Nonlinear Optimization Problem Research Articles

Adaptive sliding mode control with nonlinear MPC-based obstacle avoidance using LiDAR for an autonomous surface vehicle under disturbances

This manuscript proposes a path-following and collision avoidance guidance and control system for an autonomous surface vehicle operating in cluttered environments under perturbations. A nonlinear model predictive control strategy handles the state and control constraints for obtaining a guidance law. The approach considers the vessel sideslip angle to provide the desired heading angle, satisfying path-following and obstacle avoidance. Additionally, to solve the nonlinear optimization problem for obtaining the guidance law in real-time, the acados software package is used. Furthermore, in order to guarantee that the vehicle tracks the guidance in presence of model uncertainties and external disturbances, a robust adaptive sliding mode low-level controller is adopted. This controller is finite time convergent, and dynamically adapts its control gain to use only the necessary control inputs. Furthermore, a low-memory and computationally-light obstacle detection strategy employing a LiDAR sensor is presented and proven to run in real-time. Simulation and experimental tests conducted using a customized prototype validate the advantages of the proposed strategy and demonstrate the effectiveness of the path following while evading multiple obstacles in the presence of external disturbances and model discrepancies.

A novel optimization framework using frequency-based substructuring for estimation of linear bolted joint stiffness and damping

Estimating the stiffness and damping coming from a joint is a major challenge. This work proposes a novel framework to estimate the unknown parameters using frequency-based substructuring and a maximum a posteriori optimization approach. The idea is to minimize the discrepancy between the measured responses of an assembled system and the responses obtained by coupling the measured responses of the individual substructures with a joint model. The system’s dynamics are assumed to be linear. However, the FRFs of the coupled system (which drive the objective function) change nonlinearly with the linear joint parameters, making it a nonlinear optimization problem. Therefore, the joint parameters are estimated iteratively using the Gauss minimization method, a well-established linearization scheme. The framework is numerically validated on simple mdof systems, and the method’s effectiveness is tested by adding noise to the generated data. To demonstrate the framework’s applicability on real structures, joint stiffness and damping of a known benchmark structure are estimated based on real measurements. Two different underlying joint models are applied. The first assumes just flexibility at the connection. The second assumes the joint is a separate substructure. The algorithm can find realistic stiffness parameters and a range for the damping parameters providing a good compromise between the measurements in both cases.

An Advanced Control Method for Aircraft Carrier Landing of UAV Based on CAPF–NMPC

This paper investigates a carrier landing controller for unmanned aerial vehicles (UAVs), and a nonlinear model predictive control (NMPC) approach is proposed considering a precise motion control required under dynamic landing platform and environment disturbances. The NMPC controller adopts constraint aware particle filtering (CAPF) to predict deck positions for disturbance compensation and to solve the nonlinear optimization problem, based on a model establishment of carrier motion and wind field. CAPF leverages Monte Carlo sampling to optimally estimate control variables for improved optimization, while utilizing constraint barrier functions to keep particles within a feasible domain. The controller considers constraints such as fuel optimization, control saturation, and flight safety to achieve trajectory control. The advanced control method enhances the solution, estimating optimal control sequences of UAV and forecasting deck positions within a moving visual field, with effective trajectory tracing and higher control accuracy than traditional methods, while significantly reducing single-step computation time. The simulation is carried out using UAV “Silver Fox”, considering several scenarios of different wind scales compared with traditional CAPF–NMPC and the nlmpc method. The results show that the proposed NMPC approach can effectively reduce control chattering, with a landing error in rough marine environments of around 0.08 m, and demonstrate improvements in trajectory tracking capability, constraint performance and computational efficiency.

A dual scale spatio‐temporal control for complex distributed parameter systems

AbstractIt is very difficult to control the distributed parameter system (DPS) for consistent spatial performance. In this paper, a dual scale spatio‐temporal control is designed for the DPS to achieve a consistent performance across the entire workspace under unknown exogenous disturbances. First, the generalized “spatial observer” should be designed under spectral decomposition/synthesis, to act as the nominal model of the DPS, through which the spatial mismatch between the model and the process could be estimated. Then a distributed disturbance compensator can be constructed to suppress all the undesirable spatial disturbance through the inner loop, and the compensated system will become closer to the nominal model and easier to control. Third, a dual controller will be designed in the outer loop for the final consistent spatial performance. Since the spatial performance is mainly affected by the dominant dynamics on the slow scale, a convex optimization algorithm can be effectively established in terms of nonlinear matrix inequalities for the dual controller. In this way, the nonlinear optimization problem is solved by converting it into the problem of a linear matrix inequalities with constraints. Besides, the spatio‐temporal state variable of the controlled plant is demonstrated to converge in Hilbert space. The feasibility and effectiveness of the proposed control method are verified by temperature control of a catalytic rod, a benchmark in chemical reactors.

