Outer Approximation Algorithm Research Articles

Fairness-based user association and resource blocks allocation in satellite–terrestrial integrated networks

To meet the futuristic communications needs, a satellite–terrestrial integrated network (STIN) has been proposed and is a strong contender amongst emerging architectures. In our STIN model, we have considered a satellite-based base station (BS), dovetailed with a terrestrial N-tier heterogeneous network (HetNets). Our work considers jointly admission control of user equipment (UE), power allocation , fairness-based user association (UA), and fair spectrum resource allocation to UEs in STIN. With throughput maximization as an objective, considering such an environment, has not been investigated in the past. The formulated problem is a mixed integer non-linear programming (MINLP) problem that is Non-deterministic Polynomial-time Hard (NP-hard) and to achieve an optimal solution, it requires an exhaustive search. But, the computational load of exhaustive search increases exponentially as the number of UEs increases. Therefore, to obtain a near-optimal solution having low computational load an outer approximation algorithm (OAA) is proposed. To evaluate the proposed algorithm, extensive simulation work has been performed. The effectiveness of the proposed approach is verified by the results in terms of fairness in UA, fairness in resource block (RB) allocation, and throughput in the downlink (DL).

Radio resource allocation for energy efficiency maximization in satellite–terrestrial integrated networks

Satellite–Terrestrial Integrated Networks (STIN), where a satellite access network liaising with terrestrial networks, is useful not only in proffering seamless coverage but also in improving the backhaul capacity for heavy traffic/dense network scenarios. Therefore, an energy-efficient STIN is envisioned to be a valued gap filler both in public safety networks and in the provision of high-speed data services with ubiquitous coverage in remote areas. STIN necessitate admission control, user association, optimal power distribution and spectrum resource allocation to attain the desired quality-of-service standards. This paper investigates joint admission control, user association and power distribution for ensuring fairness while associating users in STIN and fairness in the allocation of spectrum resources to associated users in STIN with an overall objective to maximize the energy efficiency of STIN. The reviewed problem is a concave fractional programming problem which by utilizing Charnes–Cooper transformation is converted into a concave optimization problem. Subsequently, the concave optimization problem is resolved via the proposed outer approximation algorithm. The performance of the ϵ-optimum solution is extensively evaluated via the execution of different system parameters including number of users, user association, user fairness and resource block fairness.

Service Chain Caching and Workload Scheduling in Mobile Edge Computing

The computing and storage resources of an edge node in a mobile edge computing (MEC) system are limited. One edge computing node cannot cache all the services supported by the entire MEC system. A service chain caching and scheduling strategy should be designed to improve the service processing capacities of the edge computing nodes under high computing load conditions. This article combines the open Markov queuing network for the first time to model the process of edge node analysis and processing of computationally intensive tasks as a mixed integer nonlinear programming problem. Moreover, a mathematical model for the service chains is established. We analyze the average service response time during task execution to optimize the performance of service chain caching and scheduling. Experiments show that the outer approximation (OA) algorithm obtains better results than MindtPy and Multistart. The average response time obtained by the OA algorithm is 2.2% and 6.9% lower than those obtained by the MindtPy and Multistart algorithms, respectively. The OA algorithm using this mathematical model can effectively reduce the average service response time and improve the resource utilization of the edge server.

Characterizing Temporal-Coupled Feasible Region of Active Distribution Networks

With an increasing penetration of distributed energy resources (DERs), it has become a public interest regarding how to efficiently manage large-scale DERs into wholesale markets. The existing literature has focused on various coordination methods including iteration-based (e.g., Lagrangian relaxation) and projection-based (e.g., vertex enumeration) ones, which may not work for the cases with complicated constraints due to curse of dimensionality. In this article, we propose an efficient projection-based method that adapts to characterizing the feasible region of active distribution networks with temporal-coupled constraints. The problem of finding the temporal-coupled feasible region is formulated as a max–min optimization program, aiming to determine the largest set allowing zero-valued slack variables. Based on Duality Principle, the max–min model is transformed into a solvable mixed-integer linear programming that can be efficiently optimized by off-the-shelf solvers. In contrast to the existing methods searching for a vertex from interior points, we develop an outer progressive approximation algorithm that generates feasibility cuts for each iteration. Case studies based on a modified IEEE 33-bus feeder and Caracas 141-bus system demonstrate the effectiveness and efficiency of our proposed method in characterizing the feasible region with large-scale temporal-coupled constraints.

