Nonconvex Mixed-integer Nonlinear Programming Problem Research Articles

Globally optimal sequencing of optimal reactive dispatch control adjustments to minimize operational losses in transmission systems by graph shortest path, parallel computing, and dynamic programming

Minimizing operational losses in transmission systems through the Optimal Reactive Dispatch (ORD), a non-convex mixed-integer nonlinear programming problem, is crucial for operational cost reduction, resource optimization, and greenhouse gas emission mitigation. Besides all intricacies associated with solving ORDs, transmission system operators encounter the challenge of determining sequences in which ORD control adjustments must be implemented before significant changes occur in generators scheduled power output and system loading. Sequencing ORD control adjustments, in spite of not being novel, remains modestly scrutinized in the literature. This paper introduces a two-phase framework that tackles the globally optimal sequencing of n ORD control adjustments over n! potential paths by solving the ORD to minimize operational losses in transmission systems in the first phase, and optimally sequencing ORD control adjustments employing fast power flow calculations, graph shortest path, parallel computing, and dynamic programming in the second phase. We discuss the framework’s second phase asymptotic time complexity, which is exponential over factorial for brute-force approaches, and its capability to guarantee globally optimal paths toward minimal operational losses determined in the framework’s first phase. ORD control adjustments for transmission systems with up to 27 controllable variables are benchmarked against two mixed-integer nonlinear programming solvers: BARON, a global non-convex solver, and Knitro, a local solver (assuming convexity around local optima). Globally optimal sequences of ORD control adjustments over n! potential paths (more than 1028 for sequencing 27 control adjustments) and average algorithm runtimes validate the straightforward application and, more importantly, effectiveness of such a comprehensive framework.

Online Joint Optimization of Virtual Network Function Deployment and Trajectory Planning for Virtualized Service Provision in Multiple-Unmanned-Aerial-Vehicle Mobile-Edge Networks

The multiple-unmanned-aerial-vehicle (multi-UAV) mobile edge network is a promising networking paradigm that uses multiple resource-limited and trajectory-planned unmanned aerial vehicles (UAVs) as edge servers, upon which on-demand virtual network functions (VNFs) are deployed to provide low-delay virtualized network services for the requests of ground users (GUs), who often move randomly and have difficulty accessing the Internet. However, VNF deployment and UAV trajectory planning are both typical NP-complete problems, and the two operations have a strong coupling effect: they affect each other. Achieving optimal virtualized service provision (i.e., maximizing the number of accepted GU requests under a given period T while minimizing the energy consumption and the cost of accepting the requests in all UAVs) is a challenging issue. In this paper, we propose an improved online deep reinforcement learning (DRL) scheme to tackle this issue. First, we formulate the joint optimization of the two operations as a nonconvex mixed-integer nonlinear programming problem, which can be viewed as a sequence of one-frame joint VNF deployment and UAV-trajectory-planning optimization subproblems. Second, we propose an online DRL based on jointly optimizing discrete (VNF deployment) and continuous (UAV trajectory planning) actions to solve each subproblem, whose key idea is establishing and achieving the coupled influence of discrete and continuous actions. Finally, we evaluate the proposed scheme through extensive simulations, and the results demonstrate its effectiveness.

Energy-efficient BS sleeping, user association, and resource allocation in full-duplex ultra-dense networks

Full-duplex ultra-dense network (FD-UDN) is a promising technology in 5G mobile networks for handling the increase in network capacity. However, interference management and energy efficiency are two of the most important challenges that must be addressed. With the small base stations (SBSs) densification, employing sleeping strategy coupled with resource management becomes an effective approach to manage interference and power consumption in FD-UDN. To the best of our knowledge, this aspect has not been addressed in previous work. To this end, we develop a framework to optimize the BS sleeping and resource management with the aim of maximizing energy efficiency and maintaining the quality of service (QoS) requirements of users. The problem is formulated as a non-convex mixed-integer non-linear programming problem, which is difficult to handle. Employing the Dinkelbach method, the objective function of the problem is converted to an equivalent parametric subtractive form. Then, the problem is decoupled into two sub-problems: user association and resource allocation, as well as BSs on/off switching. The former is solved using the iterative reweighted lq-norm minimization (IRM) method, and the latter is solved using the Lagrangian dual method and the constrained concave-convex procedure (CCCP). The simulation results demonstrate that the proposed method is more effective than traditional ones in simultaneously improving the EE, reducing power consumption, and keeping fewer SBSs active, especially in high network loads.

