Deterministic Optimization Problem Research Articles

Robust trajectory planning for non-cooperative target rendezvous based on closed-loop uncertainty analysis

A robust trajectory planning method for non-cooperative target rendezvous is investigated based on closed-loop uncertainty analysis. The proposed method addresses the issue that uncertainties affect the optimality of objective functions and the reliability of constraints. Considering multiple uncertainties, a stochastic optimal rendezvous model with a robustness performance index and chance constraints is presented with line-of-sight dynamics. The guidance model, angles-only navigation model and control model are derived, and uncertainty propagation equations for closed-loop GN&C system is proposed by using linear covariance analysis. The stochastic optimal rendezvous model is transformed into a robust optimal rendezvous model by using 3-σ of the random variables. The genetic algorithm is used to solve the robust trajectory planning problem, and a series of examples demonstrate that the robust maneuver solution can be significantly better than the solution corresponding to deterministic trajectory optimization problem.

Online dynamic multi-user computation offloading and resource allocation for HAP-assisted MEC: an energy efficient approach

Nowadays, the paradigm of mobile computing is evolving from a centralized cloud model towards Mobile Edge Computing (MEC). In regions without ground communication infrastructure, incorporating aerial edge computing nodes into network emerges as an efficient approach to deliver Artificial Intelligence (AI) services to Ground Devices (GDs). The computation offloading and resource allocation problem within a HAP-assisted MEC system is investigated in this paper. Our goal is to minimize the energy consumption. Considering the randomness and dynamism of the task arrival of GDs and the quality of wireless communication, stochastic optimization techniques are utilized to transform the long-term dynamic optimization problem into a deterministic optimization problem. Subsequently, the problem is further decomposed into three sub-problems which can be solved in parallel. An online Energy Efficient Dynamic Offloading (EEDO) algorithm is proposed to address these problems. Then, we conduct the theoretical performance analysis for EEDO. Finally, we carry out parameter analysis and comparative experiments, demonstrating that the EEDO algorithm can effectively reduce system energy consumption while maintaining the stability of the system.

Uncertainty-resilient constrained rendezvous trajectory optimization via stochastic feedback control and unscented transformation

This paper proposes an innovative approach for constrained rendezvous trajectory optimization (CRTO) using stochastic optimal feedback control and unscented transform (UT) uncertainty quantification. The method overcomes limitations of traditional deterministic CRTO solutions that suffer from uncontrolled terminal state error growth and severely deteriorated path constraint satisfaction when initial state uncertainty, dynamics uncertainty, and navigation uncertainty are considered. The approach involves constructing a stochastic optimal feedback control (SOFC) problem with chance constraints and introducing linear feedback control to regulate both the mean and variance of terminal state errors. UT is employed to approximately quantify the state errors and their propagation, transforming the SOFC problem into an unconstrained deterministic optimization (UDO) problem. Differential dynamic programming (DDP) is then used to solve the UDO problem. The obtained stochastic optimal solution provides a robust rendezvous trajectory and a corresponding explicit closed-loop guidance law, improving terminal accuracy and path constraint satisfaction in the presence of various considered uncertainties. The influence of different term weights in the objective function on terminal accuracy and path constraint satisfaction is also studied. The effectiveness of the proposed approach is verified through Monte Carlo simulation, demonstrating the robustness of the closed-loop control strategy. The results highlight the potential of the method in enhancing terminal accuracy and path constraint satisfaction in uncertain rendezvous scenarios.

