Outer Approximation Research Articles

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.

Zonotopic set-membership state estimation for switched systems

This paper proposes a new approach for set-membership state estimation of switched discrete-time linear systems subject to bounded disturbances and noises. A zonotopic outer approximation of the state estimation domain is computed and a new criterion is proposed to reduce the size of the zonotope at each sample time. The zonotopic set-membership estimator design for switched systems is provided within the LMI framework. The extension of the proposed scheme to deal with unknown inputs is also presented. An application to vehicle lateral dynamics state estimation is provided. Simulation results demonstrate the effectiveness of the proposed algorithm and highlight its advantages over the existing methods.

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.

Admissible Region of Renewable Generation Ensuring Power Flow Solvability in Distribution Networks

The uncertainty of renewable generation makes the operating status of distribution systems more volatile, as fully controllable resource at the distribution level is rare. A dispatchable region refers to the set that consists of all admissible patterns of nodal renewable power injections under which the power flow is solvable without violating security bound constraints. This article studies the dispatchable region in distribution networks under alternating current power flow model. A rank minimization problem is proposed to test power flow solvability under a fixed nodal power injection pattern, providing basic operation to construct the exact dispatchable region. A sequential low-order semidefinite programming procedure is developed to solve the problem. Furthermore, based on a global outer approximation of the second-order conic relaxation of the distflow model, a linear programming-based polyhedral projection algorithm is developed to calculate an outer approximation of the dispatchable region. The projection algorithm is also applied to the traditional linearized distflow model. Combining the feasibility test procedure, it is shown that the intersection of the respective dispatchable regions obtained from two linearized power flow models produces a fairly accurate approximation for the true dispatchable region under the exact nonlinear distflow model. The proposed method is an extension of existing studies on security assessment for distribution systems under uncertainty.

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.

Rolling horizon-based heuristics for solving a production-routing problem with price-dependent demand

This paper studies a production-routing problem with price-dependent demand (PRP-PD). The problem is formulated as a mixed-integer nonlinear program (MINLP) and solved within an outer approximation (OA) framework where the primal and master problems are solved iteratively. In order to solve the computationally expensive master problem, which is a mixed-integer program (MIP), four different rolling horizon-based heuristics are developed. The efficiency of the proposed heuristic algorithms is benchmarked against the existing methods in the literature to solve the PRP-PD for both linear and nonlinear demand functions. Our results show that by adopting proper freezing and simplification strategies, the rolling horizon-based heuristic can solve the large problem instances significantly faster than the existing methods in the literature, with little to no reduction in solution quality. For large problem instances with linear demand functions, the proposed heuristic provides solutions within 1.5–3 min. The quality of these solutions deviates at most 3.5% from the solutions provided by the existing exact methods, which take more than 2 h. Our solutions are also higher in quality compared to the near-optimal solutions provided in the literature, and they are achieved with a 90% reduction in computational time. With nonlinear demand functions, one-third of the instances require more than 2 h to be solved by the exact methods in the literature, while they are solved within 0.5–3.5 min using our heuristic, with less than 2% deviation in solution quality.

An outer approximation bi-level framework for mixed categorical structural optimization problems

In this paper, mixed categorical structural optimization problems are investigated. The aim is to minimize the weight of a truss structure with respect to cross-section areas, materials, and cross-section type. The proposed methodology consists of using a bi-level decomposition involving two problems: master and slave. The master problem is formulated as a mixed-integer linear problem where the linear constraints are incrementally augmented using outer approximations of the slave problem solution. The slave problem addresses the continuous variables of the optimization problem. The proposed methodology is tested on three different structural optimization test cases with increasing complexity. The comparison to state-of-the-art algorithms emphasizes the efficiency of the proposed methodology in terms of the optimum quality, computation cost, as well as its scalability with respect to the problem dimension. A challenging 120-bar dome truss optimization problem with 90 categorical choices per bar is also tested. The obtained results showed that our method is able to solve efficiently large-scale mixed categorical structural optimization problems.