Efficient resource allocation for hybrid nonorthogonal multiple access based heterogeneous networks beyond fifth‐generation

AbstractFifth‐generation (5G) and future beyond fifth‐generation (B5G) networks will require immense capacity due to the high rise in the number of multimedia applications and mobile devices. The heterogeneous networks (HetNets) in 5G and B5G can increase the heterogeneity and network throughput because macro base station‐only (MBS‐only) networks cannot satisfy the drastic increase in capacity demands. Hybrid nonorthogonal multiple access (H‐NOMA) can accommodate the increasing number of multimedia applications and mobile devices in 5G and B5G. H‐NOMA in B5G can also improve spectral efficiency. Before this, the researchers had not considered user clustering with H‐NOMA in HetNets. This article investigates the user clustering with downlink H‐NOMA in HetNet to maximize the network throughput in 5G and B5G. The formulated mathematical problem optimizes the considered key performance indicators (KPIs), that is, users admission in clusters, users association with base stations, power allocation to users, and network throughput. At the same time, it also satisfies the minimum transmit power and data rate requirements of users. The formulated problem is a mixed‐integer nonlinear programming (MINLP) problem. We have proposed an‐optimal algorithm, that is, outer approximation algorithm (OAA) to solve the MINLP problem because the complexity of the optimal exhaustive search algorithm (ESA) increases exponentially with an increase in the number of users. H‐NOMA with user clustering in HetNet is evaluated with extensive simulations to show its effectiveness regarding network throughput in 5G and B5G. We have also considered the proposed framework in the MBS‐only network and compared its performance with HetNet. The results verify that the proposed framework in HetNet performs better than the MBS‐only network. The complexity of the‐optimal algorithm is calculated that gives‐optimal results within. We have also made a complexity analysis of the proposed OAA and ESA and concluded that the complexity of the proposed OAA algorithm is less than the ESA.

Hub location problem considering spoke links with incentive-dependent capacities

This study proposes a hub-and-spoke (HS) network in which the capacity of a spoke link depends on the incentive provided to the transporters. The incentives influence the capacities of the spoke links, which in turn affect the hub selections and flow assignment. A mixed integer convex programming model is developed to optimize the proposed HS network. The model minimizes total transport cost by locating p hubs, routing flows between origin–destination (OD) pairs and determining the incentives on spoke links. A customized outer-approximation algorithm is presented to provide exact solutions to the developed model, which is difficult to solve optimally within acceptable computational times for large-scale instances. An approximation algorithm based on the stepwise linearization method is then developed to provide suboptimal solutions. The developed model and algorithms are validated in computational experiments with different problem scales. The experiments demonstrate that the proposed HS network tends to route the flows on more spoke links, balance the transport costs on the potential paths between OD pairs and serve more demands with lower transport costs, when compared with the HS network with fixed capacities.

Optimal plate design problem in steel production

In this paper, we address a new optimal plate design problem in steel production, in which slab selection is jointly considered. On the basis of the underlying features, the problem is formulated as a mixed integer nonlinear programming (MINLP) model with generalized disjunctive programming (GDP) constraints. A logic-based outer approximation (L-OA) algorithm is proposed to solve the problem. Specifically, a two-stage heuristic method is designed to initialise the L-OA algorithm. Numerical results are presented to demonstrate that the proposed L-OA algorithm and the heuristic method are effective and computationally efficient.

A Norm Minimization-Based Convex Vector Optimization Algorithm

We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization.

Multicriterion Resource Management in Energy-Harvested Cooperative UAV-Enabled IoT Networks

Cooperative communication by employing unmanned aerial vehicle (UAV)-based relays with radio-frequency (RF) energy harvesting (EH) has been emerged as a prominent solution to provide extended coverage, connectivity, capacity, energy efficiency, and reliability in the future Internet-of-Things (IoT) systems. For successful integration of UAV relays in IoT networks, efficient radio resource management (RRM) is critical. We developed a multicriterion framework for energy-efficient RRM in a cooperative IoT network. We considered UAVs as relays, onboard EH facilities and deployed to relay the messages from a satellite terminal to the network IoT devices. We adopted a power splitting (PS)-based EH scheme, i.e., PS relaying protocol, for RF-EH at UAV relays. We formulate a joint optimization problem for IoT device selection, UAV relay assignment, source power allocation, and PS ratio selection. In our multicriterion framework, we consider three conflicting objectives by applying a weighted-sum method: 1) maximizing the network sum rate; 2) maximizing the number of IoT devices to be served; and 3) minimizing the carbon dioxide emissions. We propose an outer approximation algorithm (OAA) to solve the formulated problem, which is a mixed-integer nonlinear programming (MINLP) problem. Simulation results of the proposed algorithm are compared with two existing solutions, namely, the nonlinear optimization by mesh adaptive direct search (NOMAD) algorithm and an evolutionary algorithm (EA). The performance of the NOMAD algorithm is better in terms of computational complexity. However, the simulation results reveal the supremacy of the proposed OAA in terms of network sum rate, the number of selected IoT devices, and network utility.

