Integer Optimization Problem Research Articles

Optimizing spatial distribution of wastewater-based disease surveillance to advance health equity

In 2022, the US Centers for Disease Control and Prevention commissioned the National Academies of Sciences, Engineering, and Medicine to assess the role of community-level wastewater-based epidemiology (WBE) beyond COVID-19. WBE is recognized as a promising mechanism for promptly identifying infectious diseases, including COVID-19 and other novel pathogens. An important conclusion from this initiative is the critical importance of maintaining equity and expanding access to fully realize the benefits of wastewater surveillance for marginalized communities. To address this need, we propose an optimization framework that strategically allocates wastewater monitoring resources at the wastewater treatment plant (WWTP) level, ensuring more effective and equitable distribution of surveillance efforts to serve underserved populations.The purpose of the framework is to obtain a balanced spatial distribution, inclusive population coverage, and efficient representation of disadvantaged groups in the allocation of resources for WBE. Furthermore, the framework concentrates on areas with high population density and gives priority to vulnerable regions, as well as identifying signals that display significant variations from other monitored sources. The optimization objective is to maximize a weighted combination of these critical factors. This problem is formulated as an integer optimization problem and solved using simulated annealing. We evaluate various scenarios, considering different weighting factors, to optimize the allocation of WWTPs with monitoring systems. This optimization framework provides an opportunity to enhance WBE by providing customized monitoring strategies created to address specific priorities and situations, thus enhancing the decision-making processes in public health responses.

Two-stage robust optimization for optimal operation of hybrid hydrogen–electricity storage system considering ladder-type carbon trading

The prominent problems of renewable energy curtailment and its uncertainty have become a hot topic. To the end, with consideration of environmental friendliness, energy utilization efficiency and operation cost, this paper proposes a hybrid hydrogen–electricity storage system (HHES) operation framework comprising assorted types of coupling devices and carbon capture system (CCS) under ladder-type carbon trading (LCT) mechanism. By considering energy balance constraints, equipments’ operation constraints and erratic rates of RESs and multiple loads, a scheduling model with source-load uncertainties is developed with the goal of minimizing the system’s operation cost and carbon emission. Then, a two-stage tri-level robust model is built to find the optimal scheduling plan under multiple uncertainties given the various sources and loads. The first level is to make an optimal dispatch plan and the second level is to find the worst scenario under the given uncertainty set. The third level aims at enhancing renewable energy accommodation rate and reducing load shedding rate in the worst case. Since the model is formed as a tri-level mixed integer optimization problem, the Column and Constraint Generation (CCG) method is adopted to achieve the optimal result. Finally, numerical analyses are performed. The results show that the proposed approach realizes a flexible dispatch of multiple energy and a high utilization level of multiple storages. Furthermore, this paper studies the influence of uncertainty and uncertainty budget parameters with the comparison of the deterministic model and robust models with different uncertainties. The results show that the proposed HHES has strong robustness and can provide a good reference for energy dispatch.

HF skywave communication for air-to-ground networks: A cooperative transmission framework

High-Frequency (HF) communication is widely used for long-distance transmission in remote and disaster areas. However, the dynamic nature of the ionosphere and multipath propagation in the HF channel pose significant challenges to designing efficient and robust communication systems. In this paper, we propose a Fixed Station (FS) and frequency matching method, as well as a power allocation method, to improve the sum-rate of air-to-ground HF communication networks. We derive optimal power allocation among users that share the same frequency, based on which a modified water-filling algorithm is used to solve the power allocation problem in multi-user scenarios, while a low-complexity algorithm is proposed to solve the integer optimization problem of frequency-FS matching. Simulation results demonstrate that the proposed algorithm outperforms the naive algorithm, indicating its effectiveness.

A novel approach to pessimistic bilevel problems. An application to the rank pricing problem with ties

This paper introduces a novel method to address the pessimistic approach to the bilevel problem. It consists of considering a lexicographic biobjective optimization problem at the lower level. To emphasize the significance of this approach, we implement it in the context of the Rank Pricing Problem with Ties. This problem can be formulated as a bilevel problem that inherently demands the use of the pessimistic approach. Considering the properties of the lexicographic biobjective problem involved, we formulate this problem as a single level mixed integer optimization problem, deriving also valid values for the big-Ms involved and valid inequalities for this formulation. The computational experiment carried out confirms the relevance of the proposed method.