Zonotopic set-membership state estimation for multirate systems with dynamic event-triggered mechanisms

This paper is concerned with the dynamic event-triggered set-membership state estimation issue for a class of multirate networked systems subject to unknown but bounded disturbance and noise. To economize the limited communication resource when performing the desired networked estimation task, a dynamic event-triggered transmission mechanism is proposed to significantly reduce the frequency of sensor data transmissions. A key issue of the addressed problem is to construct a zonotopic outer approximation set to bound the set of states that are consistent with the disturbed system model and the noisy measured outputs. With the help of a prediction-update-correction method, we design a zonotope that is guaranteed to enclose all possible system states due to the unknown disturbance and noise. Then, the size of such a zonotope is minimized by properly designing an introduced correlation matrix at each iteration. Subsequently, the estimation performance is analyzed under the developed zonotopic estimation method. A sufficient condition is further established to ensure the boundedness of the size of the obtained zonotope. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed method.

A linear outer approximation of line losses for DC-based optimal power flow problems

This paper proposes a novel and simple linear model to capture line losses for use in linearized DC models, such as optimal power flow (DC-OPF) and security-constrained economic dispatch (SCED). The Line Loss Outer Approximation (LLOA) model implements an outer approximation of the line losses lazily and typically terminates in a small number of iterations. Experiments on large-scale power systems demonstrate the accuracy and computational efficiency of LLOA and contrast it with classical line loss approaches. The results seem to indicate that LLOA is a practical and useful model for real-world applications, providing a good tradeoff between accuracy, computational efficiency, and implementation simplicity. In particular, the LLOA method may have significant advantages compared to the traditional loss factor formulation for multi-period, stochastic optimization problems where good reference points may not be available. The paper also provides a comprehensive overview and evaluation of approximation methods for line losses.

Naturally restricted subsets of nonsignaling correlations: typicality and convergence

It is well-known that in a Bell experiment, the observed correlation between measurement outcomes – as predicted by quantum theory – can be stronger than that allowed by local causality, yet not fully constrained by the principle of relativistic causality. In practice, the characterization of the set Q of quantum correlations is carried out, often, through a converging hierarchy of outer approximations. On the other hand, some subsets of Q arising from additional constraints [e.g., originating from quantum states having positive-partial-transposition (PPT) or being finite-dimensional maximally entangled (MES)] turn out to be also amenable to similar numerical characterizations. How, then, at a quantitative level, are all these naturally restricted subsets of nonsignaling correlations different? Here, we consider several bipartite Bell scenarios and numerically estimate their volume relative to that of the set of nonsignaling correlations. Within the number of cases investigated, we have observed that (1) for a given number of inputs ns (outputs no), the relative volume of both the Bell-local set and the quantum set increases (decreases) rapidly with increasing no (ns) (2) although the so-called macroscopically local set Q1 may approximate Q well in the two-input scenarios, it can be a very poor approximation of the quantum set when ns>no (3) the almost-quantum set Q~1 is an exceptionally-good approximation to the quantum set (4) the difference between Q and the set of correlations originating from MES is most significant when no=2, whereas (5) the difference between the Bell-local set and the PPT set generally becomes more significant with increasing no. This last comparison, in particular, allows us to identify Bell scenarios where there is little hope of realizing the Bell violation by PPT states and those that deserve further exploration.

Reachable set estimation for spacecraft relative motion based on bang-bang principle

For spacecraft formation flight, the information of relative motion reachable set is very important, which can be used to predict the operating boundary of adjacent spacecraft and thus to ensure the safety of spacecraft operation. In this paper, we aim at developing a numerical method to approximate the reachable set for spacecraft relative motion. In particular, we focus on the quality of the approximation and the computational cost. Based on the bang-bang control principle, a polyhedral approximation algorithm is proposed to compute the reachable set of a relative motion spacecraft system. An inner approximation and an outer approximation of the reachable set for the system can be obtained. We prove that the approximation quality measured in Hausdorff distance can be guaranteed. The method is easy to implement and has low computational cost. Finally, the effectiveness of the algorithm is demonstrated by experimental simulation.

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.

