Nonconvex Nonlinear Programming Research Articles

A Novel Swarm Exploring Varying Parameter Recurrent Neural Network for Solving Non-Convex Nonlinear Programming.

Aiming at solving non-convex nonlinear programming efficiently and accurately, a swarm exploring varying parameter recurrent neural network (SE-VPRNN) method is proposed in this article. First, the local optimal solutions are searched accurately by the proposed varying parameter recurrent neural network. After each network converges to the local optimal solutions, information is exchanged through a particle swarm optimization (PSO) framework to update the velocities and positions. The neural network searches for the local optimal solutions again from the updated position until all the neural networks are searched to the same local optimal solution. For improving the global searching ability, wavelet mutation is applied to increase the diversity of particles. Computer simulations show that the proposed method can solve the non-convex nonlinear programming effectively. Compared with three existing algorithms, the proposed method has advantages in accuracy and convergence time.

A Novel Swarm-Exploring Neurodynamic Network for Obtaining Global Optimal Solutions to Nonconvex Nonlinear Programming Problems.

A swarm-exploring neurodynamic network (SENN) based on a two-timescale model is proposed in this study for solving nonconvex nonlinear programming problems. First, by using a convergent-differential neural network (CDNN) as a local quadratic programming (QP) solver and combining it with a two-timescale model design method, a two-timescale convergent-differential (TTCD) model is exploited, and its stability is analyzed and described in detail. Second, swarm exploration neurodynamics are incorporated into the TTCD model to obtain an SENN with global search capabilities. Finally, the feasibility of the proposed SENN is demonstrated via simulation, and the superiority of the SENN is exhibited through a comparison with existing collaborative neurodynamics methods. The advantage of the SENN is that it only needs a single recurrent neural network (RNN) interact, while the compared collaborative neurodynamic approach (CNA) involves multiple RNN runs.

Adaptive MPC for Burst Incident Management in Water Distribution Networks

This paper investigates an adaptive control strategy which takes advantage of remotely actuated pumps and valves to improve the performance of water distribution networks (WDN) during burst incidents. The proposed strategy relies on model predictive control (MPC) to switch between problem formulations corresponding to normal operating conditions and burst conditions. The optimal control problems are formulated as non-convex non-linear programs (NLP), where the action of fixed-speed pumps and control valves (non-return and bi-directional) is modeled exactly and burst remedial interventions are represented using time-dependent connectivity matrices. We propose an efficient solution method based on reformulations and approximations of the non-convex objectives and constraints and apply the proposed strategy to the adaptive control of an operational WDN during a burst incident. The numerical experiments show that the computed burst control settings allow to partially or completely restore customer supply following the detection of a burst while maintaining minimum water levels in the network tanks and without increasing the cost of pumping operations. The results suggest that the proposed adaptive control strategy can be implemented alone, to mitigate the impact of bursts incidents in WDNs at minimum cost, or to complement traditional remedial interventions.

Optimal capacity sizing of park‐and‐ride lots with information‐aware commuters

We study capacity sizing of park‐and‐ride lots that offer services to commuters sensitive to congestion and parking availability information. The goal is to determine parking lot capacities that maximize the total social welfare for commuters whose parking lot choices are predicted using the multinomial logit model. We formulate the problem as a nonconvex nonlinear program that involves a lower and an upper bound on each lot's capacity, and a fixed‐point constraint reflecting the effects of parking information and congestion on commuters' lot choices. We show that except for at most one lot, the optimal capacity of each lot takes one of three possible values. Based on analytical results, we develop a one‐variable search algorithm to solve the model. We learn from numerical results that the optimal capacity of a lot with a high intrinsic utility tends to be equal to the upper bound. By contrast, a lot with a low or moderate‐sized intrinsic utility tends to attain an optimal capacity on its effective lower bound. We evaluate the performance of the optimal solution under different choice scenarios of commuters who are shared with real‐time parking information. We learn that commuters are better off in an average choice scenario when both the effects of parking information and congestion are considered in the model than when either effect is ignored from the model.