Resource Allocation for UAV-RIS-Assisted NOMA-Based URLLC Systems

This work focuses on maximizing the sum rate of ultra-reliable low-latency communication (URLLC) systems by leveraging unmanned aerial vehicle-mounted reconfigurable intelligent surface (UAV-RIS) to provide short packet services for users based on the non-orthogonal multiple access (NOMA) protocol. To optimize the sum rate of system, a joint optimization is performed with respect to the power allocation, UAV position, decoding order, and RIS phase shifts. As the original problem is a non-convex integer optimization problem, it is challenging to obtain the optimal solution. Therefore, approximate solutions are derived using successive convex approximation (SCA), slack variables, and penalty-based methods. The simulation results demonstrate the superiority of the proposed resource allocation algorithm compared with the benchmark algorithm with orthogonal multiple access (OMA) scheme. In addition, this work emphasizes the performance gap between the proposed communication system and the traditional Shannon communication system in terms of throughput and the performance capacity sacrificed to achieve lower latency.

A hybrid quantum–classical algorithm for mixed-integer optimization in power systems

Mixed Integer Linear Programming (MILP) can be considered the backbone of the modern power system optimization process, with a large application spectrum, from Unit Commitment and Optimal Transmission Switching to verifying Neural Networks for power system applications. The main issue of these formulations is the computational complexity of the solution algorithms, as they are considered NP-Hard problems. Quantum computing has been tested as a potential solution towards reducing the computational burden imposed by these problems, providing promising results, motivating the can be used to speedup the solution of MILPs. In this work, we present a general framework for solving power system optimization problems with a Quantum Computer (QC), which leverages mathematical tools and QCs’ sampling ability to provide accelerated solutions. Our guiding applications are the optimal transmission switching and the verification of neural networks trained to solve a DC Optimal Power Flow. Specifically, using an accelerated version of Benders Decomposition , we split a given MILP into an Integer Master Problem and a linear Subproblem and solve it through a hybrid “quantum–classical” approach, getting the best of both worlds. We provide 2 use cases, and benchmark the developed framework against other classical and hybrid methodologies, to demonstrate the opportunities and challenges of hybrid quantum–classical algorithms for power system mixed integer optimization problems.

Using Optimization Techniques in Grammatical Evolution

The Grammatical Evolution technique has been successfully applied to a wide range of problems in various scientific fields. However, in many cases, techniques that make use of Grammatical Evolution become trapped in local minima of the objective problem and fail to reach the optimal solution. One simple method to tackle such situations is the usage of hybrid techniques, where local minimization algorithms are used in conjunction with the main algorithm. However, Grammatical Evolution is an integer optimization problem and, as a consequence, techniques should be formulated that are applicable to it as well. In the current work, a modified version of the Simulated Annealing algorithm is used as a local optimization procedure in Grammatical Evolution. This approach was tested on the Constructed Neural Networks and a remarkable improvement of the experimental results was shown, both in classification data and in data fitting cases.

Augmenting bi-objective branch and bound by scalarization-based information

While branch and bound based algorithms are a standard approach to solve single-objective (mixed-)integer optimization problems, multi-objective branch and bound methods are only rarely applied compared to the predominant objective space methods. In this paper we propose modifications to increase the performance of multi-objective branch and bound algorithms by utilizing scalarization-based information. We use the hypervolume indicator as a measure for the gap between lower and upper bound set to implement a multi-objective best-first strategy. By adaptively solving scalarizations in the root node to integer optimality we improve both, upper and lower bound set. The obtained lower bound can then be integrated into the lower bounds of all active nodes, while the determined solution is added to the upper bound set. Numerical experiments show that the number of investigated nodes can be significantly reduced by up to 83% and the total computation time can be reduced by up to 80%.