Coalitional Game Based Resource Allocation in D2D-Enabled V2V Communication

The joint resource block (RB) allocation and power optimization problem is studied to maximize the sum-rate of the vehicle-to-vehicle (V2V) links in the device-to-device (D2D)-enabled V2V communication system, where one feasible cellular user (FCU) can share its RB with multiple V2V pairs. The problem is first formulated as a nonconvex mixed-integer nonlinear programming (MINLP) problem with constraint of the maximum interference power in the FCU links. Using the game theory, two coalition formation algorithms are proposed to accomplish V2V link partitioning and FCU selection, where the transferable utility functions are introduced to minimize the interference among the V2V links and the FCU links for the optimal RB allocation. The successive convex approximation (SCA) is used to transform the original problem into a convex one and the Lagrangian dual method is further applied to obtain the optimal transmit power of the V2V links. Finally, numerical results demonstrate the efficiency of the proposed resource allocation algorithm in terms of the system sum-rate.

Height-Fixed UAV Enabled Energy-Efficient Data Collection in RIS-Aided Wireless Sensor Networks

Wireless sensor nodes (SNs) are usually energy-limited, a low energy efficiency of SN may be caused by the occlusion of transmission link in the unmanned aerial vehicle (UAV)-enabled data collection. Reconfigurable intelligent surfaces (RISs) provide potential communication opportunities by reflecting and enhancing the signal, thus increasing SNs’ energy efficiency. This paper investigates energy-efficient data collection aided by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> RIS elements, where a height-fixed and flight time-constrained UAV is considered as the collector, a tractable air-to-ground channel model is established considering the cascaded RIS channel and time-varying trajectory of UAV, and energy consumption of SNs is minimized. Specifically, coupling two intractable fourth-order moment variables, the air-to-ground channel model is processed into an easily computable expression through the Gamma function. Furthermore, based on the proposed two RIS deployment schemes, maximum energy consumption of SNs minimization problem is formulated as a non-convex mixed-integer nonlinear programming (NMNP) problem with optimization of UAV trajectory and SNs wake-up schedules. Additionally, through the slack technique and Taylor expansion, the NMNP problem is decoupled as linear programming and quadratically constrained quadratic programming, and an effective algorithm is designed to obtain the sub-optimal solution. Simulations prove the validity of our algorithm and RIS deployment insights are given for SNs energy saving.

Incentive Mechanism and Resource Allocation for Collaborative Task Offloading in Energy-Efficient Mobile Edge Computing

We investigate a mobile edge computing (MEC) system that supports collaborative task offloading, allowing busy users to offload their tasks to the server and idle users, or perform them locally. However, executing computing tasks consumes device energy, making idle users unwilling to perform other users' tasks due to limited battery capacity. Additionally, the components of the total energy expenditure of the MEC system supporting collaborative task offloading are complex, necessitating appropriate resource allocation and offloading strategies to minimize the system's total energy consumption. To address these challenges, we propose a computing resource sharing auction (CRSA) algorithm to motivate idle users to participate in task offloading. Then, we establish a non-convex mixed-integer nonlinear programming (MINLP) problem to minimize the total energy consumed by the system. By utilizing the McCormick and continuous relaxation (CR) approaches, we develop a low-complexity resource allocation algorithm. Finally, the numerical results demonstrate the effectiveness of the proposed mechanism and resource allocation algorithm.

Pump scheduling optimization in water distribution system based on mixed integer linear programming

The energy consumption in water distribution systems (WDSs) is significant. Improving the efficiency of pump operation can significantly reduce energy costs. However, optimal pump operation is a nonconvex mixed-integer nonlinear programming (MINLP) problem, which can be challenging to solve. A feasible approach is to linearize the problem and convert it into a mixed-integer linear programming (MILP) problem. However, this approach introduces many auxiliary variables, which can lead to inefficiency in finding the optimal solution due to the expanded search space. To address this issue, we propose a novel method for linearization of the original MINLP problem and a strategy that can adaptively adjust the number of piecewise linearization breakpoints. By reducing the number of auxiliary variables, our approach achieved competitive computing efficiency and the ability to save energy costs, as demonstrated in two benchmark instances. Furthermore, in a realistic large-scale WDS, our approach saved 9.83% more energy costs than the genetic algorithm and achieved a gap of only 7.36% from the lower bound.