A signomial programming-based approach for multi-echelon supply chain disruption risk assessment with robust dynamic Bayesian network

Disruption risk assessment is a primary and crucial step before taking measures to mitigate the negative impact of disruptions propagating along supply chains (SCs). Recently, robust dynamic Bayesian network (DBN) provides a valid tool for disruption risk estimation under the ripple effect in a data-scarce environment. However, existing literature has not considered such disruption risk assessment for multi-echelon SCs that are usually structurally complicated and thus vulnerable to disruptions with ripple effects. Motivated by this fact, we study the disruption risk assessment problem under the ripple effect for a multi-echelon SC with several suppliers and one manufacturer, in which only probability intervals of the suppliers’ states and those of the related disruption propagations are known. The aim is to acquire a robust risk estimation, measured by the worst-case total weighted probabilities for the manufacturer in the disrupted state over a time horizon. For the problem, a nonconvex nonlinear programming model is established to obtain the worst-case risk estimation. To efficiently solve the problem, a novel signomial programming (SP)-based approach is developed for finding near-optimal solutions. Numerical experiments on instances in the literature and our randomly generated instances are conducted to evaluate the efficiency of the proposed method. Besides, managerial insights are drawn.

Efficient Day-Ahead Scheduling of PV-STATCOMs in Medium-Voltage Distribution Networks Using a Second-Order Cone Relaxation

This paper utilizes convex optimization to implement a day-ahead scheduling strategy for operating a photovoltaic distribution static compensator (PV-STATCOM) in medium-voltage distribution networks. The nonlinear non-convex programming model of the day-ahead scheduling strategy is transformed into a convex optimization model using the second-order cone programming approach in the complex domain. The main goal of efficiently operating PV-STATCOMs in distribution networks is to dynamically compensate for the active and reactive power generated by renewable energy resources such as photovoltaic plants. This is achieved by controlling power electronic converters, usually voltage source converters, to manage reactive power with lagging or leading power factors. Numerical simulations were conducted to analyze the effects of different power factors on the IEEE 33- and 69-bus systems. The simulations considered operations with a unity power factor (active power injection only), a zero power factor (reactive power injection only), and a variable power factor (active and reactive power injections). The results demonstrated the benefits of dynamic, active and reactive power compensation in reducing grid power losses, voltage profile deviations, and energy purchasing costs at the substation terminals. These simulations were conducted using the CVX tool and the Gurobi solver in the MATLAB programming environment.

Delay-Tolerant UAV-Assisted Communication: Online Trajectory Design and User Association

UAV-assisted communication is a promising paradigm to enhance the performance of wireless networks. In this paper, we consider the network randomness in the delay-tolerant UAV-assisted communication systems. We aim to optimize both the UAV trajectory and user association in an online manner to achieve a finite queueing delay at the users' buffers with the minimum energy consumption of the UAVs. We first formulate our problem as a stochastic dynamic programming problem that integrates the future expected energy consumption and delay into its objective function and constraints, respectively. To deal with the inaccurate knowledge of future user demand and mobility, we apply the Lyapunov optimization technique and convert it into a series of online joint UAV trajectory and user association optimization problems. However, each online problem is a non-convex non-linear integer programming problem, which is very challenging to solve in general. As such, we first eliminate the coupling relationship of user association variables in each online problem. By exploiting the special graph structure of the problem, we can decompose the joint problem into inner optimization and outer optimization problems. Then, we propose the online UAV mobility and user association selection (UMUAS) scheme, which includes two steps to solve the inner and outer problems. Simulation results show that our online UMUAS scheme achieves the near-optimal performance of the energy-delay tradeoff, which also has a 44.2% improvement over the greedy scheme and a 51.3% improvement over the interference-free scheme. Moreover, it achieves the highest energy efficiency among the three benchmark schemes.

Resource Allocation in Backscatter-Assisted Wireless Powered MEC Networks With Limited MEC Computation Capacity

