Nonlinear Constraints Research Articles

Short-term load distribution model for cascade giant hydropower stations with complex hydraulic and electrical connections

The cascade giant hydropower station system (CGHSS) has been widely applied in countries with abundant water resources. However, the dense hydropower stations and power grids bring complex hydraulic and power connections, including the backwater effect, multiple hydroplants sharing one reservoir and one-reservoir multiple dispatch power grid, which leads to modelling difficulties and inefficient model solving. To address these challenges, a short-term load distribution (SLD) model considering complicated hydraulic and electricity constraints is developed. First, the discharge of each hydroplant and the downstream forebay water level are used as inputs to construct a multivariate nonlinear function, and the average tailwater levels are obtained. Second, a head constraint considering the shared water level is added to the model to calculate the head of each hydroplant, making the output more accurate. Finally, the nonlinear constraints are linearly reconstructed using the Big-M method, triangulation technique and special ordered sets to reduce the model solution time. The SLD model is applied to the Xiluodu-Xiangjiaba cascade hydropower system in the Jinsha River. The results show that the developed model reduces water consumption by 4.02 %, 4.70 %, 5.79 % and 5.28 % compared to the constant proportional distribution model and the real operation, generating a more executable SLD scheme.

Life-cycle production optimization with nonlinear constraints using a least-squares support-vector regression proxy

When nonlinear constraints such as field liquid or water production rate, injection pressures, etc., as functions of time need to be honored in addition to linear ones, the life-cycle production optimization problem, a component of a closed loop reservoir management, becomes challenging and computationally expensive to perform using a high-fidelity reservoir simulator with the existing gradient-based methods using the adjoint or stochastic approximate gradient methods. Therefore, the objective of this study is to present computationally efficient methods for deterministic production optimization under nonlinear constraints using a kernel-based machine learning method where the cost function is the net present value (NPV). We use the least-squares support-vector regression (LSSVR) to approximate the NPV function. To achieve computational efficiency, we generate a set of output values of the NPV and nonlinear constraint functions, which are field liquid production rate (FLPR) and water production rate (FWPR) in this study, by running the high-fidelity simulator for a broad set of input design variables (well controls) and then using the collection of input/output data to train LSSVR proxy models to replace the high-fidelity simulator to compute NPV and nonlinear state constraint functions during iterations of sequential quadratic programming (SQP). To obtain improved (higher) estimated optimal NPV values, we use the existing so-called iterative sampling refinement (ISR) method to update the LSSVR proxy so that the updated proxy remains predictive toward promising regions of search space during the optimization. Direct and indirect ways of constructing LSSVR-based NPVs as well as different combinations of input data, including nonlinear state constraints and/or the bottomhole pressures (BHPs) and water injection rates, are tested as alternative feature spaces. The results obtained from our proposed LSSVR-based optimization methods are compared with those obtained from our in-house stochastic simplex approximate gradient (StoSAG)-based line-search SQP programming (LS-SQP-StoSAG) algorithm that uses directly a high-fidelity simulator to compute the gradients of the objective function and the nonlinear state functions with StoSAG for the Brugge reservoir model. The results show that nonlinear constrained optimization with the LSSVR ISR with SQP is computationally 3.25 fold more efficient than LS-SQP-StoSAG. In addition, the results show that constructing NPV indirectly using the field liquid and water rates for a waterflooding problem where inputs come from LSSVR proxies of the nonlinear state constraints requires significantly fewer training samples than the method constructing NPV directly from the NPVs computed from a high-fidelity simulator. LSSVR has advantages for its computational efficiency which is the main goal in our research, and robustness against overfitting especially for the cases where we have limited data. However, DNNs and random forest require large training size and require much more computational resources and longer training times. Also, Random Forests and Gradient Boosting Machines can be prone to overfitting and become computationally intensive.