A novel discrete zeroing neural network for online solving time-varying nonlinear optimization problems.

To reduce transportation time, a discrete zeroing neural network (DZNN) method is proposed to solve the shortest path planning problem with a single starting point and a single target point. The shortest path planning problem is reformulated as an optimization problem, and a discrete nonlinear function related to the energy function is established so that the lowest-energy state corresponds to the optimal path solution. Theoretical analyzes demonstrate that the discrete ZNN model (DZNNM) exhibits zero stability, effectiveness, and real-time performance in handling time-varying nonlinear optimization problems (TVNOPs). Simulations with various parameters confirm the efficiency and real-time performance of the developed DZNNM for TVNOPs, indicating its suitability and superiority for solving the shortest path planning problem in real time.

Irreversible reinsurance: minimization of capital injections in presence of a fixed cost

AbstractWe propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cramér-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model’s parameters.

MPCC: strong stability of weakly nondegenerate S-stationary points

ABSTRACT In this paper, we consider the class of mathematical programmes with complementarity constraints (MPCC). Specifically, we focus on strong stability of M- and S-stationary points for MPCC. Kojima introduced this concept for standard nonlinear optimization problems. It refers to several well-posedness properties of the underlying problem. Besides its topological definition, the challenge is to state an algebraic characterization of strong stability. We obtain such a description for S-stationary points whose components of Lagrange vectors corresponding to bi-active constraints do not mutually vanish. We call these points weakly nondegenerate. Moreover, we show that a particular constraint qualification is necessary for strong stability.

Development of Steady-State and Dynamic Mass and Energy Constrained Neural Networks for Distributed Chemical Systems Using Noisy Transient Data.

The paper presents the development of algorithms for mass and energy constrained neural network models that can exactly conserve the overall mass and energy of distributed chemical process systems, even though the noisy transient data used for optimal model training violate the same. In contrast to approximately satisfying mass and energy balance constraints of a system by soft penalization of objective function, algorithms have been developed for solving equality-constrained nonlinear optimization problems, thus providing the guarantee of exactly satisfying the system mass and energy conservation laws. For developing dynamic mass-energy constrained network models for distributed systems, hybrid series and parallel dynamic-static neural networks have been leveraged. The developed algorithms for solving both the training and forward problems are validated using both steady-state and dynamic data in the presence of various noise characteristics. The developed data-driven algorithms are flexible to exactly satisfy mass and energy balance constraints for dynamic chemical processes if the system holdup information is available. The proposed network structures and algorithms are applied to the development of data-driven lumped and distributed models of an adiabatic superheater/reheater system, a nonisothermal continuous stirred tank reactor, as well as an electrically heated plug-flow reactor system where one form of energy gets transformed to another. It has been observed that the mass-energy constrained neural networks yield a root mean squared error of <1% with respect to the system truth for the case studies evaluated in this work.

Expansion planning of hybrid electrical and thermal systems using reconfiguration and adaptive bat algorithm

_ This study introduces a comprehensive model for the concurrent expansion planning of various energy systems and their associated equipment. The need to reduce network costs, emissions, losses, and feeder loading, as well as to enhance network reliability and voltage profile, mandates the utilization of proper multi-objective planning models that respect all network constraints. The introduced framework includes units for generating both electrical and thermal energies. The model leverages conventional expansion alternatives such as the installation of new lines, network reconfiguration, rewiring, and the addition of new thermal and electrical generating units to the network. Expansion planning involves determining the optimal time, location, and type of new installations to meet future energy demands while minimizing costs and emissions. Reconfiguration refers to altering the network topology to improve reliability and reduce losses. The proposed expansion planning is formulated as a discrete, nonlinear, and non-convex optimization problem, which is solved using the Self Adaptive Learning Bat Algorithm (SALBA). This algorithm improves convergence speed and increases the diversity of the search population, enhancing the likelihood of finding the global optimum. Numerical simulations of the proposed methodology on two modified standard IEEE test systems corroborate the efficacy and feasibility of the suggested approach. Key innovations include the comprehensive modeling for concurrent expansion planning, the use of an advanced optimization algorithm, and a focus on reducing costs, emissions, losses, and feeder loading while enhancing network reliability and voltage profile.

