Lagrangian Relaxation Approach Research Articles

Lagrangian-Relaxation-Based Self-Repairing Mechanism for Wi-Fi Networks

Wi-Fi was developed to support software-defined networks that are crucial for achieving high quality of service for next generation wireless networks. In this software-defined networking technology, network resources are utilized to reduce the effects of several influential factors such as the limited number of nonoverlapping channels, co-channel interference, rapidly changing distribution of customers, and access point (AP) failure. These factors cause irregular network traffic and service unavailability in some users. In this paper, we address the problems causing the AP failure and interference. The studied problems were modeled as linear and nonlinear mathematical programming problems. The Lagrangian relaxation approach, which is a type of divide-and-conquer method, was used to solve the load distribution problems near optimally. A self-repairing heuristic multiple-level load-balancing traffic adjustment was proposed to manage the problems pertaining to the AP failure. The delay difference metric, which is defined as the difference between tolerable delay and transmission delay, was used to evaluate the difference between the upper bound and lower bound of the delay. Moreover, the metric was used to estimate the improvement ratio between the existing methods and the proposed method. Modifiable transmission power ranges (which involve the cell breathing method) and association managements are adjusted to minimize the narrow gap and improve the self-repairing performance. Thus, resources are appropriately allocated to users, and adequate QoS is attained.

Lagrangian relaxation for distribution networks with cross-docking centre

This paper proposes a cross-docking in logistics network that aims to reduce the transportation costs. The proposed strategy eliminates need of inventory for storing commodities. It consists of two main stages. During first echelon, the number and location of cross-dock warehouses are determined. In second echelon, the allocation of warehouse as cross-dock to distribution centres is determined. Each warehouse has limited capacity and distribution centre can supplied by only one cross-dock. In this paper, the problem is mathematical formulated using a mixed-integer programming model. This formulation is used the concept of cross-docking allocation and commodities distribution. The Lagrangian relaxation approach is proposed to solve logistics network problem. Experimental results reveal that the proposed approach provides optimal solution in a reasonable time.

Lagrangian relaxation for distribution networks with cross-docking centre

This paper proposes a cross-docking in logistics network that aims to reduce the transportation costs. The proposed strategy eliminates need of inventory for storing commodities. It consists of two main stages. During first echelon, the number and location of cross-dock warehouses are determined. In second echelon, the allocation of warehouse as cross-dock to distribution centres is determined. Each warehouse has limited capacity and distribution centre can supplied by only one cross-dock. In this paper, the problem is mathematical formulated using a mixed-integer programming model. This formulation is used the concept of cross-docking allocation and commodities distribution. The Lagrangian relaxation approach is proposed to solve logistics network problem. Experimental results reveal that the proposed approach provides optimal solution in a reasonable time.

Service network design with equilibrium-driven demands

We study a service network design problem in which the network operator wishes to determine facility locations and sizes in order to satisfy the demand of the customers while balancing the cost of the system with a measure of quality-of-service faced by the customers. We assume customers choose the facilities that meet demand, in order to minimize their total cost, including costs associated with traveling and waiting. When having demand served at a facility, customers face a service delay that depends on the total usage (congestion) of the facility. The total cost of meeting a customer’s demand at a facility includes a facility-specific unit travel cost and a function of the service delay. When customers all minimize their own costs, the resulting distribution of customer demand to facilities is modeled as an equilibrium. This problem is motivated by several applications, including supplier selection in supply chain planning, preventive healthcare services planning, and shelter location-allocation in disaster management. We model the problem as a mixed-integer bilevel program that can be reformulated as a nonconvex mixed-integer nonlinear program. The reformulated problem is difficult to solve by general-purpose solvers. Hence, we propose a Lagrangian relaxation approach that finds a candidate feasible solution along with a lower bound that can be used to validate the solution quality. The computational results indicate that the method can efficiently find feasible solutions, along with bounds on their optimality gap, even for large instances.

Resource Allocation and Multisession Routing Algorithms in Coordinated Multipoint Wireless Communication Networks

In heterogeneous wireless networks, coordinated multipoint technology facilitates coordination and mutual cooperation among various types of networks and applications. The resource requirements of various applications differ considerably under the constraint of limited resources so that the allowable delay is considered. This study considered a centralized management scheme involving fourth-generation (4G) mobile networks Evolved Node B, access points, and back-end coordinated servers. The operator controls the flow and performs routing selection procedures through long-term evolution advanced and Wi-Fi in accordance with the 5G wireless networks. The problem was constructed as a mathematical programming model. A Lagrangian relaxation (LR) approach was adopted to maximize the revenue minus the penalty costs as objective to evaluate the resource allocation, tolerable delay, and multisession routing algorithm. The primal problem was decomposed into seven independent subproblems through LR procedures, and the combination of a subgradient method and a routing algorithm was used to obtain approximate solutions. A novel algorithm for flow control involving multisession routing is proposed for handling resource allocation. The algorithm involving a near-optimal approach was designed to solve the insufficient resource problem. The solution can be used by operators for improving resource utilization in next-generation wireless networks.