Chance-constrained UAM traffic flow optimization with fast disruption recovery under uncertain waypoint occupancy time

Trajectory-based operations offer a promising solution for effective urban air mobility (UAM) traffic management with conflict-free four-dimensional trajectories. However, these trajectories generated in strategic phases by existing methods could be significantly disrupted due to uncertainties such as flight delays and trajectory deviations. This paper develops a chance-constrained UAM traffic flow management (UTFM) optimization model with fast disruption recovery to solve observed disrupted trajectories and improve their resilience. Our model introduces a novel concept of waypoint occupancy time to cope with the probabilistic separation constraints induced by variables, notably urban wind field. We then convert the probabilistic constraint into a deterministic one, incorporating risk-bounded separation guarantees derived from flight experiment data. Furthermore, we develop a hierarchical stochastic search algorithm for solving the redefined deterministic optimization problem. Our comprehensive numerical studies demonstrate the model's effectiveness in restoring disrupted flights and resolving conflicts within seconds. Additionally, our reliability testing showcases the resilience of the model in managing disruptions, even at levels as high as 50%, and its adaptability in addressing varying levels of uncertainty risk. We further demonstrate the ability of the UTFM model to capture tail risks across Gaussian and non-Gaussian uncertainty distributions. Lastly, our scalability analysis highlights the potential capacity of the model to support up to 3,000 flights per hour in Singapore's urban airspace below 400 ft. This study introduces an adaptable framework to facilitate the modelling of robust traffic flow management and performance-based separation for a variety of eVTOL types under uncertainties.

Exploratory modelling and analysis to support decision-making under deep uncertainty: A case study from defence resource planning and asset management

Emerging approaches and tools in the literature for model-based decision support under deep uncertainty are rapidly growing. Exploratory modelling and analysis is one of the unifying ideas behind these approaches. This research leverages exploratory modelling and analysis in the context of a case study within defense resource planning for the Royal Australian Navy. We use a multi-method approach that combines exploratory modelling and analysis and a simulation model. Our research analyses a strategy from a deterministic optimization problem to assess the impact of uncertainties and to identify different model behaviours exhibited under uncertainties and the uncertainty space in which the strategy maintains reasonable performance. To achieve these objectives, we draw upon concepts from Robust Decision-Making, Behaviour-Based Scenario Discovery, and Feasible Scenario Space. The results are then analysed to derive decision-making insights and highlight the potential consequences of neglecting uncertainties. For the first time in defense resource planning, we compare the results from these three approaches and tools. These approaches and tools complement each other, leading to similar conclusions with unique insights. The results also reveal the misleading nature of considering the best approximate conditions in the predictive use of simulation models under deep uncertainty.

Optimum-pursuing method for constrained optimization and reliability-based design optimization problems using Kriging model

This paper proposes a new active learning method named as optimum-pursuing method (OPM) from the viewpoint of optimization theory, which aims to provide an effective tool for solving constrained optimization and reliability-based design optimization (RBDO) problems with low computation cost. It uses the cheap Kriging metamodel to replace the expensive physical response. The novelty of the proposed OPM primarily lies in two aspects. First, the OPM utilizes the advantage of the optimization theory rather than sampling technology. By using the augmented Lagrangian approach, it comprehensively considers the objective, constraints, and their relations, thereby automatic identification of important region in the vicinity of the optimum. Second, the accordingly optimum-pursuing function consists of three parts: Kriging mean, Kriging standard deviation, and merit function. Also, the target reliability surface is further considered to enhance the local accuracy of the reliability analysis. The performance of OPM is tested for both deterministic optimization and problems, in which two mathematical and three real-world engineering examples are selected to showcase the feasibility and validity. The results demonstrate that OPM is promising for solving both deterministic optimization and RBDO problems by comparing with the well-known active learning methods.

Robust transmission for multi-user OFDM-based IRS-assisted cognitive radio networks