Multi-objective energy management in a renewable and EV-integrated microgrid using an iterative map-based self-adaptive crystal structure algorithm

The use of plug-in hybrid electric vehicles (PHEVs) provides a way to address energy and environmental issues. Integrating a large number of PHEVs with advanced control and storage capabilities can enhance the flexibility of the distribution grid. This study proposes an innovative energy management strategy (EMS) using an Iterative map-based self-adaptive crystal structure algorithm (SaCryStAl) specifically designed for microgrids with renewable energy sources (RESs) and PHEVs. The goal is to optimize multi-objective scheduling for a microgrid with wind turbines, micro-turbines, fuel cells, solar photovoltaic systems, and batteries to balance power and store excess energy. The aim is to minimize microgrid operating costs while considering environmental impacts. The optimization problem is framed as a multi-objective problem with nonlinear constraints, using fuzzy logic to aid decision-making. In the first scenario, the microgrid is optimized with all RESs installed within predetermined boundaries, in addition to grid connection. In the second scenario, the microgrid operates with a wind turbine at rated power. The third case study involves integrating plug-in hybrid electric vehicles (PHEVs) into the microgrid in three charging modes: coordinated, smart, and uncoordinated, utilizing standard and rated RES power. The SaCryStAl algorithm showed superior performance in operation cost, emissions, and execution time compared to traditional CryStAl and other recent optimization methods. The proposed SaCryStAl algorithm achieved optimal solutions in the first scenario for cost and emissions at 177.29 €ct and 469.92 kg, respectively, within a reasonable time frame. In the second scenario, it yielded optimal cost and emissions values of 112.02 €ct and 196.15 kg, respectively. Lastly, in the third scenario, the SaCryStAl algorithm achieves optimal cost values of 319.9301 €ct, 160.9827 €ct and 128.2815 €ct for uncoordinated charging, coordinated charging and smart charging modes respectively. Optimization results reveal that the proposed SaCryStAl outperformed other evolutionary optimization algorithms, such as differential evolution, CryStAl, Grey Wolf Optimizer, particle swarm optimization, and genetic algorithm, as confirmed through test cases.

ZNN Continuous Model and Discrete Algorithm for Temporally Variant Optimization With Nonlinear Equation Constraints via Novel TD Formula

ZNN Continuous Model and Discrete Algorithm for Temporally Variant Optimization With Nonlinear Equation Constraints via Novel TD Formula

A novel machine learning method for the design optimization of diamond waveguides fabricated by femtosecond laser writing

We report on a novel machine learning method for the design optimization of femtosecond (fs) laser written dielectric waveguides. Experimental results previously obtained from the optical characterization of fs laser written depressed cladding diamond waveguides have been used to form statistically generated regression models. Design variables such as core diameter and number of written tracks were varied to both minimize the propagation loss as well as to establish a full-factorial experimental design. The regression models were used to conduct a multi-objective optimization study to optimize the competing objectives such as maximizing the refractive index contrast while minimizing the propagation loss and V-number by using a genetic algorithm. Optimization was subject to a nonlinear Rayleigh range constraint to ensure that the structure was in the waveguiding regime. Results from the optimization revealed the optimum variables to achieve low-loss and nearly single-mode guiding for a fs laser written diamond waveguide. Using the solution sets of design parameters resulting from the optimization study and their corresponding objective function values, important correlations between the design parameters and the objective functions have been revealed. With this regard, it has been shown that the number of written tracks is a much more dominant parameter, when compared to core diameter, during the design of a fs laser written circular depressed cladding diamond waveguide. The proposed method should be applicable not only to diamond waveguides but also to a wide range of dielectric waveguides fabricated by fs laser writing.

Mathematical Model for Multi Depot Simultaneously Pick Up and Delivery Vehicle Routing Problem with Stochastic Pick Up Demand

In classical vehicle routing problem (VRP), customer demands are known with certainty. On the other hand, in real life problems, customer demands may change over time. Therefore, in the classical VRP, the assumption that customer demands are stochastic should be taken into account. In order to meet the demands of the customers faster and to reduce the fuel consumption and carbon emissions, the companies meet the distribution and collection demands of the customers simultaneously. Customers' distribution requirements can be predicted, but it is impossible to predict in advance the product requirements they will send for recycling. Hence in this study, a mathematical programming model is developed for the multi-depot simultaneously pick up and delivery vehicle routing problem, under the assumption that customers' picking demands are stochastic. However, There are non-linear constraints in the developed model. Thereby firstly the stochastic model is linearized, and then the effectiveness of the model is analyzed. The efficiency of the linearized model is determined by creating test problems.