Reliability-Centered Maintenance Scheduling Considering Failure Rates Uncertainty: A Two-Stage Robust Model

Reliability-centered maintenance (RCM) as a subset of the reliability assessment problems, is a powerful tool for enhancing reliability, and subsequently achieving the customers’ higher satisfaction. However, deterministic RCM models could scarcely guarantee global optimality and feasibility of their solutions with real-world uncertain failure rates. In this paper, a robust model is proposed to consider the impact of failure rate uncertainty in the RCM scheduling problem, followed by a two-stage robust model development. The two-stage robust model is solved using Benders decomposition (BD) and outer approximation (OA) algorithms. The proposed model and solution method are applied to both educational RBTS2 system and real-size Finnish Distribution System, followed by demonstration of a higher financial and reliability performance, compared to that of the deterministic model. Sensitivities of the costs, available budget, uncertainty interval and computational time are analyzed and discussed, to conclude the paper.

Formula omitted] control for continuous-time Markov jump nonlinear systems with piecewise-affine approximation

This paper investigates the problem of stability, H∞ performance analysis, and H∞ control for continuous-time Markov jump nonlinear systems, where the nonlinear subsystems are approximated by the piecewise-affine technique. The proposed Markov jump piecewise-affine systems contain different modes and regions, both of which are determined by Markov chains and piecewise-affine partitions, respectively. A new admissible adjacent region switching paths (AARSPs) algorithm is proposed for the first time in the continuous-time domain to decrease the conservatism of the complete adjacent region switching paths (CARSPs) algorithm. This new algorithm optimizes the path selection conditions of the next instantaneous time region switching in the CARSPs algorithm, and effectively reduces the computational complexity and the conservatism of the CARSPs algorithm. Furthermore, a state-feedback piecewise-linear controller is designed by means of the ellipsoidal outer approximation estimation method, such that the corresponding closed-loop system is stochastically stable and has a guaranteed H∞ performance index. Finally, the effectiveness and practicability of both the AARSPs algorithm and the piecewise-linear control strategy are fully demonstrated via two illustrative examples including a class of tunnel diode circuit systems.

A combined and robust modal-split/traffic assignment model for rail and road freight transport

Searching for achieving an ambitious reduction in greenhouse gas emissions, the EU has set as a goal a modal shift in freight transport of 30% by rail for the near future. In this context, it is vital to use modal choice models road versus rail to assess the shippers’ acceptance of the actions promoting the use of rail. This paper develops a combined model for jointly evaluating modal split and railway freight flows, addressed to the case when a modal split based on a random utility model is available, and some of its coefficients may present a non-negligible variability. To this end, after the initial deterministic formulation a robust counterpart of the model is developed. The model, formulated as a non-linear integer programming problem, is oriented to a multi-carrier environment and includes constraints to consider the interactions between the different types of flows on the railway network, allowing a good evaluation of the cost types of the carriers and the network capacity. An algorithmic solution based on the outer approximation method is shown to provide accurate solutions in a reasonable computational time for the robust and non-robust versions of the model. Examples centered on a section of the Trans-European Transport Network, the TEN-T Core network corridors, are reported to test the model’s applicability. Results show that this model can be a helpful tool for analyzing the possible shippers’ response to the different railway carriers’ services competing with the road.

Kleene Algebra to Compute Invariant Sets of Dynamical Systems

In this paper, we show that a basic fixed point method used to enclose the greatest fixed point in a Kleene algebra will allow us to compute inner and outer approximations of invariant-based sets for continuous-time nonlinear dynamical systems. Our contribution is to provide the definitions and theorems that will allow us to make the link between the theory of invariant sets and the Kleene algebra. This link has never be done before and will allow us to compute rigorously sets that can be defined as a combination of positive invariant sets. Some illustrating examples show the nice properties of the approach.

A Convex Reformulation and an Outer Approximation for a Large Class of Binary Quadratic Programs

Binary Quadratic Program with Variable Partitioning Constraints The binary quadratic program with variable partitioning constraints is a very general class of optimization problems that is very difficult to solve because of the nonconvexity and integrality of the variables and is ubiquitous, among others, in network design, computer vision, and transportation and logistics. In their article, “A convex reformulation and an outer approximation for a large class of binary quadratic programs,” Rostami et al. show how to transform such a nonconvex challenging problem into a convex bilinear program with decomposable structure. The authors develop a branch-and-cut algorithm based on outer approximation cuts, in which the cuts are generated on the fly by efficiently solving separation subproblems. The results of their computational experiments on different problems confirm the efficacy of the solution methods in solving large-scale problem instances.

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.