AI-Based Resource Allocation in End-to-End Network Slicing Under Demand and CSI Uncertainties

Network slicing (NwS) is one of the main technologies in the fifth-generation of mobile communication and beyond (5G+). One of the important challenges in the NwS is information uncertainty which mainly involves demand and channel state information (CSI). Demand uncertainty is divided into three types: number of users requests, amount of bandwidth, and requested virtual network functions workloads. Moreover, the CSI uncertainty is modeled by three methods: worst-case, probabilistic, and hybrid. In this paper, our goal is to maximize the utility of the infrastructure provider by exploiting deep reinforcement learning (DRL) algorithms in end-to-end NwS resource allocation under demand and CSI uncertainties. Enhanced mobile broadband (eMBB) requires high data rates. The uncertainties we argued above have a direct negative impact on the data rate and our objective function. Therefore, we focus primarily on eMBB. Additionally, we also consider ultra-reliable low latency communications (uRLLC) and massive machine-type communication (mMTC). The proposed formulation is a non-convex mixed-integer non-linear programming problem. To perform resource allocation in problems that involve uncertainty, we need a history of previous information. To this end, we use a recurrent deterministic policy gradient (RDPG) algorithm, a recurrent and memory-based approach in DRL. Then, we compare the RDPG method in different scenarios with soft actor-critic (SAC), deep deterministic policy gradient (DDPG), distributed, and greedy algorithms. The simulation results show that the SAC method is better than the DDPG, distributed, and greedy methods, respectively. Moreover, the RDPG method out performs the SAC approach on average by 70%.

Joint Terminal Pairing and Multi-Dimensional Resource Allocation for Cooperative Computation in a WP-MEC System

The combination of multi-access edge computing (MEC) and wireless power transfer (WPT) can potentially improve system performance and prolong terminals’ battery lifetime. In this paper, we study cooperative computation in a wireless powered multi-access edge computing (WP-MEC) system. In this system, there are an access point (AP) deployed with an MEC server and multiple cellular terminals that are powered by harvested energy from a power station. The computational task of each terminal is partially completed locally and partially offloaded to the AP and an nearby idle terminal. In order to minimize energy consumption in the system, we jointly optimize terminal pairing and multi-dimensional resource (i.e., computing power and communication resource) allocation. A non-convex mixed-integer non-linear programming (MINLP) problem is formulated in this paper. To solve this challenging problem, we propose a bilevel optimization approach, in which the multi-dimensional resource allocation scheme is designed under given terminal pairing relationships in the inner level, and then the pairing relationships between terminals are obtained in the outer level based on the obtained multi-dimensional resource allocation scheme. Extensive numerical results show that the proposed bilevel optimization scheme can reduce energy consumption considerably compared to some baseline schemes.

Latency Minimization for Advanced RSMA-Enabled Wireless Caching Networks

The collaboration between wireless edge caching and rate splitting multiple access (RSMA) can significantly reduce the perceived latency of users. In this letter, an advanced RSMA transmission scheme is proposed with the consideration of users’ content requests. In traditional RSMA, multiple replicas of the same messages are transmitted when users request the same contents, which cause the redundant interference. The advanced RSMA is proposed to reduce such redundant interference. Based on the newly designed transmission paradigm, we investigate the joint cache placement and RSMA resource management problem, aiming to minimize the average latency of all users. The formulated problem is a non-convex mixed-integer nonlinear programming (MINLP) problem. By leveraging its structural features, we decompose the original minimization problem into two types of subproblems, i.e., the long-term caching decision problem and the short-term RSMA resource allocation problem. The caching decision problem is tackled by the dynamic programming approach. Whilst the RSMA resource allocation problem is solved by an alternating optimization method with one-dimensional search. Extensive simulation results validate the superiority of our proposed scheme compared to various benchmark strategies.

