Relaxation-based Algorithm Research Articles

A Novel Affine Relaxation-Based Algorithm for Minimax Affine Fractional Program

This paper puts forward a novel affine relaxation-based algorithm for solving the minimax affine fractional program problem (MAFPP) over a polyhedron set. First of all, some new variables are introduced for deriving the equivalence problem (EP) of the MAFPP. Then, for the EP, the affine relaxation problem (ARP) is established by using the two-stage affine relaxation method. The method provides a lower bound by solving the ARP in the branch-and-bound searching process. By subdividing the output space rectangle and solving a series of ARPs continuously, the feasible solution sequence generated by the algorithm converges to a global optimal solution of the initial problem. In addition, the algorithmic maximum iteration in the worst case is estimated by complexity analysis for the first time. Lastly, the practicability and effectiveness of the algorithm have been verified by numerical experimental results.

Relaxation-Based Radiometric Normalization for Multitemporal Cross-Sensor Satellite Images

Multitemporal cross-sensor imagery is fundamental for the monitoring of the Earth’s surface over time. However, these data often lack visual consistency because of variations in the atmospheric and surface conditions, making it challenging to compare and analyze images. Various image-normalization methods have been proposed to address this issue, such as histogram matching and linear regression using iteratively reweighted multivariate alteration detection (IR-MAD). However, these methods have limitations in their ability to maintain important features and their requirement of reference images, which may not be available or may not adequately represent the target images. To overcome these limitations, a relaxation-based algorithm for satellite-image normalization is proposed. The algorithm iteratively adjusts the radiometric values of images by updating the normalization parameters (slope (α) and intercept (β)) until a desired level of consistency is reached. This method was tested on multitemporal cross-sensor-image datasets and showed significant improvements in radiometric consistency compared to other methods. The proposed relaxation algorithm outperformed IR-MAD and the original images in reducing radiometric inconsistencies, maintaining important features, and improving the accuracy (MAE = 2.3; RMSE = 2.8) and consistency of the surface-reflectance values (R2 = 87.56%; Euclidean distance = 2.11; spectral angle mapper = 12.60).

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

Improved approximation algorithms for solving the squared metric k-facility location problem

The squared metric k-facility location problem is a frequently encountered generalization of the k-means problem, where a specific cost should be paid for opening each facility. The current best approximation ratio for this problem is 44.473+ϵ, which was obtained using a local search algorithm. We advance the state-of-the-art for the problem by devising a Lagrangian relaxation-based algorithm that achieves an improved approximation guarantee of 36.342+ϵ. Our improvement comes from a new deterministic rounding approach, which exploits the properties of the squared metric.

A Traveler Incentive Program for Promoting Community-Based Ridesharing

Traffic congestion has become a serious issue around the globe, partly owing to single-occupancy commuter trips. Ridesharing can present a suitable alternative for serving commuter trips. However, there are several important obstacles that impede ridesharing systems from becoming a viable mode of transportation, including the lack of a guarantee for a ride back home as well as the difficulty of obtaining a critical mass of participants. This paper addresses these obstacles by introducing a traveler incentive program (TIP) to promote community-based ridesharing with a ride back home guarantee among commuters. The TIP program allocates incentives to (1) directly subsidize a select set of ridesharing rides and (2) encourage a small, carefully selected set of travelers to change their travel behavior (i.e., departure or arrival times). We formulate the underlying ride-matching problem as a budget-constrained min-cost flow problem and present a Lagrangian relaxation-based algorithm with a worst-case optimality bound to solve large-scale instances of this problem in polynomial time. We further propose a polynomial-time, budget-balanced version of the problem. Numerical experiments suggest that allocating subsidies to change travel behavior is significantly more beneficial than directly subsidizing rides. Furthermore, using a flat tax rate as low as 1% can double the system’s social welfare in the budget-balanced variant of the incentive program.

Discrete Semantics-Guided Asymmetric Hashing for Large-Scale Multimedia Retrieval