Mixed Integer Linear Programming for Maintenance Scheduling in Power System Planning

This paper discussed the merit of mixed-integer linear programming (MILP)-based approach against Lagrangian relaxation (LR)-based approach in solving generation and transmission maintenance scheduling problem. MILP provides a straightforward solution by formulating coupling constraints equations so that these sub-problems can be solved simultaneously without involving multipliers. In LR-based approach, generation and transmission maintenance scheduling, and security-constrained unit commitment have been solved individually and the integration was realized through a series of multipliers which has caused computational burden to the system. Numerical case studies were evaluated on the 6-bus system. A comparative study is carried out between the MILP and LR approaches. Simulation results indicate that the maintenance schedule derived by the proposed MILP approach outperforms the LR in terms of operational cost savings and gap tolerance. The operating cost could be saved up to 5% and the gap tolerance achieved is 0.01% as compared to 0.14% by LR.

Green-blood supply chain network design: Robust optimization, bounded objective function & Lagrangian relaxation

Abstract The aim of this research is to consider the issue of sustainability in designing a blood supply chain network by presenting a multi objective mixed integer mathematical programming model that aims to simultaneously minimize the total cost of the supply chain network and the total environmental impacts of the activities of the supply chain network. As the nature of supplying the blood by the donors and also demand for the blood product are uncertain, a robust optimization approach is applied in the model to deal with this type of uncertainty. To convert the proposed multi objective model into a single objective one, the bounded objective function method is used. Then, as the presented mathematical model is a complicated mixed integer linear programming model, an algorithm based on the Lagrangian relaxation approach is proposed to solve the model. At the end, a computational study is done to present the competency of the proposed Lagrangian relaxation algorithm.

A Lagrangian relaxation approach for stochastic network capacity expansion with budget constraints

In this paper, we consider capacity expansion for network models subject to uncertainty and budget constraints. We use a scenario tree approach to handle the uncertainty and construct a multi-stage stochastic mixed-integer programming model for the network capacity expansion problem. We assume that permanent capacity and spot market capacity are available, which can be used permanently or temporarily by the decision maker respectively. By relaxing the budget constraints, we propose a heuristic Lagrangian relaxation method to solve the problem. Two algorithms are developed to find tight upper bounds for the Lagrangian relaxation procedure. The experimental results show superior performance of the proposed Lagrangian relaxation method compared with CPLEX.

Lagrangian relaxation approach to minimize makespan for hybrid flow shop scheduling problem with unrelated parallel machines

This research addresses scheduling problem of n jobs on a Hybrid Flow Shop (HFS) with unrelated parallel machines in each stage. A monolithic mixed integer linear programming (MILP) model is presented to minimize the maximum completion time (makespan). As the research problem is shown to be strongly NP-hard, a Lagrangian relaxation (LR) algorithm is developed to handle the HFS scheduling problem. We design two approaches, simplification of subproblems and dominance rules, to solve the subproblems which are generated in each iteration. For evaluation purposes, numerical experiments with small and large size problems are randomly generated with up to 50 jobs and four stages. The experimental results show that the Lagrangian relaxation approaches outperform the MILP model with respect to CPU time. Furthermore, from the results, it can be conclude that the simplification of subproblems shows slightly better solutions in comparison with dominance rules to solve the subproblems.

Lagrangian relaxation method for optimizing delay of multiple autonomous guided vehicles

ABSTRACTIn this paper, a scheduling and routing based on AGVs processing and waiting times and the existing paths in a jobshop manufacturing system is proposed. A mathematical formulation is developed to schedule and route multiple AGVs handling jobs to shops so that the total delay of AGVs including the earliness and tardiness is minimized. A Lagrangian relaxation method is developed and a sub-gradient algorithm is composed to update the iterations in the searching process of method. The results show that the method is efficient in larger sizes problems while exact method cannot obtain the solutions in reasonable time. Two different problems are solved using two algorithms of Lagrangian relaxation and linear relaxation. Statistical comparisons showed much better performance of Lagrangian relaxation approach in a negligible run time. For larger sizes that exact method cannot be obtained even in long run times, Lagrangian relaxation approach is useful and efficient.

