the piano-score difficulty measurement (PSDM) problem. SSA is adapted to the joint optimization scheme of the PSDM problem with the advantages of only one main controlling parameter and less computation complexity involving the gradual SSA movement approach to balance global exploration and local exploitation in an unknown and complex search space. The joint-optimization mechanism by SSA achieves an overall accuracy of 78.80% and 60.68% on two datasets of 677 and 2040 piano pieces with difficulty levels of four and nine, respectively. The results of recognition accuracy obviously validate the distinguished performance of our joint-optimization scheme compared to the successive optimization and joint optimization by other seven optimization algorithms in terms of the PSDM problem.

A Method for Calculating Aggregate Flexibility Parameters of Virtual Power Plant Based on Improved Column and Constraint Generation Algorithm

The virtual power plant (VPP) can aggregate massive, dispersed, and heterogeneous distributed energy resources (DERs) and provide an effective energy management way for an active distribution system (ADS). In this paper, three parameters of upper and lower limits of active power output, ramping, and discharging energy is used to describe the aggregate flexibility of VPP. Based on this, a two-stage robust optimization (RO) model with dispatch instructions (DI) from the superior power grid (SPG) as uncertainty variables was constructed. Considering that the range of DI from the SPG changes during the iteration of the column and constraint generation (C&CG) algorithm, a position set is proposed to record the boundary information of the uncertain set of DI issued by the SPG when the sub-problem (SP) finds the worst scenario in each iteration, and the corresponding constraints are added to the master problem (MP). For the solution of SP, the bilevel max-min model is firstly transformed into a single-level bilinear programming (BLP) model by the duality theory and then solved by the concave-convex process (CCP) iterative method. Compared with transforming the BLP problem to a mixed integer programming (MILP) problem based on Karush-Kuhn-Tucker (KKT) condition, it can not only meet the engineering accuracy but also greatly save the solution time. Finally, the effectiveness of the proposed model and algorithm was verified on a modified IEEE 33-bus system.

Improving energy aware nanosatellite task scheduling by a branch-cut-and-price algorithm

CubeSats are small satellite platforms that have become increasingly popular for space research due to their small size, modular design, and affordability. However, their limited power and communication capabilities present significant challenges for the development and operation of these satellites. One of the main challenges is to effectively manage the limited power resources available on CubeSats for supporting the various tasks and payloads of the satellite. Addressing these challenges requires careful task scheduling that considers the power requirements and importance of different tasks and the trade-offs between power consumption and other factors such as data rate, volume, and accuracy. Decision-making in task scheduling for CubeSats often involves solving large mixed-integer programming (MIP) problems, which can be computationally intensive and time-consuming. To address this challenge, we propose a branch-cut-and-price algorithm for CubeSats that improves upon previous approaches in the literature. We apply dual stabilization to column generation and propose new valid inequalities; resort to an effectively implemented dynamic programming (DP) algorithm to generate multiple columns in parallel; and rely on a new branching strategy based on pseudocosts to reduce the number of nodes and improve the overall performance of the method. Our results demonstrate the effectiveness of these enhancements for solving a complex task scheduling problem in the context of CubeSats compared to the state-of-the-art approach, and further highlight its potential to improve the efficiency and accuracy of mission planning and operations for these small satellite platforms.

Combining local branching and descent method for solving the multiple‐choice knapsack problem with setups

AbstractIn this paper, the multiple‐choice knapsack problem with setups is tackled with an iterative method, where both local branching and descent method cooperate. First, an iterative procedure is designed for solving a series of mixed integer programming problems combined with a special reduced subproblem; that is, a combined model built by injecting some valid cardinality constraints. Second, the local branching‐based learning strategy is embedded into an iterative search to mimic the variable neighborhood descent method, such that the local branching strategy drives the search process for enhancing the quality of the solutions. Third, the proposed method is experimentally analyzed on benchmark instances extracted from the literature, where its provided (lower) bounds are compared to those reached by methods published in the literature and the Cplex solver. Finally, its performance is evaluated by providing a statistical analysis.

Limiting flexibility in nonparametric efficiency evaluations: An ex post k-centroid clustering approach

An attractive feature of standard data envelopment analysis (DEA) is that decision-making units (DMUs) are put in their best possible light. This is accomplished by not imposing any restrictions on how the inputs and outputs can be weighted together. Unfortunately, this flexibility also has drawbacks. The underlying best practice production structure is typically very complex, which can make it difficult to understand efficiency evaluations. This flexibility may also motivate DMUs to pick specialized input‒output combinations simply to appear more efficient, particularly in applications where they are rewarded based on their DEA performance. In this paper, we therefore propose an approach that retains the fundamental principles of DEA while reducing flexibility in the evaluations. Our approach entails performing ex post k-centroid clustering of DMUs according to their input‒output mix patterns. The resulting mixed-integer programming problem seeks to put the centroids (as prototypes of their generated clusters) in their best possible light. This clustering enables us to work with a less complex underlying technology, which simplifies the evaluations and enhances their interpretability. Furthermore, it limits the possibility of DMUs inflating their performance by selecting extreme input‒output combinations. We demonstrate our approach with a simple numerical example and a complex data set involving Brazilian electricity distribution companies, providing clear and realistic illustrations of its application.