Data-adaptive robust coordinated optimization of dynamic active and reactive power flow in active distribution networks

The uncertainty of wind power brings security and economic issues in active distribution networks. As an effective solution to these problems, the active/reactive optimization aims at minimizing the total operation cost and network loss by controlling multiple resources. This paper proposes a two-stage data-adaptive robust optimization model for the dynamic active-reactive power flow. The dynamic correlation between the uncertainties of active and reactive power is analyzed at two kinds of power factor modes of wind turbines and then is described by a linear affine decision rule. To characterize these uncertainties, a data-adaptive multi-band uncertainty set is constructed based on historical data, which allows the number of bands and the corresponding weight coefficients to be adjusted under different confidence levels. Subsequently, the conic relaxation and linearization techniques are performed to reformulate the robust optimization model as a mixed-integer second-order cone programming problem. A two-level solution framework, consisting of a column-and-constraint generation algorithm and an outer approximation algorithm, is employed to solve this problem. Simulation results on a modified IEEE 33-bus system indicate that the variable power factor mode is more suitable for describing the dynamic correlation than the constant power factor mode. Besides, the proposed model and method also show better practicality and solution efficiency than the existing ones.

Energy Efficiency and Throughput Maximization Using Millimeter Waves–Microwaves HetNets

The deployment of millimeter waves can fulfil the stringent requirements of high bandwidth and high energy efficiency in fifth generation (5G) networks. Still, millimeter waves communication is challenging because it requires line of sight (LOS). The heterogeneous network (HetNet) of millimeter waves and microwaves solves this problem. This paper proposes a millimeter -microwaves heterogeneous HetNet deployed in an indoor factory (InF). In InF, the manufacturing and production are performed inside big and small halls. We consider non standalone dual-mode base stations (DMBS) working on millimeter waves and microwaves. We analyze the network in terms of throughput and energy efficiency (EE). We formulate mixed-integer-non-linear-programming (MINLP) to maximize the throughput and EE of the network. The formulated problem is a complex optimization problem and hard to solve with exhaustive search. We propose a novel outer approximation algorithm (OAA) to solve this problem, and the proposed algorithm OAA achieves optimal solution at β = 10−3. At this β, the average throughput value obtained is approximately 50 Mbps, whereas the value of EE is 4.4 Mbits/J. We also compare the performance of OAA with the mesh-adaptive-direct-search-algorithm (NOMAD), and the experimental results verify that OAA outperforms NOMAD in terms of throughput and EE maximization. We also compare the performance of OAA with particle swarm optimization (PSO), genetic algorithm (GA), and many others optimization algorithms. Experimental results verify that OAA outperforms all other algorithms.

A New Stochastic Model for Bus Rapid Transit Scheduling with Uncertainty

Nowadays, authorities of large cities in the world implement bus rapid transit (BRT) services to alleviate traffic problems caused by the significant development of urban areas. Therefore, a controller is required to control and dispatche buses in such BRT systems.. However, controllers are facing new challenges due to the inherent uncertainties of passenger parameters such as arrival times, demands, alighting fraction as well as running time of vehicles between stops. Such uncertainties may significantly increase the operational cost and the inefficiencies of BRT services. In this paper, we focus on the controller’s perspective and propose a stochastic mixed-integer nonlinear programming (MINLP) model for BRT scheduling to find the optimal departure time of buses under uncertainty. The objective function of the model consists of passenger waiting and traveling time and aims to minimize total time related to passengers at any stop. From the modeling perspective, we propose a new method to generate scenarios for the proposed stochastic MINLP model. Furthermore, from the computational point of view, we implement an outer approximation algorithm to solve the proposed stochastic MINLP model and demonstrate the merits of the proposed solution method in the numerical results. This paper accurately reflect the complexity of BRT scheduling problem and is the first study, to the best of our knowledge, that presents and solves a mixed-integer nonlinear programming model for BRT scheduling.