Optimal reliable design of energy-efficient Wireless Body Area Networks

A Wireless Body Area Network (WBAN) is a network of wireless devices that are used to monitor the health of individuals or provide them with information pertinent to their personal health. A WBAN typically consists of multiple sensors and a central sink and it enables the tracking of multiple body signals without the need for cables or other wired connections, allowing for a more comfortable user experience. The used sensors are extremely small and lightweight, making them well-suited for long-term medical monitoring. In this paper, we consider the joint problem of device positioning and data routing in WBANs leveraging relay-based multi-hop communication to address network energy consumption and reliable data transmission, which are two fundamental challenges in the WBAN design. The deployment of a WBAN is the most important factor that impacts both the network lifetime and reliability. On the other side, routing in multi-hop WBANs is of paramount importance for efficient communications between sensors and the centralized hub. For this reason, we define and assign to each relay a concave and non-increasing function (called reliability function) which permits to decrease consistently congestion occurrence in the network by balancing relays loads. In this work, we consider the problem of designing an energy-aware routing on WBANs while ensuring that all data are delivered to the sink with a sufficiently large probability. This problem is formulated as a non-convex mixed-integer non-linear programming problem and proven to be NP-hard. We first present an efficient linearization technique for the problem and then propose lower- and upper-bounding schemes for the problem, which can be applied within a cutting plane algorithm-based heuristic to solve the problem. Finally, we evaluate the proposed model with a real topology scenario and compare its performance with the most notable methods presented in the literature. Numerical results show that our model, independently of patient body movement, can transfer data through reliable paths from biosensors to the sink while further offering these options: (1) the probability that each biosensor successfully transfers its data to the sink is adjustable according to the desired application; (2) congestion is limited through a careful balancing of incoming traffic to relays, and (3) a small number of relays is installed, while ensuring an energy-efficient routing of data.

Optimal design-for-control of self-cleaning water distribution networks using a convex multi-start algorithm

The provision of self-cleaning velocities has been shown to reduce the risk of discolouration in water distribution networks (WDNs). Despite these findings, control implementations continue to be focused primarily on pressure and leakage management. This paper considers the control of diurnal flow velocities to maximize the self-cleaning capacity (SCC) of WDNs. We formulate a new optimal design-for-control problem where locations and operational settings of pressure control and automatic flushing valves are jointly optimized. The problem formulation includes a nonconvex objective function, nonconvex hydraulic conservation law constraints, and binary variables for modelling valve placement, resulting in a nonconvex mixed integer nonlinear programming (MINLP) optimization problem. Considering the challenges with solving nonconvex MINLP problems, we propose a heuristic algorithm which combines convex relaxations (with domain reduction), a randomization technique, and a multi-start strategy to compute feasible solutions. We evaluate the proposed algorithm on case study networks with varying size and degrees of complexity, including a large-scale operational network in the UK. The convex multi-start algorithm is shown to be a more robust solution method compared to an off-the-shelf genetic algorithm, finding good-quality feasible solutions to all design-for-control numerical experiments. Moreover, we demonstrate the implemented multi-start strategy to be a fast and scalable method for computing feasible solutions to the nonlinear SCC control problem. The proposed method extends the control capabilities and benefits of dynamically adaptive networks to improve water quality in WDNs.

Energy-Efficient Resource Allocation for Short Packet Transmission in MISO Multicarrier NOMA

In this article, we investigate energy efficiency (EE) maximization of short packet transmission (SPT) in a downlink multiple-input single-output (MISO) multi-carrier non-orthogonal multiple access (MC-NOMA) communication system. We use the ratio of the effective throughput to the power as a performance metric to formulate a non-convex mixed-integer nonlinear programming (MINLP) problem. Then, a low-complexity three-step optimization framework is proposed for MINLP optimization. Under a given maximum transmit power and channel blocklength of each subcarrier, a block coordinate descent (BCD) based method is proposed to jointly optimize the power allocation coefficient, transmission rate, and transmit power within a single subcarrier. Then, we transform the subcarrier allocation problem into a weighted bipartite graph matching problem and obtain the optimal subcarrier allocation scheme using the Kuhn-Munkres (KM) algorithm. Finally, a dynamic programming (DP) method is proposed to obtain the optimal channel blocklength allocation scheme. The simulation results validate the effectiveness of the proposed three-step optimization framework and indicate that the MC-NOMA system achieves higher EE and higher effective throughput than the orthogonal frequency-division multiple access (OFDMA) system in SPT.