Cross-modal hashing technology is a key technology for real-time retrieval of large-scale multimedia data in real-world applications. Although the existing cross-modal hashing methods have achieved impressive accomplishment, there are still some limitations: (1) some cross-modal hashing methods do not make full consider the rich semantic information and noise information in labels, resulting in a large semantic gap, and (2) some cross-modal hashing methods adopt the relaxation-based or discrete cyclic coordinate descent algorithm to solve the discrete constraint problem, resulting in a large quantization error or time consumption. Therefore, in order to solve these limitations, in this paper, we propose a novel method, named Discrete Semantics-Guided Asymmetric Hashing (DSAH). Specifically, our proposed DSAH leverages both label information and similarity matrix to enhance the semantic information of the learned hash codes, and the ℓ2,1 norm is used to increase the sparsity of matrix to solve the problem of the inevitable noise and subjective factors in labels. Meanwhile, an asymmetric hash learning scheme is proposed to efficiently perform hash learning. In addition, a discrete optimization algorithm is proposed to fast solve the hash code directly and discretely. During the optimization process, the hash code learning and the hash function learning interact, i.e., the learned hash codes can guide the learning process of the hash function and the hash function can also guide the hash code generation simultaneously. Extensive experiments performed on two benchmark datasets highlight the superiority of DSAH over several state-of-the-art methods.

UAV-Assisted Intelligent Reflecting Surface Symbiotic Radio System

This paper investigates a symbiotic unmanned aerial vehicle (UAV)-assisted intelligent reflecting surface (IRS) radio system, where the UAV is leveraged to help the IRS reflect its own signals to the base station, and meanwhile enhance the UAV transmission by passive beamforming at the IRS. First, we consider the weighted sum bit error rate (BER) minimization problem among all IRSs by jointly optimizing the UAV trajectory, IRS phase shift matrix, and IRS scheduling, subject to the minimum primary rate requirements. To tackle this complicated problem, a relaxation-based algorithm is proposed. We prove that the converged relaxation scheduling variables are binary, which means that no reconstruct strategy is needed, and thus the UAV rate constraints are automatically satisfied. Second, we consider the fairness BER optimization problem. We find that the relaxation-based method cannot solve this fairness BER problem since the minimum primary rate requirements may not be satisfied by the binary reconstruction operation. To address this issue, we first transform the binary constraints into a series of equivalent equality constraints. Then, a penalty-based algorithm is proposed to obtain a suboptimal solution. Numerical results are provided to evaluate the performance of the proposed designs under different setups, as compared with benchmarks.

Collaborative real-time optimization strategy for train rescheduling and track emergency maintenance of high-speed railway: A Lagrangian relaxation-based decomposition algorithm

To ensure the efficient operation and low maintenance costs of high-speed railway lines, a collaborative real-time optimization strategy for train rescheduling and track emergency maintenance is proposed in this paper. This strategy not only makes online decisions on the track emergency maintenance, but simultaneously deals with the delays caused by the track emergency and disturbances. Based on a bi-directional railway line, a mixed-integer nonlinear optimization model is established, in which the decision variables mainly include the track maintenance intervention type, the end time of track maintenance, arrival/departure time, arrival/departure orders, stopping plans, and train cancellations. The proposed nonlinear model is converted to an equivalent linear model by a linearization method to reduce the computational burden. Moreover, a Lagrangian relaxation-based decomposition algorithm under a rolling horizon framework is designed to satisfy the real-time performance further. Particularly, the rolling horizon framework divides the whole time horizon into three stages according to the status of the maintenance work, i.e., the maintenance decision-making stage, the maintenance stage, and the maintenance completion stage. Furthermore, the Lagrangian relaxation-based algorithm decomposes the original large-scale optimization problem into several smaller sub-problems, which can be computed in parallel to improve the solution procedure. This designed algorithm not only reduces the computation effort for the real-time implementation, but also realizes online feedback correction and improves the robustness of the control strategy. Several numerical experiments are carried out based on the data of Beijing-Shanghai high-speed railway to demonstrate the feasibility and effectiveness of the proposed strategy.

Robust design of blood supply chains under risk of disruptions using Lagrangian relaxation

Emergency supply of blood in disasters is a crucial task for humanitarian aid. In this paper, we present a bi-objective robust optimization model for the design of blood supply chains that are resilient to disaster scenarios. The proposed two-stage stochastic optimization model aims at minimizing the time and cost of delivering blood to hospitals after the occurrence of a disaster, while considering possible disruptions in blood facilities and transportation routes. A Lagrangian relaxation-based algorithm is developed that is capable of solving large-scale instances of the model. We apply this framework to a real case study of blood banks in Jordan.