Computationally efficient solution of mixed integer model predictive control problems via machine learning aided Benders Decomposition

Mixed integer Model Predictive Control (MPC) problems arise in the operation of systems where discrete and continuous decisions must be taken simultaneously to compensate for disturbances. The efficient solution of mixed integer MPC problems requires the computationally efficient online solution of mixed integer optimization problems, which are generally difficult to solve. In this paper, we propose a machine learning-based branch and check Generalized Benders Decomposition algorithm for the efficient solution of such problems. We use machine learning to approximate the effect of the complicating variables on the subproblem by approximating the Benders cuts without solving the subproblem, therefore, alleviating the need to solve the subproblem multiple times. The proposed approach is applied to a mixed integer economic MPC case study on the operation of chemical processes. We show that the proposed algorithm always finds feasible solutions to the optimization problem, given that the mixed integer MPC problem is feasible, and leads to a significant reduction in solution time (up to 97% or 50×) while incurring small error (in the order of 1%) compared to the application of standard and accelerated Generalized Benders Decomposition.

On application of linear fractional programming problem using e-education system

This article focuses on an area of nonlinear programming problems known as linear fractional programming problems with multiple objectives. When tackling real-world linear fractional optimization problems, ambiguity and uncertainty in decision-making are inherent. This research aims to present a simple and computationally quick approach to solving multiple objective linear fractional programming problems with all decision variables and parameters described in terms of crisp. The proposed solution algorithm is based primarily on the fuzzy-based technique, and a membership function strategy. To resolve the multi-objective linear fractional programming problem, first consider the problem as a single objective function and along with the fuzzy programming model obtain the optimal solution using LINGO software. LINGO is a software application primarily used for solving linear, nonlinear, and integer optimization problems Moreover, an e-education setup problem demonstrates the steps of the proposed method.

Unit Commitment Considering Electric Vehicles and Renewable Energy Integration—A CMAES Approach

Global fossil fuel consumption and associated emissions are continuing to increase amid the 2022 energy crisis and environmental pollution and climate change issues are becoming even severer. Aiming at energy saving and emission reduction, in this paper, a new unit commitment model considering electric vehicles and renewable energy integration is established, taking into account the prediction errors of emissions from thermal units and renewable power generations. Furthermore, a new binary CMAES, dubbed BCMAES, which uses a signal function to map sampled individuals is proposed and compared with eight other mapping functions. The proposed model and the BCMAES algorithm are then applied in simulation studies on IEEE 10- and IEEE 118-bus systems, and compared with other popular algorithms such as BPSO, NSGAII, and HS. The results confirm that the proposed BCMAES algorithm outperforms other algorithms for large-scale mixed integer optimization problems with over 1000 dimensions, achieving a more than 1% cost reduction. It is further shown that the use of V2G energy transfer and the integration of renewable energy can significantly reduce both the operation costs and emissions by 5.57% and 13.71%, respectively.

Fair resource allocation and SBS selection for MC-NOMA-based HetNets supporting D2D communications

This paper studies the application of downlink multi-carrier non-orthogonal multiple access (MC-NOMA) and device-to-device (D2D) in heterogeneous networks (HetNets). In the considered model, the small base stations (SBSs) can reuse the sub-channels (SCs) of the macro base stations (MBS). The paper aims to maximize the system sum-rate while managing co-tier and co-channel interference by optimizing joint SBS selection, fair SC allocation, and power allocation, under the constraints of guaranteeing a certain amount of resources to all cellular users (CUs)and D2D pairs and satisfying CUs’ minimum signal-to-interference-plus-noise ratio (SINR) requirements. However, obtaining an efficient exact solution is challenging due to the non-convexity of the mixed integer optimization problem. To deal with this issue, the problem is decoupled into two sub-problems: 1- SBS selection and fair SC allocation problem and 2- power allocation problem. Then, a four-sided many-to-one matching method is proposed to solve the SBS selection and fair SC allocation problem jointly. The performance of the SC fair allocation algorithm is investigated in terms of the system quality of services (QoS) using a proposed mathematical expression. Furthermore, a novel dynamic search operator is introduced to enhance the performance of the matching-based algorithm. Next, the non-convex power allocation problem is transformed into a deep reinforcement learning (DRL) based framework, and the deep deterministic policy gradient (DDPG) approach is employed to solve the problem. Moreover, a joint SBS selection, SC allocation, and power allocation iterative algorithm is proposed to improve the system sum-rate. Simulation results validate our analysis and indicate the superiority of the proposed algorithms in terms of the system sum-rate compared to the benchmark algorithms.