Performance Tradeoff of MVNOs in OFDMA-Based Virtualized Wireless Networks

In this paper, we analyze the tradeoff between the profits gained by mobile virtual network operators (MVNOs) in an orthogonal frequency division multiple access (OFDMA)-based virtualized wireless networks (VWNs). In this respect, MVNOs rent the network resources from a mobile network operator (MNO) to create virtual resources based on allocated rates and the cost due to allocated transmit powers in two different strategies: resource-based isolation strategy and rate-based isolation strategy. In resource-based isolation strategy, it is assumed that the whole bandwidth in each base station is divided equally between MVNOs whereas in rate-based isolation strategy, the whole bandwidth in each base station is allocated dynamically between MVNOs. We then formulate a multi-objective optimization problem (MOOP) to show the tradeoff between the maximum profit of each MVNO. To solve the formulated MOOP, we adopt the weighted Tchebycheff approach, which can provide the complete Pareto optimal region for nonconvex problems. Thus, by developing such formulation, we will be able to find the best profit gained by MVNOs. We also employ monotonic optimization to obtain the global optimal solutions of the non-convex problems by applying polyblock outer approximation algorithm. Our numerical results show the importance of the aforementioned tradeoffs and confirm that the proposed geometric programming (GP) based algorithm achieves excellent performance as compared to the monotonic method.

Energy Efficient Resource Allocation for H-NOMA Assisted B5G HetNets

The resource allocation solution offered based on Non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) schemes are sub-optimal to address the challenging quality of service (QoS) and higher data rate viz-a-viz energy efficiency (EE) requirements in 5th generation (5G) cellular networks. In this work, we maximize the EE using user equipment (UE) clustering (UE-C) with downlink hybrid NOMA (H-NOMA) assisted beyond 5G (B5G) HetNets. We formulate an optimization problem incorporating UE admission in a cluster, UE association with a base station (BS), and power allocation assisted by H-NOMA, i.e., OMA and NOMA schemes in the macro base station (MBS) only and heterogeneous networks (HetNets) environments. The problem formulated is a type of non-linear concave fractional programming (CFP) problem. The Charnes-Cooper transformation (CCT) is applied to the formulated non-linear CFP problem to convert it into a concave optimization, i.e., mixed-integer non-linear programming (MINLP) problem. A two-phase ϵ-optimal outer approximation algorithm (OAA) is used to solve the MINLP problem. The simulation results show that H-NOMA with HetNets outperforms H-NOMA with MBS only in terms of UE admission, UE association, throughput, and EE.

Optimal lot-sizing of an integrated EPQ model with partial backorders and re-workable products: an outer approximation

In this paper, we develop and optimise the lot-sizing policy in an integrated Economic Production Quantity (EPQ) model with partial backorders and re-workable products considering linear and fixed backordering costs. Time intervals, number of lots, and lot size are the decision variables of our model. The cost function includes the set-up costs, the holding costs, the goodwill loss costs, the fixed backorder costs, the backorder costs, the costs of reworking, the production costs, the disposal costs, and the screening costs. The profit function consists of the sale profits obtained from the sold products. The goal is to minimise the cost function and maximise the profit function under stochastic constraints simultaneously. A Lexicographic method is applied for integrating the conflicting objective functions. The integrated objective function, with stochastic constraints, is a Mix Integer Nonlinear Programming (MINLP) model. Accordingly, an Outer Approximation (OA) algorithm is provided for optimal lot-sizing the integrated EPQ model. Two numerical examples and a real large-scale example demonstrate the decent and acceptable performance of the presented OA with respect to the optimality criteria such as optimum solutions, complementarity, and taken iterations.

Integrating stochastic programming and reliability in the optimal synthesis of chemical processes

Plant availability and operating uncertainties are critical considerations for the design and operation of chemical processes as they directly impact service level and economic performance. This paper proposes a two-stage stochastic programming GDP (Generalized Disjunctive Programming) model with reliability constraints to deal with both the exogenous and endogenous uncertainties in process synthesis, where the reliability model is incorporated into the flowsheet superstructure optimization. The proposed stochastic programming model anticipates the market uncertainties through scenarios for selecting the optimal flowsheet topology, equipment sizes and operating conditions, while considering the impact of selecting parallel units for improving plant availability. An improved logic-based outer approximation algorithm is applied to solve the resulting hybrid GDP model, which effectively avoids numerical difficulties with zero flows and provides high quality design solutions. The applicability of the proposed modeling framework and the efficiency of solution strategy are illustrated with two well-known conceptual design case studies: methanol synthesis process and toluene hydrodealkylation process. The model, which integrates reliability (endogenous uncertainty) and exogenous uncertainty, shows the best economic performance with the increasing operational flexibility and plant availability.