A robust interpolated model predictive control based on recurrent neural networks for a nonholonomic differential-drive mobile robot with quasi-LPV representation: computational complexity and conservatism

This paper presents an improved Model Predictive Control (MPC) for path tracking of a nonholonomic mobile robot with a differential drive. Nonlinear dynamics and nonholonomic constraints make the optimisation problem of MPC for the robot challenging. Nonlinear dynamics of the robots are expressed by a Linear Parameter Varying (LPV), and a Recurrent Neural Network (RNN) solves the constrained optimisation problem, providing optimal velocities. Moreover, an interpolation-based approach has been introduced to augment the region of attraction. The algorithm ensures stability in the presence of bounded disturbances through the inclusion of free control moves in the control law. The controller efficiency has been evaluated in two scenarios in a hospital setting. The simulation results illustrate that the proposed method performs better than nonlinear MPC and standard LPV-based MPC in terms of computational cost, disturbance rejection, and region of attraction.

GQL-based bound-preserving and locally divergence-free central discontinuous Galerkin schemes for relativistic magnetohydrodynamics

This paper develops novel and robust central discontinuous Galerkin (CDG) schemes of arbitrarily high-order accuracy for special relativistic magnetohydrodynamics (RMHD) with a general equation of state (EOS). These schemes are provably bound-preserving (BP), meaning they consistently preserve the upper bound for subluminal fluid velocity and the positivity of density and pressure, while also (locally) maintaining the divergence-free (DF) constraint for the magnetic field. For 1D RMHD, the standard CDG method is exactly DF, and its BP property is proven under a condition achievable by the BP limiter. For 2D RMHD, we design provably BP and locally DF CDG schemes based on the suitable discretization of a modified RMHD system, which is the relativistic analogue of Godunov's symmetrizable form of the non-relativistic MHD system (Godunov, 1972 [19]). A key novelty in our schemes is the meticulous discretization of additional source terms in the modified RMHD equations, so as to precisely counteract the influence of divergence errors on the BP property across overlapping meshes. Notably, we provide rigorous proofs of the BP property for our CDG schemes and first establish the theoretical connection between BP and discrete DF properties on overlapping meshes for RMHD. Owing to the absence of explicit expressions for primitive variables in terms of conserved variables, the constraints of physical bounds are strongly nonlinear, making the BP proofs highly nontrivial. We overcome these challenges through technical estimates within the geometric quasilinearization (GQL) framework (Wu and Shu, 2023 [49]), which equivalently converts the nonlinear constraints into linear ones. Furthermore, we introduce a new 2D cell average decomposition on overlapping meshes, which relaxes the theoretical BP CFL constraint and reduces the number of internal nodes, thereby enhancing the efficiency of the 2D BP CDG method. Finally, we implement the proposed CDG schemes for extensive RMHD problems with various EOSs, demonstrating their robustness and effectiveness in challenging scenarios like ultra-relativistic blasts and jets in strongly magnetized environments.

Model predictive control of a wave-to-wire wave energy converter system with non-linear dynamics and non-linear constraints using a tailored pseudo-spectral method

A novel pseudo-spectral method is proposed to tackle the non-linear control problem of a wave energy converter (WEC) integrated with an all-electrical power-take-off (PTO). The WEC hydrodynamic model is constructed using linear wave theory together with a non-linear viscous damping term. The PTO model encompasses a permanent magnet synchronous generator (PMSG) whose local field weakening mechanism imposes non-linear operational constraints on the WEC’s motion and PTO torque. The energy-maximizing control problem of this wave-to-wire model is non-linear and has not been investigated before. In comparison to pseudo-spectral methods previously studied, the proposed pseudo-spectral method is exempt from the assumption of ideal incoming wave prediction. This is made possible by the introduction of a novel mapping function to convert the finite control horizon onto the Gaussian quadrature interval. This mapping function increases more slowly with time at the beginning of the control horizon where wave prediction attains greater precision. As a result, the collocation points are utilized to the advantage of the better approximation of the objective function, thus better overall control performance.