In this paper, we consider a backscatter-assisted wireless powered mobile edge computing (MEC) network, where multiple Internet-of-Things (IoT) nodes harvest energy from the energy signals transmitted by a power beacon (PB) and utilize the harvested energy for local computing and task offloading via hybrid backscatter communication (BackCom) and active transmission (AT). Considering the limited computation capacity of the MEC server and the quality-of-service (QoS) and energy-causality constraints per IoT node, we propose two resource allocation schemes to maximize the total computation bits of all the IoT nodes and the system computation energy efficiency (EE), respectively, by jointly optimizing the computation frequency and time of the MEC server and each IoT node, the transmit power of the PB and each IoT node, and the BackCom power reflection coefficient and the time for energy harvesting (EH), BackCom, and AT of each IoT node. The non-convex computation bits maximization problem is transformed to a convex one by introducing a series of auxiliary variables and proof by contradiction, and then solved by the existing convex tools. The system computation EE maximization is a non-convex nonlinear programming problem. We propose a two-layer iterative algorithm to solve it optimally and devise a reduced-complexity iterative algorithm to solve it sub-optimally by leveraging the block coordinate decent technique. Computer simulations validate the convergence of the proposed iterative algorithms and their superior performance over the benchmark schemes in terms of the computation bits or EE.

An Enhanced Second-Order Cone Programming-Based Evaluation Method on Maximum Hosting Capacity of Solar Energy in Distribution Systems with Integrated Energy

In order to adjust to the change of the large-scale deployment of photovoltaic (PV) power generation and fully exploit the potentialities of an integrated energy distribution system (IEDS) in solar energy accommodation, an evaluation method on maximum hosting capacity of solar energy in IEDS based on convex relaxation optimization algorithm is proposed in this paper. Firstly, an evaluation model of maximum hosting capacity of solar energy for IEDS considering the electrical-thermal comprehensive utilization of solar energy is proposed, in which the maximization of PV capacity and solar collector (SC) capacity are fully considered. Secondly, IEDS’s potential in electricity, heat, and gas energy coordinated optimization is fully exploited to enhance the hosting capacity of solar energy in which the electric distribution network, heating network, and natural gas network constraints are fully modeled. Then, an enhanced second-order cone programming (SOCP)-based method is employed to solve the proposed maximum hosting capacity model. Through SOCP relaxation and linearization, the original nonconvex nonlinear programming model is converted into the mixed-integer second-order cone programming model. Meanwhile, to ensure the exactness of SOCP relaxation and improve the computation efficiency, increasingly tight linear cuts of distribution system and natural gas system are added to the SOCP relaxation. Finally, an example is given to verify the effectiveness of the proposed method. The analysis results show that the maximum hosting capacity of solar energy can be improved significantly by realizing the coordination of an integrated multi-energy system and the optimal utilization of electricity, heat, and gas energy. By applying SOCP relaxation, linearization, and adding increasingly tight linear cuts of distribution system and natural gas system to the SOCP relaxation, the proposed model can be solved accurately and efficiently.

Full Life-Cycle Optimal Battery Scheduling for Maximal Lifetime Value Considering Degradation

The dilemma between battery operating revenue and aging has not been addressed appropriately by previous studies. This paper presents a co-optimization method of battery full life-cycle scheduling and lifespan for its maximal lifetime value, involving usage limits, nonlinear aging characteristics, and time-value-of-money effects. The resulting dispatch model consists of numerous large-scale non-convex nonlinear programming (NLP) problems with the nonlinearity appearing in the equality constraint, which cannot be solved efficiently through conventional methods. Thus, we design an exact and fast solution approach including the equivalent problem transformation, a modified pegging algorithm, and a novel acceleration algorithm. The proposed methods are illustrated and validated by numerical tests using real historical data from Pecan St. Project. Test results suggest that it is not the case that the longer service life, the more battery lifetime profits can be generated. The proposed scheduling method can improve the lifetime profits than the conventional methods. For a battery with a 15-year design life, the optimal scheduling and lifespan with a 5-day resolution can be obtained within half an hour.

ESN Reinforcement Learning for Spectrum and Flight Control in THz-Enabled Drone Networks