Sparse classification: a scalable discrete optimization perspective

We formulate the sparse classification problem of n samples with p features as a binary convex optimization problem and propose a outer-approximation algorithm to solve it exactly. For sparse logistic regression and sparse SVM, our algorithm finds optimal solutions for n and p in the 10,000 s within minutes. On synthetic data our algorithm achieves perfect support recovery in the large sample regime. Namely, there exists an \(n_0\) such that the algorithm takes a long time to find an optimal solution and does not recover the correct support for \(n<n_0\), while for \(n\geqslant n_0\), the algorithm quickly detects all the true features, and does not return any false features. In contrast, while Lasso accurately detects all the true features, it persistently returns incorrect features, even as the number of observations increases. Consequently, on numerous real-world experiments, our outer-approximation algorithms returns sparser classifiers while achieving similar predictive accuracy as Lasso. To support our observations, we analyze conditions on the sample size needed to ensure full support recovery in classification. For k-sparse classification, and under some assumptions on the data generating process, we prove that information-theoretic limitations impose \(n_0 < C \left( 2 + \sigma ^2\right) k \log (p-k)\), for some constant \(C>0\).

Itinerary planning for cooperative truck platooning

A cooperative truck platoon is a set of virtually linked trucks driving with a small intra-vehicle headway enabled by connected and automated vehicle technologies. One of the primary benefits of truck platooning is energy savings due to the reduction of aerodynamic drag on the platooned vehicles. The focus of this paper is on scheduling travel itineraries of a given set of trucks to facilitate the formation of platoons to maximize energy savings. By constructing a time-expanded network, we formulate the problem as a minimum concave-cost network flow problem, and devise a few solution methods to find the optimal or high-quality solutions. The solution methods include an outer approximation algorithm for solving a mixed-integer convex minimization reformulation of the problem, a dynamic-programming-based heuristic scalable to large-scale instances, and a fast approximation algorithm with guaranteed performance for a restrictive version of the problem. All the proposed algorithms are examined and benchmarked on medium to large networks under various test scenarios. The numerical results demonstrate the efficiency of the proposed methods and their applicability in real-world settings.

Energy efficient resource allocation in cache‐enabled fog networks

AbstractTo gain high energy efficiency and low latency, fog computing can be considered as a promising enabling technology for supporting future IoT networks. Exponentially increasing data rate demand of smart IoT devices in the fifth‐Generation (5G) networks require new resource management and power allocation schemes. Fog computing involves the deployment of data storage devices (fog nodes) near the proximity of IoT nodes. The fog nodes collaborate to process requests generated from users. These nodes can store data in their storage capacity and supports in achieving quality‐of‐service (QoS) requirements. A joint node association and energy efficiency maximization problem is formulated in this article. The proposed problem is a non‐linear concave programming problem under cache size, association, and optimal power allocation constraints. A mesh adaptive direct search algorithm (MADS) is used to solve the formulated problem to find the sub‐optimal results. The efficient performance of the algorithm can be seen in comparison with more complex algorithms namely, outer approximation algorithm and exhaustive search algorithm. Extensive simulations are done to observe the detailed performance analysis of the IoT‐Fog network. As a result, energy‐efficient resource allocation is done under cache and QoS constraints with very little complexity and low computational power and delays.

Hybrid model generation for superstructure optimization with Generalized Disjunctive Programming

We propose a novel iterative procedure to generate hybrid models (HMs) within an optimization framework to solve design problems. HMs are based on first principle and surrogate models (SMs) and they may represent potential plant units embedded within a superstructure. We generate initial SMs with simple algebraic regression models and refine them by adding Gaussian Radial Basis Functions in three steps: initial SM refinement, domain exploration, and, after solving the optimal design problem, further domain exploitation, until the convergence criterion is fulfilled. The superstructure optimization problem is formulated with Generalized Disjunctive Programming and solved with the Logic-based Outer Approximation algorithm. We addressed methanol synthesis and propylene plant design problems. Compared to rigorous model-based optimal design, the proposed HMs gave the same configuration, objective function and decision variables with maximum relative differences of 1 and 7 %, respectively. A sensitivity analysis shows that the proposed strategy reduced CPU time by 33 %.