In cognitive radio networks (CRNs) employing underlay multi-user scenarios, intelligent reflecting surface (IRS) plays a crucial function in augmenting total transmission rate for cognitive radio (CR) users, while adhering to restrictions caused by the interference from secondary users (SUs) to primary users (PUs). However, due to the passive characteristics of IRS, it operates in a non-cooperative relationship with both PU and SU network, resulting in delayed information acquisition and channel errors. Therefore, in this research, considering channel uncertainties, limited interference, and interference among SUs, we put forward a robust power and subcarrier allocation strategy for orthogonal frequency division multiplexing (OFDM) with IRS-assisted CRNs in a multi-user scenario. Formulated as a resource allocation (RA) problem in an underlay spectrum sharing mode for multi-user CR, the joint optimization problem encompasses subcarrier allocation and power allocation for SUs, a phase shift matrix for IRS, and minimum rate constraints. We transform the problem using worst-case criterion into a deterministic convex optimization problem to handle the constraints of bounded channel uncertainties. By utilizing matching theory, variable substitution, scaling methods, convex-concave programming (CCP), and alternating optimization techniques, the converted non-convex problem is transformed into a convex one. Simulation results showcase that the proposed algorithm achieves significant gains in transmission rate and exhibits robustness.

Detecting the most probable transition pathway based on optimal control theory

Many natural systems exhibit transitions when external environmental conditions (such as complex noise) spark a shift to a new and sometimes quite different state. Therefore, detecting the most probable transition pathway between metastable states of a stochastic dynamical system is a significant topic. The most probable transition pathway can be treated as the minimizer of the associated Onsager-Machlup action functional. We convert this variational problem for computing the most probable transition pathway into a deterministic optimal control problem. One traditional approach for an optimal control problem is via Pontryagin's Maximum Principle, but it is challenging in high dimensional systems. In this paper, we devise a method to detect the most probable transition pathway for stochastic dynamical systems, by combining Pontryagin's Maximum Principle with a successive approximation scheme and a nested neural network technique. We validate our method with three stochastic dynamical systems, including a double well system, a Maier-Stein system, and a Nutrient-Phytoplankton-Zooplankton system. Specifically, in order to illustrate its effectiveness, we test the error and convergence of our method with an upper bound of training loss. Our work contributes to a better understanding of transition phenomena in complex systems under random fluctuations.

Data-Driven Chance-Constrained Planning for Distributed Generation: A Partial Sampling Approach

The planning of distributed energy resources has been challenged by the significant uncertainties and complexities of distribution systems. To ensure system reliability, one often employs chance-constrained programs to seek a highly likely feasible solution while minimizing certain costs. The traditional sample average approximation (SAA) is commonly used to represent uncertainties and reformulate a chance-constrained program into a deterministic optimization problem. However, the SAA introduces additional binary variables to indicate whether a scenario sample is satisfied and thus brings great computational complexity to the already challenging distributed energy resource planning problems. In this paper, we introduce a new paradigm, i.e., the partial sample average approximation (PSAA) using real data, to improve computational tractability. The innovation is that we sample only a part of the random parameters and introduce only continuous variables corresponding to the samples in the reformulation, which is a mixed-integer convex quadratic program. Our extensive experiments on the IEEE 33-Bus and 123-Bus systems show that the PSAA approach performs better than the SAA because the former provides better solutions in a shorter time in in-sample tests and provides better guaranteed probability for system reliability in out-of-sample tests. All the data used in the experiments are real data acquired from Pecan Street Inc. and ERCOT. More importantly, our proposed chance-constrained model and PSAA approach are general enough and can be applied to solve other valuable problems in power system planning and operations. Thus, this paper fits one of the journal scopes: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Distribution System Planning in Power System Planning and Implementation</i> .

Decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods

Interval multi-objective optimization problems (IMOPs) are one of the most critical optimization problems in practical applications. However, compared to deterministic multi-objective optimization problems (MOPs), there are few researchs addressing IMOP. In addition, the uncertainty contained in the problem makes the distribution of the population more challenging. Therefore, this paper proposed a decomposition-based interval multi-objective evolutionary algorithm with adaptive adjustment of weight vectors and neighborhoods (IMOEA/D-AWN). Firstly, an interval sparsity level function (ISL) is constructed to measure the density of individuals, and a comprehensive ranking of interval sparsity ranking and interval uncertainty ranking is proposed. For the purpose of improving the distribution of the population while reducing its uncertainty, based on the above comprehensive ranking, a new adaptive adjustment weight vector strategy guided by interval elite population is designed. Besides, an adaptive adjustment neighborhoods strategy is designed. This strategy adjusts individuals' neighborhood size according to the number of iterations to improve the efficiency of evolution. Finally, the IMOEA/D-AWN is evaluated on 17 interval benchmark test problems and a collaborative computation offloading optimization problem, and compared with four advanced multi-objective evolutionary algorithms with interval parameters (IMOEAs). Experimental results show that this algorithm performs well in convergence, diversity, and uncertainty.