Terahertz (THz)-band communications have been envisioned as a key technology to support ultra-high-data-rate applications in 5G-beyond (or 6G) wireless networks. Compared to the microwave and mmWave bands, the main challenges with the THz band are in its i) large path loss hence limited network coverage and ii) visible-light-like propagation characteristics hence poor support of mobility in blockage-rich environments. This paper studies quantitatively the applicability of THz-band communications in blockage-rich mobile environments, focusing on a new network scenario called FlyTera. In FlyTera, a set of hotspots mounted on flying drones collaboratively provide data streaming services to ground users, in the microwave, mmWave and THz bands. We first provide a mathematical formulation of the FlyTera control problem, where the objective is to maximize the network spectral efficiency by jointly controlling the flight of the drone hotspots, their association to the ground users, and the spectrum bands used by the users. To solve the resulting problem, which is shown to be a mixed integer nonlinear nonconvex programming (MINLP) problem, we design distributed solution algorithms based on a combination of echo state learning and reinforcement learning. An extensive simulation campaign is then conducted with SimBAG, a newly developed Simulator of Broadband Aerial-Ground wireless networks. It is shown that no single spectrum band can meet the requirements of high data rate and wide coverage simultaneously. Moreover, from the network-level point of view, THz-band communications can significantly benefit from the mobility of the flying drones, and on average 4 - 6 times higher (rather than lower) throughput can be achieved in mobile than in static environments.

Mathematical programming formulations for the alternating current optimal power flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.

A fast, fully distributed nonlinear model predictive control algorithm with parametric sensitivity through Jacobi iteration

Centralized model predictive control is impractical for many complex systems due to communication burden and robustness issues. For these systems, distributed model predictive control (DMPC) is an alternative control strategy. In DMPC, the use of nonlinear first-principle model improves the prediction accuracy. However, it also brings about computational delay due to time-consuming optimization of large, non-convex nonlinear programs, which can then degrade the control performance. In this work, a fully distributed nonlinear model predictive control (DNMPC) algorithm is developed to accelerate control feedback. The input computation procedure contains background and online stages, in which prediction–correction mode is applied. In the background stage, the future state is predicted one step forward based on the nominal plant model. Each controller optimizes its own local input and exchanges latest information with other controllers to improve decision making. After distributed optimization, the local controllers collect optimality information to prepare for future computation. When the true state is available, the state prediction error can be calculated. Each controller formulates its local sensitivity equation based on parametric sensitivity. All the sensitivity equations are solved in parallel with application of the Jacobi iterative method. After solution, the nominal optimum is updated with the correction vector and then implemented to the plant. The theoretical analysis of the proposed method is presented. Four case studies are given to demonstrate the effectiveness of the proposed algorithm.

Convex Approximation Algorithm for AC/DC Distribution Network With Energy Router

The optimal operation model of AC/DC distribution network with energy router (ER) is essentially a nonconvex nonlinear programming (NLP) problem. In order to improve the feasibility of solving the model, a convex approximation algorithm is proposed in this work. The steady-state model of ER is developed with considering the loss characteristics and multiport coordinated control strategy. It is embedded in the optimization formulations of AC/DC network as basic operating equations. Then, using second-order cone relaxation technology, the power flow equations of AC and DC distribution networks are convexly relaxed. On this basis, the highly nonlinear operating model of ER is linearized by introducing a successive approximation approach. Therefore, the original NLP problem is transformed into the convex programming problem and the solution efficiency is improved. Meanwhile, an iterative solution algorithm is developed to ensure the accuracy of the convex approximation approach. Simulation results verify the feasibility and efficiency of the proposed algorithm.

Polyhedral approximation strategies for nonconvex mixed-integer nonlinear programming in SHOT

Different versions of polyhedral outer approximation are used by many algorithms for mixed-integer nonlinear programming (MINLP). While it has been demonstrated that such methods work well for convex MINLP, extending them to solve nonconvex problems has traditionally been challenging. The Supporting Hyperplane Optimization Toolkit (SHOT) is a solver based on polyhedral approximations of the nonlinear feasible set of MINLP problems. SHOT is an open source COIN-OR project, and is currently one of the most efficient global solvers for convex MINLP. In this paper, we discuss some extensions to SHOT that significantly extend its applicability to nonconvex problems. The functionality include utilizing convexity detection for selecting the nonlinearities to linearize, lifting reformulations for special classes of functions, feasibility relaxations for infeasible subproblems and adding objective cuts to force the search for better feasible solutions. This functionality is not unique to SHOT, but can be implemented in other similar methods as well. In addition to discussing the new nonconvex functionality of SHOT, an extensive benchmark of deterministic solvers for nonconvex MINLP is performed that provides a snapshot of the current state of nonconvex MINLP.