Predeparture Flight Planning to Minimize Operating Cost for Urban Air Mobility

Urban air mobility (UAM) is envisioned to move to highly automated and high-density operations in low-altitude urban airspace in the future. Providers of services for UAM (PSU), rather than the legacy air traffic control, are anticipated to support operators with operational planning, aircraft deconfliction, conformance monitoring, and emergency information dissemination. Such services, for hundreds to thousands of simultaneous UAM operations in constrained airspace, can only be realized with automated systems. In this study, airspace and deconfliction models for generating predeparture conflict-free four-dimensional (4-D) flight trajectories are proposed, which can be further developed into an automated flight planning tool for PSU. A semistructured scalable airspace design for future UAM is proposed (i.e., a layered airspace topology with direct routes between vertiports, avoiding physical obstacles, such as buildings, obstructions, and restricted airspace) using the visibility graph method. Based on the proposed airspace design, deconfliction strategies (e.g., flight-level assignment and departure delay) are applied to obtain predeparture conflict-free 4-D trajectories of UAM operations by solving a mixed-integer programming problem, with the objective function to minimize the operating cost of UAM operations. Furthermore, sensitivity analysis is performed to investigate the impacts of three key cost parameters (electricity price, crew hourly rate, and maintenance hourly rate). The relationships of departure delay bound (maximum departure delay allowed) vs operating cost saving and departure delay bound vs delay cost to passengers are examined, as is the tradeoff between operating cost saving and passenger delay cost.

Resource optimization for energy‐efficient NOMA‐based multi‐UAV‐enabled relaying networks

AbstractOwing to the advantages of low cost, flexibility, and the transmission characteristics of air–ground (AG) channels, using the unmanned aerial vehicle (UAV) as mobile relay to assist wireless communication has received significant interest recently. A green non‐orthogonal multiple access (NOMA)‐based multi‐UAV‐enabled relaying system with different rate requirements is proposed here. To improve energy efficiency (EE) of the UAV relays, an optimization problem for user grouping, UAV trajectory design and resource allocation under the constraints of information causality constraint, UAVs' maximum service capacity, maximum transmit power constraint, and different communication rate requirements of the mobile users (MUs) is formulated. According to the grouping results, the UAVs' trajectory and power allocation by Dinkelbach method, successive convex approximation, and condensation algorithm are alternately optimized. To solve this highly coupled non‐linear mixed integer programming problem, a graph‐based grouping algorithm is proposed to reduce the relative distance between the UAV relay and the MU. Simulation results show that the proposed optimization algorithm can effectively improve the EE performance of this NOMA‐based multi‐UAV‐enabled relaying system.

Mean-Variance-VaR portfolios: MIQP formulation and performance analysis

AbstractValue-at-risk is one of the most popular risk management tools in the financial industry. Over the past 20 years, several attempts to include VaR in the portfolio selection process have been proposed. However, using VaR as a risk measure in portfolio optimization models leads to problems that are computationally hard to solve. In view of this, few practical applications of VaR in portfolio selection have appeared in the literature up to now. In this paper, we propose to add the VaR criterion to the classical Mean-Variance approach in order to better address the typical regulatory constraints of the financial industry. We thus obtain a portfolio selection model characterized by three criteria: expected return, variance, and VaR at a specified confidence level. The resulting optimization problem consists in minimizing variance with parametric constraints on the levels of expected return and VaR. This model can be formulated as a mixed-integer quadratic programming (MIQP) problem. An extensive empirical analysis on seven real-world datasets demonstrates the practical applicability of the proposed approach. Furthermore, the out-of-sample performance of the more binding optimal Mean-Variance-VaR portfolios seems to be generally better than that of the Equally Weighted and of the Mean-Variance-CVaR portfolios.

Online Learning of Stable Integer Variables in Unit Commitment Using Internal Information

In a set of similar unit commitment (UC) problems, a variable is called a stable integer variable (SIV) if its optimal value is 0 or 1 on a regular basis, which reflects an inherent pattern in UC solutions. The computational complexity can be significantly reduced by fixing SIVs with a minor chance of accuracy loss. To identify SIVs, this letter proposes a method to collect internal solution information from the initial branch-and-bound process of artificial mixed-integer programming problems generated from the given problem. A distinct advantage of this method is that because no prior offline training is needed, learning is instance-specific and has no dependency on the training set. With identified SIVs, machine learning-based acceleration is achieved. Based on 50 test cases from publicly available and practical data, the proposed method improves the percentage of solved problems within the clearing time window from 68% to 96% compared with commercial solvers, counting the total time of data collection, prediction, and optimization.