Peer-to-Peer Energy Sharing Among Smart Energy Buildings by Distributed Transaction

Efficient building energy management is essential for energy saving and green society. This paper investigates sustainable energy management for an energy building cluster with distributed transaction. The building cluster consists of several types of energy buildings, e.g., office, industrial, and commercial buildings. We first formulate utility functions for the buildings of consuming energy based on the characteristics of their controllable loads. Then a two-stage energy sharing strategy is presented. In the first stage, the total social energy cost is minimized through finding the optimal energy sharing profiles in a distributed way. In the second stage, the clearing for mutual energy sharing is modeled as a non-cooperative game, and the existence of the equilibrium of the game is illustrated and a relaxation-based algorithm is introduced to search for the equilibrium. Moreover, a real-time model for each building to overcome real-time uncertainties, such as renewable energy generation and base loads is provided. The simulation results show that the proposed energy sharing strategy is economically beneficial for the energy buildings, computationally efficient, and is promising to facilitate a sustainable regional building cluster.

An adaptive Lagrangian relaxation-based algorithm for a coordinated water supply and wastewater collection network design problem

The last century has seen an increased prevalence and duration of droughts as well as the water shortage especially in Middle East countries like Iran. This urgent situation in Iran such as Urmia Lake in the west Azerbaijan province is a motivation for us to model a new coordinated water supply and wastewater collection network design problem. Due to the uncertainty, as one of inherent sections of the water supply chain, a two-stage stochastic programming approach is used to formulate the problem. To solve the proposed model, a Lagrangian relaxation-based algorithm formulated by a new adaptive strategy is employed. This algorithm considers both upper and lower bounds of the problem to reach a performance solution. The proposed algorithm is compared with two similar algorithms from the literature to reveal its performance. As such, the efficiency of the proposed model is evaluated by some sensitivity analyses. Finally, a comprehensive discussion is provided to show the main findings and practical insights of this research.

Stochastic battery energy storage scheduling considering cell degradation and distributed energy resources

The penetration of distributed energy resources (DERs) greatly raises the risk of distribution network operation such as peak shaving and voltage stability. Battery energy storage (BES) has been widely accepted as the most potential application to cope with the challenge of high penetration of DERs. To cope with the uncertainties and variability of DERs, a stochastic BES scheduling model and its algorithm are proposed. The overall economy is achieved by fully considering the cell degradation, DERs, electricity purchasing, and active power losses. The rainflow algorithm-based cycle counting method is incorporated in the BES scheduling model to capture the cell degradation, greatly extending the expected BES lifetime and achieving a better economy. DER power scenarios are generated to consider the uncertainties and correlations based on the Copula theory. To solve the BES scheduling model, we propose a Lagrangian relaxation-based algorithm, which has a significantly reduced complexity with respect to the existing techniques. For this reason, the proposed algorithm enables much more scenarios incorporated in the BES scheduling model and better captures the DER uncertainties and correlations. Finally, numerical studies for the day-ahead BES scheduling in the IEEE 123-node test feeder are presented to demonstrate the merits of the proposed method. Results show that the actual BES life expectancy of the proposed model has increased to 4.89 times compared with the traditional ones. The problems caused by DERs are greatly alleviated by fully capturing the uncertainties and correlations with the proposed method.

Multi-Server Multi-User Multi-Task Computation Offloading for Mobile Edge Computing Networks.

This paper studies mobile edge computing (MEC) networks where multiple wireless devices (WDs) offload their computation tasks to multiple edge servers and one cloud server. Considering different real-time computation tasks at different WDs, every task is decided to be processed locally at its WD or to be offloaded to and processed at one of the edge servers or the cloud server. In this paper, we investigate low-complexity computation offloading policies to guarantee quality of service of the MEC network and to minimize WDs’ energy consumption. Specifically, both a linear programing relaxation-based (LR-based) algorithm and a distributed deep learning-based offloading (DDLO) algorithm are independently studied for MEC networks. We further propose a heterogeneous DDLO to achieve better convergence performance than DDLO. Extensive numerical results show that the DDLO algorithms guarantee better performance than the LR-based algorithm. Furthermore, the DDLO algorithm generates an offloading decision in less than 1 millisecond, which is several orders faster than the LR-based algorithm.

Evacuation Planning for Disaster Management by Using The Relaxation Based Algorithm and Route Choice Model

Research in the field of disaster management is done by utilizing information and communication technology. Where disaster management is discussed is about evacuation planning issues. The evacuation stage is a very crucial stage in the disaster evacuation process. There have been many methods and algorithms submitted for the evacuation planning process, but no one has directly addressed evacuation planning on dynamic issues concerning time-varying and volume-dependent. This research will use the Relaxation-Based Algorithm combined with the Route Choice Model to produce evacuation models that can be applied to dynamic issues related to time-varying and volume-dependent because some types of disaster will result in damage as time and evacuation paths are volume-dependent so as to adjust to the change in the number of people evacuated. Disaster data that will be used in this research is sourced from Disaster Information Management System sourced from DesInventar. The results of this study are expected to produce an evacuation planning model that can be applied to dynamic problems that take into account the time-varying and volume-dependent aspects.