Two-Warehouse Inventory Model: Interval Valued Costs, Advanced Payment, Stock-Dependent Demand, and Partial Backlogging

A great deal of inventory costs are typically unpredictable because of the unpredictability of a competitive market. The fluctating demand from customer's and the unpredictability of the market economy make it difficult for researchers and operation research practitioners to appropriately replicate an inventory problem. In order to prevail this type of unpredictability circumstances, in this paper, we have represented the inventory parameters as interval. Using this concept, we progressed a two-warehouse inventory model for deteriorating items with a fixed Shelf life, partially backlogging shortages and interval-valued deterioration rate. In addition to this the parameters like ordering cost, purchase cost, shortage cost, deterioration cost for both the rented and owned warehouse except the backlogging parameter have been considered as interval-valued. The uncertainty in the inventory parameters motivated us to view them as interval-valued in the present research. Based on the assumptions, the cost function of this problem is a highly nonlinear constraint optimization problem. Mathematica is used to tackle this nonlinear optimization problem. A numerical example has been presented to demonstrate the computational result. Then, a sensitivity analysis has been performed to study the effect of changes of different parameters of the model on the optimal policy.

A new fault detection and relay coordination technique for low voltage DC microgrid

Integration of various distributed generation units in the DC distribution network creates an outsized impact on magnitude and direction of fault current. The specific behaviour of fault current during a short circuit fault demands quick and reliable detection and isolation of fault, which poses a significant challenge in developing an effective protection and coordination scheme for DC microgrids. In this regard, a new fault detection and relay coordination algorithm is proposed for a Low voltage DC (LVDC) microgrid, which utilizes inverse time relay characteristics based on variance of fault current. While the relay responds and picks up during both internal and external faults, its operation is automatically limited upon operation of the downstream circuit breaker without the need for any communication channel. The efficacy of the proposed strategy is evaluated by modelling a mesh-connected DC microgrid network using PSCAD/EMTDC software. The protection coordination problem is developed as a constraint non-linear optimization problem to determine optimal relay settings. The proposed approach has been tested by simulating various types of faults at different fault locations and fault resistances along with various microgrid operating modes. The results suggest that the proposed strategy stands out by optimizing relay operating time and providing backup protection while upholding relay coordination. A comparative assessment of the proposed technique with existing ones proves its effectiveness in terms of swift relay tripping time, better immunity against noise, and independency against microgrid operating mode and topology.

A gradient-based optimization algorithm to solve optimal control problems with conformable fractional-order derivatives

In this paper, we solve numerically a class of fractional optimal control problems in the conformable sense. We first propose a nonlinear fractional optimal control problem subject to general equality and inequality constraints. Then, based on a novel numerical integration scheme, we present a discretization method for the governed fractional-order system as well as the cost and constraint functionals, which yields a discretized approximate problem. We further derive the gradients of the cost and constraint functionals in regard to the discretized controls. On the basis of this result, a numerical optimization approach to solve the approximate problem is developed. Finally, three non-trivial example problems are solved to illustrate the applicability and effectiveness of the developed approach.

Local Energy Community to Support Hydrogen Production and Network Flexibility

This paper deals with the optimal scheduling of the resources of a renewable energy community, whose coordination is aimed at providing flexibility services to the electrical distribution network. The available resources are renewable generation units, battery energy storage systems, dispatchable loads, and power-to-hydrogen systems. The main purposes behind the proposed strategy are enhancement of self-consumption and hydrogen production from local resources and the maximization of the economic benefits derived from both the selling of hydrogen and the subsidies given to the community for the shared energy. The proposed approach is formulated as an economic problem accounting for the perspectives of both community members and the distribution system operator. In more detail, a mixed-integer constrained non-linear optimization problem is formulated. Technical constraints related to the resources and the power flows in the electrical grid are considered. Numerical applications allow for verifying the effectiveness of the procedure. The results show that it is possible to increase self-consumption and the production of green hydrogen while providing flexibility services through the exploitation of community resources in terms of active and reactive power support. More specifically, the application of the proposed strategy to different case studies showed that daily revenues of up to EUR 1000 for each MW of renewable energy generation installed can be obtained. This value includes the benefit obtained thanks to the provision of flexibility services, which contribute about 58% of the total.