Probabilistic wind power expansion planning of bundled wind-thermal generation system with retrofitted coal-fired plants using load transfer optimization

The bundled wind-thermal generation system (BWTGS) is an attractive option to integrate wind energy to actively participate in power systems. With the greatly enhanced penetration of wind energy in BWTGS, wind curtailment against sharp wind power fluctuations is inevitable. Thus, our study is stimulated by whether and to what extent we can maximize wind power penetration with proper provision of dispatchable loads against wind curtailment, and this is exactly the problem we aim to solve in this paper. Here, we initially focus on the flexible characteristics of retrofitted coal-fired units in BWTGS and the load transfer optimization (LTO) for high voltage distribution networks (HVDNs). Then, an appropriate probabilistic wind power capacity expansion planning method for BWTGS with retrofitted coal-fired units using LTO can be reformulated as a mixed-integer second-order cone programming problem. Moreover, since load and wind power data have inherited uncertainty, we extend this proposed model to be a probabilistic expanding model. This contributes to better decision-making for the short-term wind capacity expansion under quantified uncertainty. Numerical results of the practical HVDNs demonstrate the effectiveness of the proposed method.

Bi-level transmission expansion planning considering prohibited operating zones and multi-fuel units

Transmission expansion planning (TEP) provides a solution to economically and securely expand the transmission network. In this context, the bi-level frameworks are receiving increased attention considering a DC or (linear/convex) AC economic dispatch at the lower level. However, the prohibited operating zones (POZ) and multi-fuel units (MFU) are ignored to be modeled at the lower level, while it plays a vital role in the output results. To this end, this article presents a bi-level TEP model that considers POZ and MFU. Incorporating these two issues into the lower level’s economic dispatch problem introduces more complexity to the optimization problem but provides more realistic optimal plans. Unlike the typical approaches to solving bi-level programming (BP), the formulated problem cannot be solved by Karush–Kuhn–Tucker (KKT) conditions or strong duality theory (SDT) due to binary variables at the lower level. To this end, an iterative-based reformulation-and-decomposition (R&D) technique is utilized to find the optimal solutions to the bi-level mixed-integer programming (BMIP) problem. The presented methodology is applied to the IEEE 118-bus system. The numerical results corroborate the effectiveness of the proposed structure and highlight that modeling POZ and MFU is an essential task in the TEP problems.

Multi-area transmission maintenance scheduling for two-level markets

With the increasing renewable energy and expanding trans-regional transmission, higher requirements are placed on the outage maintenance scheduling for high-voltage power transmission networks. In this paper, a bi-level multi-area joint maintenance scheduling optimization model is established for high-voltage transmission lines taking into account the consumption of renewable energy under two-level market environment. This model is able to achieve the following in terms of the maintenance scheduling: (i) It adapts to the optimization objectives and modes of different electricity spot markets between and within areas; (ii) The goals of maintenance willingness and consumption of renewable energy are balanced; (iii) By using the Kuhn–Tucker method, the bi-level model is transformed into a single-level mixed integer programming problem with complementarity constraints for the purpose of solving. The method proposed in this paper is verified by numerical examples, which reveal how the proposed model can improve the capability of multi-area power system to accommodate renewable energy and better arrange the maintenance resources.

Multiobjective Design of Gravity-Assist Trajectories via Graph Transcription and Dynamic Programming

Multiple gravity-assist (MGA) trajectory design requires the solution of a mixed-integer programming problem to find the best sequence among all possible combinations of candidate planets and dates for spacecraft maneuvers. Current approaches require computing times rising steeply with the number of control parameters, and they strongly rely on narrow search spaces. Moreover, the challenging multiobjective optimization needs to be tackled to appropriately inform the mission design with full extent of launch opportunities. This paper describes a methodology based upon a trajectory model to transcribe the mixed-integer space into a discrete graph made by grids of interconnected nodes. The model is based on Lambert arc grids obtained for a range of departure dates and flight times between two planets. A Tisserand-based criterion selects planets to pass by. Dynamic programming is extended to multiobjective optimization of MGA trajectories and used to explore the graph, guaranteeing Pareto optimality with only moderate computational effort. Robustness is ensured by evaluating the relationship between graph and mixed-integer spaces. Missions to Jupiter and Saturn alongside challenging comet sample return transfers involving long MGA sequences are discussed. These examples illustrate the robustness and efficiency of the proposed approach in capturing globally optimal solutions and wide Pareto fronts on complex search spaces.