Fault-prognosability, [formula omitted]-step prognosis and [formula omitted]-step predictive diagnosis in partially observed petri nets by means of algebraic techniques

In this paper, we explore some issues relevant to fault monitoring in discrete event systems modeled by partially observed LPNs with, possibly indistinguishable observable events and acyclic unobservable subnet. Firstly, we address the (offline) fault-prognosability analysis problem. Subsequently, we tackle the two online problems of K-step fault prognosis and K-step predictive diagnosis. We propose algebraic formulations and solutions to these problems. Namely, a necessary and sufficient condition for fault-prognosability is established based on solving an integer optimization problem. The proposed approach is applicable for bounded Petri nets. As for the K-step prognosis and K-step predictive diagnosis, algebraic approaches based on state estimation on a sliding horizon are elaborated to produce relevant verdicts. The established results for K-step prognosis and K-step predictive diagnosis are applicable for both bounded and unbounded Petri nets.

In search of maximum non-overlapping codes

AbstractNon-overlapping codes are block codes that have arisen in diverse contexts of computer science and biology. Applications typically require finding non-overlapping codes with large cardinalities, but the maximum size of non-overlapping codes has been determined only for cases where the codeword length divides the size of the alphabet, and for codes with codewords of length two or three. For all other alphabet sizes and codeword lengths no computationally feasible way to identify non-overlapping codes that attain the maximum size has been found to date. Herein we characterize maximal non-overlapping codes. We formulate the maximum non-overlapping code problem as an integer optimization problem and determine necessary conditions for optimality of a non-overlapping code. Moreover, we solve several instances of the optimization problem to show that the hitherto known constructions do not generate the optimal codes for many alphabet sizes and codeword lengths. We also evaluate the number of distinct maximum non-overlapping codes.

3D Trajectory Planning for Real-Time Image Acquisition in UAV-Assisted VR

Nowadays, unmanned aerial vehicles (UAVs), empowered with the capability of high-definition image transmission, are used to capture the rapidly changing physical environment by leveraging its high flexibility to reconstruct an immersive realistic virtual environment for metaverse users. In this paper, we consider a novel UAV-assisted image acquisition system where a UAV is dispatched to take off from an initial location to capture real-time images of multiple ground targets and then transfer the captured images back to the ground user for virtual environment reconstruction. We aim to minimize the time for the UAV to complete the image acquisition task by optimizing the three-dimensional UAV trajectory under the constraints of image quality, information causality and energy consumption. To this end, we first formulate the investigated scenario into a mixed integer optimization problem, which, however, is difficult to solve due to the infinite time-varying variables closely coupled with each other. Then, a three-stage progressive algorithm is proposed to obtain an efficient solution to the formulated mixed integer optimization problem, where the constraints of image quality, information causality and energy consumption can be sequentially satisfied. Finally, comprehensive performance evaluation is conducted to verify the effectiveness of the proposed three-stage progressive trajectory design algorithm, and the results show that the proposed algorithm significantly outperforms the benchmark schemes.

Pleiotropic genetic association analysis with multiple phenotypes using multivariate response best-subset selection

Genetic pleiotropy refers to the simultaneous association of a gene with multiple phenotypes. It is widely distributed in the whole genome and can help to understand the common genetic mechanism of diseases or traits. In this study, a multivariate response best-subset selection (MRBSS) model based pleiotropic association analysis method is proposed. Different from the traditional genetic association model, the high-dimensional genotypic data are viewed as response variables while the multiple phenotypic data as predictor variables. Moreover, the response best-subset selection procedure is converted into an 0-1 integer optimization problem by introducing a separation parameter and a tuning parameter. Furthermore, the model parameters are estimated by using the curve search under the modified Bayesian information criterion. Simulation experiments show that the proposed method MRBSS remarkably reduces the computational time, obtains higher statistical power under most of the considered scenarios, and controls the type I error rate at a low level. The application studies in the datasets of maize yield traits and pig lipid traits further verifies the effectiveness.