A Lagrangian relaxation approach to fuzzy robust multi-objective facility location network design problem

This study considers a multi-objective combined budget constrained facility location/network design problem (FL/NDP) in which the system uncertainty is considered. The most obvious practical examples of the problem are territorial designing and locating of academies, airline networks, and medical service centers. In order to assure the network reliability versus uncertainty, an efficient robust optimization approach is applied to model the proposed problem. The formulation is minimizing the total expected costs, including, transshipment costs, facility location (FL) costs, fixed cost of road/link utilization as well as minimizing the total penalties of uncovered demand nodes. Then, in order to consider of several system uncertainty, the proposed model is changed to a fuzzy robust model by suitable approaches. An efficient Sub-gradient based Lagrangian relaxation algorithm is applied. In addition, a practical example is studied. At the following, a series of experiments, including several test problems, is designed and solved to evaluate of the performance of the algorithm. The obtained results emphasize that considering of practical factors (e.g., several uncertainties, system disruptions, and customer satisfaction) in modelling of the problem can lead to significant improvement of the system yield and subsequently more efficient utilization of the established network.

Incorporating Aggregate Diversity in Recommender Systems Using Scalable Optimization Approaches

The success of a recommender system is generally evaluated with respect to the accuracy of recommendations. However, recently diversity of recommendations has also become an important aspect in evaluating recommender systems. One dimension of diversity is called aggregate diversity, which refers to the diversity of items in the recommendation lists of all users and can be defined with different metrics. The maximization of both accuracy and the aggregate diversity simultaneously renders a multiobjective optimization problem that can be handled by different approaches. In this paper, after providing a thorough analysis of the multiobjective optimization approaches for this problem, we propose a new model that takes into account both accuracy and aggregate diversity. Different from previous works, our model is specifically designed to incorporate distributional diversity metrics, which measure how evenly the items are distributed in the recommendation lists of users. To solve the large-scale instances, we propose a column generation algorithm and a Lagrangian relaxation approach based on the decomposition of the model. We present the results of the mathematical models and the performance of the proposed methodology that are obtained by computational experiments on real-world data sets. These results reveal that our model successfully captures the trade-off between the objectives and reaches very high levels of distributional diversity.

Fast Adaptation of Activity Sensing Policies in Mobile Devices

With the proliferation of sensors, such as accelerometers, in mobile devices, activity and motion tracking has become a viable technology to understand and create an engaging user experience. This paper proposes a fast adaptation and learning scheme of activity tracking policies when user statistics are unknown a priori, varying with time, and inconsistent for different users. In our stochastic optimization, user activities are required to be synchronized with a backend under a cellular data limit to avoid overcharges from cellular operators. The mobile device is charged intermittently using wireless or wired charging for receiving the required energy for transmission and sensing operations. First, we propose an activity tracking policy by formulating a stochastic optimization as a constrained Markov decision process (CMDP). Second, we prove that the optimal policy of the CMDP has a threshold structure using a Lagrangian relaxation approach and the submodularity concept. We accordingly present a fast Q-learning algorithm by considering the policy structure to improve the convergence speed over that of conventional Q-learning. Finally, simulation examples are presented to support the theoretical findings of this paper.

Lagrangian relaxation for the reliable shortest path problem with correlated link travel times

Finding a reliable shortest path (RSP) in a stochastic network is a fundamental problem in transportation science. Link travel time correlation significantly affects path reliability, but also greatly increases the complexity of the RSP problem due to the quadratic form of the standard deviation term. Lagrangian relaxation (LR) based on problem reformulation, which only needs to solve a series of shortest path problems, has been recognized as an efficient method to obtain near-optimal RSPs with the optimality gap guarantee. This paper proposes a novel LR approach based on a new convex problem reformulation, and new methods to update Lagrangian multipliers and handle negative cycles of the resulting shortest path problems. Different from existing LR approaches, which adopt the classical subgradient method to solve the dual problem, a constraint generation (CG) algorithm and a subgradient projection (SP) algorithm are proposed to update Lagrangian multipliers effectively, and both algorithms are further modified to handle negative cycles. We also reveal the connection between different reformulations of the RSP problem and show that the proposed approach has a smaller duality gap than existing ones. Experiments on real transportation networks validate the effectiveness of the proposed approach in terms of convergence rate, run time, duality gap and optimality by comparison with the existing LR approaches and the outer approximation algorithm.

Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems

We consider the piecewise linear multicommodity network flow problem with the addition of a constraint specifying that the total flow on each arc must be an integer. This problem has applications in transportation and logistics, where total flows might represent vehicles or containers filled with different products. We introduce formulations that exploit this integrality constraint by adapting to our problem a technique known as discretization that has been used to derive mixed-integer programming models for several combinatorial optimization problems. We enhance the discretized models either by adding valid inequalities derived from cut-set inequalities or by using flow disaggregation techniques. Since the size of the formulations derived from discretization and flow disaggregation rapidly increases with problem dimensions, we develop an efficient and effective Lagrangian relaxation method to compute lower and upper bounds. We perform computational results on a large set of randomly generated instances that allow us to compare the relative efficiency of the different modeling alternatives (flow disaggregation plus addition of cut-set inequalities with or without discretization), when used within the Lagrangian relaxation approach.