Robust convex optimization for reentry glide trajectory using polynomial chaos

Aiming at the reentry glide trajectory design of hypersonic vehicles with uncertain parameters, a robust trajectory optimization method is proposed to enhance the anti-interference ability and process reliability. Firstly, a robust trajectory optimization model for re-entry gliding under uncertain conditions is constructed, which is transformed into a deterministic trajectory optimization problem with dimension expansion by using an uncertainty propagation algorithm based on the Gaussian quadrature and non-intrusive polynomial chaos. Then, the convex optimization technology is used to convexity and discretize the problem, and a deterministic trajectory optimization strategy based on the sequential convex optimization algorithm is designed to achieve a fast solution for the high-dimensional deterministic problem. Finally, the hypersonic glide reentry of the American X-33 is chosen as an example for the numerical simulation. The results show that comparing with the traditional deterministic trajectory optimization algorithm, the present method can effectively reduce the impact of random interference on the reentry glide trajectory and improve the reliability and robustness of the trajectory.

Joint Optimization of Secure Over-the-Air Computation and Reliable Multicasting Assisted by a MIMO Untrusted Two-Way Relay

In this paper, we investigate joint optimization of secure AirComp and reliable multicasting assisted by a multiple-input multiple-output untrusted two-way relay, where artificial noise is employed at the access point (AP) to interfere the relay for ensuring secure AirComp. We aim to minimize the computation distortion at the AP by jointly designing the transmit variables of all the nodes and the aggregation variables at the AP and relay, under the secure AirComp constraint, the reliable multicasting constraint, and the transmit power constraints of all the nodes. We consider two scenarios that perfect and imperfect channel state information (CSI) are available. For the former, the formulated optimization problem is highly nonconvex, and we propose an effective block coordinate descent (BCD)-penalty successive convex approximation (penaltySCA) method to solve the nonconvex problem. For the latter, we model the imperfect CSI by using worst-case criterion and the formulated robust optimization problem is much more challenging than its counterpart with perfect CSI. To solve the robust problem effectively, we first transform it to a deterministic optimization problem by employing some powerful mathematical lemmas, and then apply the proposed BCD-penaltySCA method to solve the reformulated deterministic problem. The proposed methods are shown by simulations to significantly reduce the computation distortion compared with other benchmarks under considering secure AirComp and reliable multicasting.

Optimization Method of Photovoltaic Microgrid Energy Storage System Based on Price-based DR

Extreme events lead to an increasing number of power outages in the distribution network. It is currently the most effective method to restore power supply after distribution network failure to connect distributed photovoltaic to the distribution network in the form of microgrids. However, the randomness of distributed PV output and load is the biggest obstacle limiting its development. Therefore, an optimization method of photovoltaic microgrid energy storage system (ESS) based on price-based demand response (DR) is proposed in this paper. Firstly, based on the influence of the uncertainty of the time of use (TOU) and load on the price-based DR, a price-based DR model is built. Then, taking the minimum difference between photovoltaic output and load demand as the optimization objective, the optimal operation model of price-based DR based on the fuzzy chance constrained program (FCCP) is established, and an optimization model of photovoltaic microgrid ESS based on FCCP is obtained. Finally, the fuzzy chance optimization problem is converted into a deterministic optimization problem through constraint equivalent treatment. The proposed method is verified by numerical simulations.