Bayesian expectation maximization-maximization for robust estimation in proton exchange membrane fuel cells: A comparative study

In this paper, we present a novel approach based on Bayesian Expectation Maximization-Maximization (BEMM) to address this challenge. Unlike traditional optimization methods, which may struggle with high-dimensional and nonlinear optimization problems, BEMM offers a robust framework that combines the benefits of Bayesian inference with the flexibility of expectation maximization and maximization techniques. By iteratively updating parameter estimates based on observed data and maximizing the likelihood of the model, BEMM effectively navigates the solution space to converge on accurate estimates of the unspecified variables in Proton Exchange Membrane Fuel Cells (PEMFCs)models. Through extensive experimentation and comparison with other metaheuristic techniques, including Arithmetic Optimization Algorithm (AOA), Gravitational Search Algorithm (GSA), Flower Pollination Algorithm (FPA), and Biogeography-Based Optimization (BBO), we demonstrate the superior performance of our BEMM approach. Our results show that BEMM outperforms these alternative methods in terms of precision, convergence speed, and stability across various scenarios involving different numbers of unspecified variables. The implications of our findings are significant for both researchers and practitioners in the field of PEMFC modeling and optimization. By providing a more efficient and reliable method for estimating model parameters, our approach can facilitate more accurate predictions of PEMFC performance, leading to better-informed decision-making in the design, operation, and optimization of PEMFC systems. Furthermore, the robustness and versatility of BEMM make it well-suited for a wide range of optimization problems beyond PEMFC modeling, highlighting its potential impact across various domains of engineering and scientific research. In the sensitivity analysis, as the population size increases from 10 to 40, there is a significant improvement in solution quality by approximately 100 %. However, beyond a population size of 40, the marginal gains diminish, with only marginal improvements of less than 1 % observed despite further increases in population size up to 200.

LPI-based resource allocation strategy for multiple targets tracking in CMIMO radar system with array division

Considering the scenario of multiple targets equipped with interceptors in the modern electronic warfare environment, an LPI-based Resource Allocation Strategy (LPIRAS) for multiple targets tracking (MTT) in collocated MIMO (CMIMO) radar is proposed in this paper, where subarray number is exploited in the low probability of intercept (LPI) optimization problem for the first time. As the problem is formulated as a nonconvex and nonlinear optimization problem, the array power, target-beam assignment, subarray number, and subarray-beam assignment are corporately managed through a methodology incorporating the two-stage search algorithm (TSA), permutation-based target swarm generation algorithm (PTSGA) and adaptive extra subarray assignment algorithm (AESAA), resulting the LPIRAS. Finally, numerical simulations show the proposed strategy is effective and real-time. In comparison to other benchmarks, the LPIRAS achieves superior LPI performance.

Nonlinear Optimal Control Based on FBDEs and its Application to AGV.

This article focuses on solving a finite-horizon nonlinear optimal control problem by using the Pontryagin's maximum principle. In practical applications, linearization is a common approach for solving nonlinear dynamical systems. However, it is not universally applicable due to various reasons, such as instability and low accuracy. In contrast to linearization, the inherent challenge in directly solving the above nonlinear optimal control problem lies in addressing the highly coupled nonlinear forward and backward differential equations. In order to address this problem, an equivalent relationship is established between these equations and a new optimization problem. By exploiting the inherent relationship between supervised learning and an optimization problem from the view of a dynamical system, a deep neural network framework is constructed for describing the new optimization problem. Furthermore, a numerical algorithm for optimal control, which is very powerful for a large variety of nonlinear dynamical systems, is implemented by training a deep residual network. Finally, the effectiveness of the algorithm is demonstrated by solving a trajectory tracking control problem for automatic guided vehicle. The obtained results reveal that the proposed control scheme can achieve high-precision tracking.