A novel Lagrangian relaxation level approach for scheduling steelmaking-refining-continuous casting production

A Lagrangian relaxation (LR) approach was presented which is with machine capacity relaxation and operation precedence relaxation for solving a flexible job shop (FJS) scheduling problem from the steelmaking-refining-continuous casting process. Unlike the full optimization of LR problems in traditional LR approaches, the machine capacity relaxation is optimized asymptotically, while the precedence relaxation is optimized approximately due to the NP-hard nature of its LR problem. Because the standard subgradient algorithm (SSA) cannot solve the Lagrangian dual (LD) problem within the partial optimization of LR problem, an effective deflected-conditional approximate subgradient level algorithm (DCASLA) was developed, named as Lagrangian relaxation level approach. The efficiency of the DCASLA is enhanced by a deflected-conditional epsilon-subgradient to weaken the possible zigzagging phenomena. Computational results and comparisons show that the proposed methods improve significantly the efficiency of the LR approach and the DCASLA adopting capacity relaxation strategy performs best among eight methods in terms of solution quality and running time.

Lagrangian and branch-and-cut approaches for upgrading spanning tree problems

Problems aiming at finding budget constrained optimal upgrading schemes to improve network performance have received attention over the last two decades. In their general setting, these problems consist of designing a network and, simultaneously, allocating (limited) upgrading resources in order to enhance the performance of the designed network.In this paper we address two particular optimal upgrading network design problems; in both cases, the sought-after layout corresponds to a spanning tree of the input network and upgrading decisions are to be taken on nodes. We design Mixed Integer Programming-based algorithmic schemes to solve the considered problems: Lagrangian relaxation approaches and branch-and-cut algorithms. Along with the designed algorithms, different enhancements, including valid inequalities, primal heuristic and variable fixing procedures, are proposed.Using two set of instances, we experimentally compare the designed algorithms and explore the benefits of the devised enhancements. The results show that the proposed approaches are effective for solving to optimality most of the instances in the testbed, or manage to obtain solutions and bounds giving very small optimality gaps. Besides, the proposed enhancements turn out to be beneficial for improving the performance of the algorithms.

Eco-reliable path finding in time-variant and stochastic networks

This paper addresses a route guidance problem for finding the most eco-reliable path in time-variant and stochastic networks such that travelers can arrive at the destination with the maximum on-time probability while meeting vehicle emission standards imposed by government regulators. To characterize the dynamics and randomness of transportation networks, the link travel times and emissions are assumed to be time-variant random variables correlated over the entire network. A 0–1 integer mathematical programming model is formulated to minimize the probability of late arrival by simultaneously considering the least expected emission constraint. Using the Lagrangian relaxation approach, the primal model is relaxed into a dualized model which is further decomposed into two simple sub-problems. A sub-gradient method is developed to reduce gaps between upper and lower bounds. Three sets of numerical experiments are tested to demonstrate the efficiency and performance of our proposed model and algorithm.

Reliable Back-up Facility in Distribution Network

The designing of reliable facility location is an important way to evade against unexpected facility disruptions. In this paper, a facility allocation for distribution supply chain is designed that intend to achieve the effectiveness of system under site dependent facility disruptions. The proposed mixed integer model is implemented in commercial optimization solver CPLEX 12.6.3. The Lagrangian relaxation approach is used to solve the model. It is compared with CPLEX solver on randomly generated twenty test instances. The performance gap and computational time is used to measure the performance of proposed model. The experimental results reveal that the proposed model has capability to generate an optimal solution.

Efficient data replica placement for sensor clouds

The authors address the problem of determining a replica placement scheme for sensor data in a cache network constructed by a number of cache nodes in edge networks of the Internet, with the purpose of minimising the total costs for data placement and access. They formulate the placement problem as a mixed integer programming (MIP) problem and then propose several heuristic approaches to solve the problem. They first relax the MIP problem to a linear programming (LP) problem and then round the fractional solution values of the LP problem to integers. Next, they employ a Lagrangian relaxation approach to find a solution for each data item without considering node capacity and then rearrange the placement at each node to satisfy the capacity constraint. For comparison, they also propose a greedy approach that determines a replica placement at a node by using the node local information. They show that the former two approaches lead to good performance that is only <8% worst than the theoretical lower bound.