The complexity of geometric scaling

Geometric scaling, introduced by Schulz and Weismantel in 2002, solves the integer optimization problem max⁡{c⋅x:x∈P∩Zn} by means of primal augmentations, where P⊂Rn is a polytope. We restrict ourselves to the important case when P is a 0/1-polytope. Schulz and Weismantel showed that no more than O(nlog2⁡n‖c‖∞) calls to an augmentation oracle are required. This upper bound can be improved to O(nlog2⁡‖c‖∞) using the early-stopping policy proposed in 2018 by Le Bodic, Pavelka, Pfetsch, and Pokutta. Considering both the maximum ratio augmentation variant of the method as well as its approximate version, we show that these upper bounds are essentially tight by maximizing over a n-dimensional simplex with vectors c such that ‖c‖∞ is either n or 2n.

Multiagent Reinforcement Learning-Based Orbital Edge Offloading in SAGIN Supporting Internet of Remote Things

We investigate a computing task scheduling problem in space-air-ground integrated network (SAGIN) for Internet of remote Things (IoRT). In the considered scenario, the unmanned aerial vehicles (UAVs) collect computing tasks from IoRT devices and make offloading decisions, in which the tasks can be computed at the UAVs or offloaded to the low earth orbital (LEO) satellite. The optimization objective is to design offloading strategies for each UAV that maximum the number of tasks satisfies delay constraint and reduces the energy consumption of UAVs. We formulate this problem as a nonlinear integer optimization problem, and remodel it as a stochastic game to solve it. To copy with the dynamic and complexity environment, we propose a learning-based orbital edge offloading (LOEF) approach which can coordinate all UAVs to learn the optimal offloading strategies. This method is based on the Actor-Critic framework of multi-agent reinforcement learning, the Actor observes the environment and output the current offloading decision, the Critic evaluates the action of the Actor and coordinate the actions of all UAV to increase the reward of system. The simulation results show that the proposed offloading strategy converges fast, increases the number of satisfying the delay constraints tasks and the energy utilization of UAVs compared with the baseline strategies.

Power distribution network configuration applying the corridor method

In this paper, an optimization scheme based on the Corridor Method (CM) matheuristic is proposed for power distribution network configuration with the objective of client balancing between feeders. The problem is formulated as a mixed integer optimization problem (MIP) referring to a variant of the constrained spanning forest problem on a network. The problem is classified as NP-complete in its decision version, and the proposed matheuristic optimization approach is aimed to find high quality solutions in a short computational time on large real instances. According to the characteristics of the power distribution networks, suitable modeling solutions are adopted to retain a radial structure, satisfy power flow and voltage constraints, and reduce the reconfiguration efforts requiring a limited number of maneuvers on the electrical network. Computational experiments on different instances related to a real network have been conducted. The results demonstrated the soundness and the effectiveness of the proposed solution with respect to a baseline represented by a MIP formulation solved by a state-of-the-art commercial solver: the scalability towards large-size cases has been also considered. These results show that the proposed configuration method can effectively and efficiently solve the considered problem giving adequate support to the decision-makers.

Generating collective counterfactual explanations in score-based classification via mathematical optimization

Due to the increasing use of Machine Learning models in high stakes decision making settings, it has become increasingly important to have tools to understand how models arrive at decisions. Assuming an already trained Supervised Classification model, post-hoc explanations can be obtained via so-called counterfactual analysis: a counterfactual explanation of an instance indicates how this instance should be minimally modified so that the perturbed instance is classified in the desired class by the given Machine Learning classification model.Most of the Counterfactual Analysis literature focuses on the single-instance single-counterfactual setting, in which the analysis is done for one single instance to provide one single counterfactual explanation. Taking a stakeholder’s perspective, in this paper we introduce the so-called collective counterfactual explanations. By means of novel Mathematical Optimization models, we provide a counterfactual explanation for each instance in a group of interest, so that the total cost of the perturbations is minimized under some linking constraints. Making the process of constructing counterfactuals collective instead of individual enables us to detect the features that are critical to the entire dataset to have the individuals classified in the desired class. Our methodology allows for some instances to be treated individually, as in the single-instance single-counterfactual case, performing the collective counterfactual analysis for a fraction of records of the group of interest. This way, outliers are identified and handled appropriately.Under some assumptions on the classifier and the space in which counterfactuals are sought, finding collective counterfactual explanations is reduced to solving a convex quadratic linearly constrained mixed integer optimization problem, which, for datasets of moderate size, can be solved to optimality using existing solvers.The performance of our approach is illustrated on real-world datasets, demonstrating its usefulness.