A Lagrangian Relaxation-based Algorithm to Solve a Home Health Care Routing Problem

Nowadays, a rapid growth in the rate of life expectancy can be seen especially in the developed countries. Accordingly, the population of elderlies has been increased. By another point of view, the number of hospitals, retirement homes along with medical staffs has not been grown with a same rate. Hence, Home Health Care (HHC) operations including a set of nurses and patients have been developed recently by both academia and health practitioners to consider elderlies’ preferences willing to receive their cares at their homes instead of hospitals or retirement homes. To alleviate the drawbacks of pervious works and make HHC more practical, this paper introduces not only a new mathematical formulation considering new suppositions in this research area but also a solution approach based on Lagrangian relaxation theory has been employed for the first time. The main strategy of used algorithm aims to fill the gap between the lower bound and upper bound of problem and finds a solution which has both optimality and feasibility properties. By generating a number of numerical examples, results show the performance of the proposed algorithm analyzed by different criteria as well as the efficiency of developed formulation through a set of sensitivity analyses.

A constraint relaxation-based algorithm for the load-dependent vehicle routing problem with time windows

We introduce the load-dependent vehicle routing problem with time windows (LDVRPTW) in this paper. Transportation costs in this new problem, unlike those in the classical vehicle routing problem with time windows (VRPTW), are calculated based on not only the travel distances but also the vehicular loads on travel arcs. To solve this challenging NP-hard problem, we design a new constraint relaxation-based algorithm. In the proposed algorithm, a new constraint relaxation is introduced, i.e., some clients are not visited by a real vehicle and instead are entrusted to an additional virtual vehicle. Based on this relaxation, we present an effective execution scheme of local search procedures. The proposed algorithm is tested on benchmark instances of several special cases of the LDVRPTW, including the VRPTW. Numerical results for different variant problems demonstrate that the algorithm consistently yields impressive results: in particular, for one special variant, namely the fuel consumption rate considered vehicle routing problem (FCR-VRP), the algorithm improves the best-known solutions found by existing state-of-the-art methods.

A Lagrangian Relaxation-based Algorithm to solve a Collaborative Water Supply Chain Network Design Problem

A Lagrangian Relaxation-based Algorithm to solve a Collaborative Water Supply Chain Network Design Problem

Solving large-scale general phase retrieval problems via a sequence of convex relaxations.

We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include, e.g., the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method referred to as convex optimization-based phase retrieval (COPR) is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems, the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast algorithm called alternating direction method of multipliers (ADMM) that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.

Parallel Jobs Scheduling with a Specific Due Date: Asemi-definite Relaxation-based Algorithm

This paper considers a different version of the parallel machines scheduling problem in which the parallel jobs simultaneously requirea pre-specifiedjob-dependent number of machines when being processed.This relaxation departs from one of the classic scheduling assumptions. While the analytical conditions can be easily statedfor some simple models, a graph model approach is required when conflicts of processor usage are present. The main decisions and solving steps are as follows, respectively. (i) Converting the scheduling problem to graph model; (ii) Dividing jobs into independent sets: in this phase, we propose a semi-definite relaxation algorithm in which we use graph coloring concept; (iii) Sequencing the independent sets as a single-machine scheduling in which jobs in such a system arejob sets formed by using a semi-definite relaxation solution and determining the problem as a schedule that minimizes the sum of the tardiness of jobs. In this regard, after grouping the jobs by a semi-definite programming relaxation algorithm, we used the rounding algorithm for graph coloring. We also proposed a variable neighborhood search algorithm for sequencing the obtained job sets in order to minimize the sum of the tardiness. Experimental results show that this methodology is interesting by obtaining good results.

Optimal solutions for the continuous p-centre problem and related -neighbour and conditional problems: A relaxation-based algorithm

This paper aims to solve large continuous p-centre problems optimally by re-examining a recent relaxation-based algorithm. The algorithm is strengthened by adding four mathematically supported enhancements to improve its efficiency. This revised relaxation algorithm yields a massive reduction in computational time enabling for the first time larger data-sets to be solved optimally (e.g., up to 1323 nodes). The enhanced algorithm is also shown to be flexible as it can be easily adapted to optimally solve related practical location problems that are frequently faced by senior management when making strategic decisions. These include the -neighbour p-centre problem and the conditional p-centre problem. A scenario analysis using variable is also performed to provide further managerial insights.