Real-Time Energy Management for Marine Applications Using Markov Approximation

All-electric and hybrid-electric ships have become the centerpiece for reducing greenhouse gas emissions and improving fuel efficiency in the maritime industry. Real-time power management of multiple power sources in both design and operation becomes critical due to the uncertainty and randomness of the vessel's power demand. Furthermore, the vessel operators may have additional requirements for operational flexibility, such as fully recharging the battery before the next trip and limiting engine switch on/off to reduce wear-and-tear, which lead to complex power management systems requiring domain-expert knowledge. In this paper, a hybrid solution for real-time power management is proposed. The model-based approach encapsulates the power system, and the data-driven approach (a machine learning technique called Markov approximation) handles the uncertainty. First, an offline deterministic optimization problem for power management is formulated. Second, the deterministic problem is transformed into a maximum weighted independent set (MWIS) problem. Next, the Markov approximation framework is applied to the transformed problem to utilize the machine learning techniques widely used in wireless communications and computing industries. Finally, a reinforcement learning-based real-time solution is proposed. Extensive simulations are performed on a ferry case study, and the results are within the proven theoretical bounds of machine learning techniques.

Age-Optimal Downlink NOMA Resource Allocation for Satellite-Based IoT Network

The upcoming satellite-based Internet of Things (S-IoT) has the capability to provide timely status updates to massive terrestrial user equipments (UEs) via non-orthogonal multiple access (NOMA), due to the worldwide coverage inherited from satellite. Considering the constrained power and storage resources while keeping the information freshness in S-IoT, we first propose three constraint conditions including average/peak power constraints, network stability and minimum requirement. Then, we formulate a long-term age of information (AoI) minimization problem under the three constraint conditions. To solve this complex long-term problem, we transform the above mentioned constraints into three queue stability problems via the Lyapunov optimization framework, thus converting our long-term multi-slot stochastic optimization problem into a series of single time slot deterministic optimization problems. Moreover, we leverage the ListNet algorithm to derive the weights of the queue backlog and channel conditions to obtain an optimized power allocation order with linear complexity. Finally, we utilize the particle swarm optimization algorithm to derive the NOMA long-term AoI minimization (AM) power allocation problem within a practical complexity, named NOMA-AM scheme. Simulation results show that the proposed NOMA-AM scheme has the lowest expected weighted sum AoI compared to several benchmark schemes.

Uncertain optimization for initial disturbance of projectile for moving tank based on stochastic programming and deep learning surrogate model

To enhance the hitting accuracy of tank with moving firing, an uncertain optimization method based on stochastic programming is adopted to reduce the initial disturbance of projectile. Firstly, the firing dynamics model of moving tank is modeled to simulate the initial disturbance of projectile. Secondly, the controllable interior ballistic parameters such as projectile structure parameters and propellant parameters are treated as random design variables. The surrogate model for the firing dynamics model of moving tank is constructed by using the neural network based on deep learning. The uncertain optimization problem is transformed into a deterministic optimization problem by stochastic programming method. Then multi-objective genetic algorithm is adopted to settle optimization model, and reasonable design interval of random design variables is obtained. Finally, a six-degree-of-freedom rigid external ballistics model is used to establish a hitting accuracy evaluation model of moving tank based on interval uncertainty analysis. Through this model, the effectiveness of the optimization method is demonstrated.