Spatial–Temporal Joint Design and Optimization of Phase-Coded Waveform for MIMO Radar

By simultaneously transmitting multiple different waveform signals, a multiple-input multiple-output (MIMO) radar possesses higher degrees of freedom and potential in many aspects compared to a traditional phased-array radar. The spatial–temporal characteristics of waveforms are the key to determining their performance. In this paper, a transmitting waveform design method based on spatial–temporal joint (STJ) optimization for a MIMO radar is proposed, where waveforms are designed not only for beam-pattern matching (BPM) but also for minimizing the autocorrelation sidelobes (ACSLs) of the spatial synthesis signals (SSSs) in the directions of interest. Firstly, the STJ model is established, where the two-step strategy and least squares method are utilized for BPM, and the L2p-Norm of the ACSL is constructed as the criterion for temporal characteristics optimization. Secondly, by transforming it into an unconstrained optimization problem about the waveform phase and using the gradient descent (GD) algorithm, the hard, non-convex, high-dimensional, nonlinear optimization problem is solved efficiently. Finally, the method’s effectiveness is verified through numerical simulation. The results show that our method is suitable for both orthogonal and partial-correlation MIMO waveform designs and efficiently achieves better spatial–temporal characteristic performances simultaneously in comparison with existing methods.

Numerical algorithms for generating an almost even approximation of the Pareto front in nonlinear multi-objective optimization problems

A multiobjective optimization problem (MOP) returns a set of non-dominated points, the so-called Pareto front. Since this set is usually infinite, it is impossible to generate it completely in practice. Therefore, a discrete approximation of the Pareto front is created. One of the most important features of this approximation is a uniform distribution of points on the full Pareto front in order to present a wide variety of solutions to the decision maker who chooses a final solution. While a few algorithms consider this property, two algorithms based on the Pascoletti-Serafini (PS) scalarization approach are proposed. In addition, six well-known test problems with convex and non-convex Pareto fronts are considered to show the effectiveness of the proposed algorithms. Their results are compared with some algorithms including Normal Constraint (NC), Benson type, Non-Dominated Sorting Genetic Algorithm-II (NSGA-II), S-Metric Selection Evolutionary Multiobjective Algorithm (SMS-EMOA), Differential Evolution (DE) with Binomial Crossover and MOEA/D-DE. The computational results on CPU time and reasonable distribution of points obtained on the Pareto front show that the presented algorithms perform better than other algorithms on these criteria. In addition, although the proposed algorithms compete closely with some algorithms in terms of CPU time, they have more non-dominated solutions and more appropriate distribution than they do in most problems.

A Novel Path-Following Method for Time-Varying Optimizations with Optimal Parametric Functions

We address a broad category of nonlinear constrained optimization problems. We then reformulate it as a time-varying optimization using continuous-time parametric functions and derive a dynamical system to track the respective optimal solution. We then re-parameterize the dynamical system according to a linear combination of parametric functions. By applying the calculus of variations, we optimize these parametric functions to minimize the optimality distance. Consequently, we develop an iterative dynamic algorithm, termed as OP-TVO, to achieve efficient convergence to the solution. We compare the performance of the proposed algorithm with the prediction correction method (PCM) in terms of optimality and computational complexity. The results demonstrate that OP-TVO effectively competes with PCM for the given class of optimization problems, suggesting a promising alternative to PCM. This work also introduces a novel paradigm for solving parametric dynamic systems.

Task offloading framework to meet resiliency demand in mobile edge computing system

With the development of 5 G mobile users are increasing massively. Some mobile applications like healthcare are latency-critical and requires real-time data processing. A preference-based task offloading framework in mobile edge computing with a device-to-device offloading (MECD2D) system has been proposed to fulfill the latency demands of such applications for minimum energy consumption ensuring resiliency. The problem is formulated as a constraint-based non-linear optimization problem which is complex. The resources are allocated in two steps. In the first step, resources are allocated based on latency demand to ensure resiliency. In the second step, allocated resources are optimized using a non-cooperative mean field game for dynamic system. To ensure the performance of the system for dynamic network, the results are executed on a real-time Shanghai dataset. The computational results indicate that the proposed algorithm performs better in terms of energy consumption. Other parameters such as throughput, network utilization and task computation are also analysed. The results are verified by performing the proposed algorithm with existing Q learning and mean-field game algorithms. The results performed on the dataset indicate an improvement in energy consumption by 5–10 %, and 10–50 % as compared to Q learning and mean-field game respectively.