Task Allocation for Energy Optimization in Fog Computing Networks With Latency Constraints

Fog networks offer computing resources of varying capacities at different distances from end users. A Fog Node (FN) closer to the network edge may have less powerful computing resources compared to the cloud, but the processing of computational tasks in FN limits long-distance transmission. How should the tasks be distributed between fog and cloud nodes? We formulate a universal non-convex Mixed-Integer Nonlinear Programming (MINLP) problem minimizing task transmission- and processing-related energy with delay constraints to answer this question. It is transformed with Successive Convex Approximation (SCA) and decomposed using the primal and dual decomposition techniques. Two practical algorithms called Energy-EFFicient Resource Allocation (EEFFRA) and Low-Complexity (LC)-EEFFRA are proposed and their effectiveness is tested for various network and traffic scenarios. Using EEFFRA/LC-EEFFRA can significantly decrease the number of computational requests with unmet delay requirements when compared with baseline solutions (from 48% to 24% for 10 MB requests). Utilizing Dynamic Voltage and Frequency Scaling (DVFS) minimizes energy consumption (by one-third) while satisfying delay requirements.

Refrigeration system synthesis based on de-redundant model by particle swarm optimization algorithm

Simultaneous optimization of refrigeration system (RS) and its heat exchanger network (HEN) leads to a large-scale non-convex mixed-integer non-linear programming (MINLP) problem. Conventionally, researchers usually adopted simplifications to confine problem scale from being too large at the cost of reducing solution space. This study established an optimization framework for the simultaneous optimization of RS and HEN. Firstly, A more comprehensive and compact model was developed to guarantee a relatively complete solution space while reducing model scale as well as its solving difficulty. In this model, a tandem arrangement of connecting sub-coolers and expansion valves was considered in the superstructure; and the pressure/temperature levels were optimized as continuous variables. On this basis, we proposed a “two-step transformation method” to equivalently transform the cross-level structure into a non-cross-level structure, and the de-redundant superstructure was established with ensuring comprehensiveness and rigor. Furthermore, the MINLP model was developed and solved by Particle Swarm Optimization algorithm. Finally, our methodology was validated to get better optimal results with less CPU time in two case studies, an ethylene RS in an existing plant and a reported propylene RS.

Joint Subcarrier and Transmission Power Allocation in OFDMA-Based WPT System for Mobile-Edge Computing in IoT Environment

Mobile-edge computing (MEC) is expected to play an important role in the next-generation of Internet-of-Things (IoT) services with artificial intelligence (AI) by providing the sustainable computation capability of resource-constrained IoT devices. Since the finite battery lifetime has been a longstanding challenge of the MEC system for IoT services, the wireless power transfer (WPT) technology has been recently developed for the MEC system in order to support the perpetual operation of IoT devices. In this article, we introduce two resource allocation problems for OFDMA-based WPT-MEC systems: 1) a max–min energy fairness (MMEF) problem and 2) a power sum maximization (PSM) problem. These problems ensure high-performance computations for AI-based applications, where the network reliability and the tremendous power consumption may be required. Moreover, we incorporate a logarithmic nonlinear energy harvesting (EH) model into our formulated problems, which lead to nonconvex mixed-integer nonlinear programming (MINLP) problems. In order to resolve these NP-hard problems, we convert the proposed problems into their equivalent convex forms by applying the continuous relaxation method. The near-optimal solutions are thereby obtained in closed-form expressions by leveraging the Lagrangian duality method for each relaxed problem. Numerical results are presented to validate the merits of the proposed algorithms of MMEF&PSM over the alternative benchmark algorithm and provide significant insights on the effects of the key system parameters, including the number of IoT devices and power transmission subcarriers.