Risk-based preventive energy management for resilient microgrids

A microgrid's energy management system (EMS) is typically formulated as a deterministic optimization problem. However, more risks and uncertainties are emerging due to the increasing complexity of the power system structure with intelligent distributed energy integration. As a result, different types of risks must be considered while implementing the EMS to improve system resiliency. In this paper, a risk-based EMS is introduced for islanded microgrids. A scenario-based optimization method is formulated for the EMS, where each system component's probability of failure (PoF) is considered in each scenario. In addition, the loads are classified into critical, semi-critical, and non-critical loads to prioritize serving the essential loads during a shortage of generation resources in microgrids. The nonlinear power flow equation of the new problem formulation is relaxed into a convex form to reduce the computational burden. In detail, the system's resiliency is improved by employing three functionalities in the EMS as a preventive measure: a) prioritizing to serve the critical loads, b) lowering the total amount of load served to mitigate the impact of the failure of any system component, and c) reducing the dependency on the generation resources with high PoF. The proposed preventive and risk-based EMS is validated against IEEE-14, 30, and 118 bus systems. The results demonstrate that the proposed EMS sheds more loads than the base EMS, based on the PoF of each system component, due to the knowledge that risks of failure of any component may induce a cascading failure and even blackout if only base EMS is provided. Moreover, the proposed EMS precisely considers the criticality of loads for curtailment to match the available system resources. The curtailment indirectly improves the system's resiliency by increasing the chance of survivability in the islanded mode with the probability of future component failure.

Convex risk-bounded continuous-time trajectory planning and tube design in uncertain nonconvex environments

In this paper, we address the trajectory planning problem in uncertain nonconvex static and dynamic environments that contain obstacles with probabilistic location, size, and geometry. To address this problem, we provide a risk-bounded trajectory planning method that looks for continuous-time trajectories with guaranteed bounded risk over the planning time horizon. Risk is defined as the probability of collision with uncertain obstacles. Existing approaches to address risk-bounded trajectory planning problems either are limited to Gaussian uncertainties and convex obstacles or rely on sampling-based methods that need uncertainty samples and time discretization. To address the risk-bounded trajectory planning problem, we leverage the notion of risk contours to transform the risk-bounded planning problem into a deterministic optimization problem. Risk contours are the set of all points in the uncertain environment with guaranteed bounded risk. The obtained deterministic optimization is, in general, nonlinear and nonconvex time-varying optimization. We provide convex methods based on sum-of-squares optimization to efficiently solve the obtained nonconvex time-varying optimization problem and obtain the continuous-time risk-bounded trajectories without time discretization. The provided approach deals with arbitrary (and known) probabilistic uncertainties, nonconvex and nonlinear, static and dynamic obstacles, and is suitable for online trajectory planning problems. In addition, we provide convex methods based on sum-of-squares optimization to build the max-sized tube with respect to its parameterization along the trajectory so that any state inside the tube is guaranteed to have bounded risk.

Interval Uncertainty Optimization Method for Electromagnetic Orbital Launcher

The performance optimization and reliability of electromagnetic orbital launchers need to take into account the uncertainties that exist in the manufacturing and service processes. Considering that it is difficult to identify the problem of the parameter probability distribution in advance for electromagnetic orbital launchers, this paper uses the interval number to describe the uncertain variables and realizes the conversion of uncertainty optimization into deterministic optimization problems based on interval sequential relations; moreover, it establishes interval uncertainty optimization methods. The converted deterministic optimization problem is solved by combining a high-precision proxy model with an optimization algorithm to search for the optimal solution set. Finally, reliability estimation is achieved by taking the armature of the electromagnetic orbital launcher as the optimization object. The computational example proves that the method is able to deal with the optimization problem of the parameter interval of an electromagnetic orbital launcher containing uncertain parameters and has good engineering applicability.

A neurodynamic algorithm for dependent joint chance constrained geometric programs

This research studies the use of copula theory to model dependencies in joint probabilistic constrained geometric programs with dependent rows. The row vectors are assumed to follow an elliptical distribution, and their dependencies are modeled through a Gumbel–Hougaard copula. We use a log transformation to convert the chance-constrained geometric program into a deterministic optimization problem. Then we solve the resulting deterministic program using a dynamical neural network. The stability and convergence of the proposed neural network approach are demonstrated. The primary characteristic of our framework is its ability to solve the dependent joint chance constrained geometric programs without resorting to any convex approximation methods. This feature sets our approach apart from the current state-of-the-art solving techniques. The neurodynamic algorithm is finally applied to solve three geometric optimization problems.