Energy-Efficient Communications in Unmanned Aerial Relaying Systems

Due to flexible deployment and low cost, an unmanned aerial relaying (UAR) system has been considered as a promising aerial communication platform to complement current terrestrial infrastructure in some scenarios like temporary hotspots, disaster areas, and complex terrains. However, practical UAR systems may not be able to provide ubiquitous and high-quality wireless connections, mainly due to the limited on-board energy of unmanned aerial vehicles (UAVs). Consequently, the UAVs are unable to hover in the sky for a long time, thus having to land frequently for energy replenishment. To save UAVs' energy, we propose a joint communication and placement design for energy-efficient communications in a multi-hop UAR system. Specifically, we formulate the Energy Consumption Rate minimization problem, called ECR, by jointly considering routing, data rate, transmission power, and UAV position. Since ECR is a non-convex nonlinear programming, we propose an iterative optimization method to solve ECR iteratively by first reducing ECR into two sub-problems ECR-C and ECR-P and then solving them sequentially at an iteration. Specifically, we develop an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -bounded approximation algorithm to solve ECR-C with a performance guarantee, and reformulate ECR-P as a global consensus optimization problem solved by an ADMM (alternating direction method of multipliers) method. Particularly, all UAVs locally update their own positions in parallel, reducing the computation and implementation complexity significantly. Extensive simulation results show the energy efficiency improvement by up to 20% compared with other existing algorithms.

A risk-sensitive task offloading strategy for edge computing in industrial Internet of Things

Edge computing has become one of the key enablers for ultra-reliable and low-latency communications in the industrial Internet of Things in the fifth generation communication systems and is also a promising technology in the future sixth generation communication systems. In this work, we consider the application of edge computing to smart factories for mission critical task offloading through wireless links. In such scenarios, although high end-to-end delays from the generation to completion of tasks happen with low probability, they may incur severe casualties and property loss and should be seriously treated. Inspired by the risk management theory widely used in finance, we adopt the Conditional Value at Risk to capture the tail of the delay distribution. An upper bound of the Conditional Value at Risk is derived through analysis of the queues both at the devices and the edge computing servers. We aim to find out the optimal offloading policy taking into consideration both the average and the worst-case delay performance of the system. Given that the formulated optimization problem is a non-convex mixed integer nonlinear programming problem, a decomposition into subproblems is performed and a two-stage heuristic algorithm is proposed. The simulation results validate our analysis and indicate that the proposed algorithm can reduce the risk in both the queueing and end-to-end delay.

Strong Convergence on the Aggregate Constraint-Shifting Homotopy Method for Solving General Nonconvex Programming

In the paper, the aggregate constraint-shifting homotopy method for solving general nonconvex nonlinear programming is considered. The aggregation is only about inequality constraint functions. Without any cone condition for the constraint functions, the existence and convergence of the globally convergent solution to the K-K-T system are obtained for both feasible and infeasible starting points under much weaker conditions.

Sample average approximation for stochastic nonconvex mixed integer nonlinear programming via outer-approximation

We propose a sample average approximation-based outer-approximation algorithm (SAAOA) that can address nonconvex two-stage stochastic programs (SP) with any continuous or discrete probability distributions. Previous work has considered this approach for convex two-stage SP (Wei and Realff in Comput Chem Eng 28(3):333–346, 2004). The SAAOA algorithm does internal sampling within a nonconvex outer-approximation algorithm where we iterate between a mixed-integer linear programming (MILP) master problem and a nonconvex nonlinear programming (NLP) subproblem. We prove that the optimal solutions and optimal value obtained by the SAAOA algorithm converge to the optimal solutions and the optimal value of the true SP problem as the sample size goes to infinity. The convergence rate is also given to estimate the sample size. Since the theoretical sample size estimate is too conservative in practice, we propose an SAAOA algorithm with confidence intervals for the upper bound and the lower bound at each iteration of the SAAOA algorithm. Two policies are proposed to update the sample sizes dynamically within the SAAOA algorithm with confidence intervals. The proposed algorithm works well for the special case of pure binary first stage variables and continuous stage two variables since in this case the nonconvex NLPs can be solved for each scenario independently. The proposed algorithm is tested with a stochastic pooling problem and is shown to outperform the external sampling approach where large scale MINLPs need to be solved.

Mathematical programming formulations for the alternating current optimal power flow problem

Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.