Residual Energy Maximization for Wireless Powered Mobile Edge Computing Systems With Mixed-Offloading

This paper studies a joint design of resource allocation and task offloading in a wireless powered mobile edge computing network involving different types of computation tasks. To deal with diverse computation tasks, we explore a mixed-offloading paradigm to support the coexistence of partial and binary offloading modes. Specifically, devices harvest energy from an access point (AP) via wireless power transfer (WPT) and utilize the harvested energy to execute their computation tasks using partial or binary offloading. Based on a practical non-linear energy harvesting model, a residual energy maximization problem is formulated by jointly optimizing the transmit power of the AP, the offloading power of devices, the time allocation on WPT and task offloading, and the task partitions and the binary offloading decisions of devices, which turn out to be a non-convex mixed-integer non-linear programming problem. Thus, we develop an efficient dual-layer optimization algorithm by decomposing the optimization problem into an inner and outer layer structure that aims to obtain resource allocation and offloading decisions. Simulation results show that our proposed scheme achieves residual energy gains compared to existing schemes.

Green resource allocation for mobile edge computing

We investigate the green resource allocation to minimize the energy consumption of the users in mobile edge computing systems, where task offloading decisions, transmit power, and computation resource allocation are jointly optimized. The considered energy consumption minimization problem is a non-convex mixed-integer non-linear programming problem, which is challenging to solve. Therefore, we develop a joint search and Successive Convex Approximation (SCA) scheme to optimize the non-integer variables and integer variables in the inner loop and outer loop, respectively. Specifically, in the inner loop, we solve the optimization problem with fixed task offloading decisions. Due to the non-convex objective function and constraints, this optimization problem is still non-convex, and thus we employ the SCA method to obtain a solution satisfying the Karush-Kuhn-Tucker conditions. In the outer loop, we optimize the offloading decisions through exhaustive search. However, the computational complexity of the exhaustive search method is greatly high. To reduce the complexity, a heuristic scheme is proposed to obtain a sub-optimal solution. Simulation results demonstrate the effectiveness of the developed schemes.

Non-convex nested Benders decomposition

We propose a new decomposition method to solve multistage non-convex mixed-integer (stochastic) nonlinear programming problems (MINLPs). We call this algorithm non-convex nested Benders decomposition (NC-NBD). NC-NBD is based on solving dynamically improved mixed-integer linear outer approximations of the MINLP, obtained by piecewise linear relaxations of nonlinear functions. Those MILPs are solved to global optimality using an enhancement of nested Benders decomposition, in which regularization, dynamically refined binary approximations of the state variables and Lagrangian cut techniques are combined to generate Lipschitz continuous non-convex approximations of the value functions. Those approximations are then used to decide whether the approximating MILP has to be dynamically refined and in order to compute feasible solutions for the original MINLP. We prove that NC-NBD converges to an varepsilon -optimal solution in a finite number of steps. We provide promising computational results for some unit commitment problems of moderate size.

A Branch and Bound Algorithm for Transmission Network Expansion Planning Using Nonconvex Mixed-Integer Nonlinear Programming Models

The branch and bound (BB) algorithm is widely used to obtain the global solution of mixed-integer linear programming (MILP) problems. On the other hand, when the traditional BB structure is directly used to solve nonconvex mixed-integer nonlinear programming (MINLP) problems, it becomes ineffective, mainly due to the nonlinearity and nonconvexity of the feasible region of the problem. This article presents the difficulties and ineffectiveness of the direct use of the traditional BB algorithm for solving nonconvex MINLP problems and proposes the formulation of an efficient BB algorithm for solving this category of problems. The algorithm is formulated taking into account particular aspects of nonconvex MINLP problems, including (i) how to deal with the nonlinear programming (NLP) subproblems, (ii) how to detect the infeasibility of an NLP subproblem, (iii) how to treat the nonconvexity of the problem, and (iv) how to define the fathoming rules. The proposed BB algorithm is used to solve the transmission network expansion planning (TNEP) problem, a classical problem in power systems optimization, and its performance is compared with the performances of off-the-shelf optimization solvers for MINLP problems. The results obtained for four test systems, with different degrees of complexity, indicate that the proposed BB algorithm is effective for solving the TNEP problem with and without considering losses, showing equal or better performance than off-the-shelf optimization solvers.