Inverse-free zeroing neural network for time-variant nonlinear optimization with manipulator applications

In this paper, the problem of time-variant optimization subject to nonlinear equation constraint is studied. To solve the challenging problem, methods based on the neural networks, such as zeroing neural network and gradient neural network, are commonly adopted due to their performance on handling nonlinear problems. However, the traditional zeroing neural network algorithm requires computing the matrix inverse during the solving process, which is a complicated and time-consuming operation. Although the gradient neural network algorithm does not require computing the matrix inverse, its accuracy is not high enough. Therefore, a novel inverse-free zeroing neural network algorithm without matrix inverse is proposed in this paper. The proposed algorithm not only avoids the matrix inverse, but also avoids matrix multiplication, greatly reducing the computational complexity. In addition, detailed theoretical analyses of the convergence performance of the proposed algorithm is provided to guarantee its excellent capability in solving time-variant optimization problems. Numerical simulations and comparative experiments with traditional zeroing neural network and gradient neural network algorithms substantiate the accuracy and superiority of the novel inverse-free zeroing neural network algorithm. To further validate the performance of the novel inverse-free zeroing neural network algorithm in practical applications, path tracking tasks of three manipulators (i.e., Universal Robot 5, Franka Emika Panda, and Kinova JACO2 manipulators) are conducted, and the results verify the applicability of the proposed algorithm.

Reduced complexity model predictive direct power control for unbalanced grid

The model predictive control (MPC) group is frequently used as a controller in PWM rectifiers because of its accuracy. Yet, the requirement of a processor with high computational efficiency for its real-time implementation has been a significant drawback. Although a few low-complexity MPCs have been reported in the past, none of such methods are validated in an unbalanced grid. Hence, this work aims to propose a reduced-complexity MPC algorithm that can also handle an unbalanced grid. The conventional MPC evaluates the cost function for all the switching vectors and then chooses the switching vector that produces the minimum cost. In contrast, by equating its first-order derivative of the cost function to 0, the proposed analytical approach directly evaluates switching vectors, resulting in a minimum value of the cost function. Thus, the controller estimates only the closest possible solutions of the cost function minimization process and abandons the rest. Furthermore, the proposed method also endorses the use of additional non-linear constraints in the cost function, which eradicates one of the major drawbacks of the previous low-complexity techniques. To ascertain the practicability of the proposed method, it is examined in the MATLAB/Simulink environment and the real-time system. The obtained results look promising as the proposed controller is able to imitate the performance of conventional MPC with less computational burden.

Geometric Design Methodology for Deployable Self-Locking Semicylindrical Structures

Due to their unique bistable characteristics, deployable self-locking structures are suitable for many engineering fields. Without changing the geometrical composition, such structures can be unfolded and locked solely by the elastic deformation of materials. However, their further applications are hampered by the lack of simple and systematic geometric design methodologies that consider arbitrary structural curvature profiles. This paper proposes such a methodology for double-layer semicylindrical grid structures to simplify their cumbersome geometric design. The proposed methodology takes joint sizes into account to ensure that the design results can be applied to actual projects without further adjustments. By introducing symmetry into the structural units (SUs) and selecting reasonable geometric parameters that describe the structural side elevation profile, a concise set of simultaneous nonlinear geometric constraint equations is established, the solution of which provides the geometric parameter values of the grid shape. On this basis, the remaining geometric parameter values, i.e., the geometric parameter values of the inner scissor-like elements (SLEs) of each SU, can be achieved from the equations derived from general SLEs. Design examples and the assembled physical grid structure indicate the feasibility and wide applicability of the proposed geometric design methodology.

An ensemble spatial prediction method considering geospatial heterogeneity

Ensemble learning synthesizes the advantages of different models and has been widely applied in the field of spatial prediction. However, the nonlinear constraints of spatial heterogeneity on the model ensemble process make it difficult to adaptively determine the ensemble weights, greatly limiting the predictive ability of the ensemble learning model. This paper therefore proposes a novel geographical spatial heterogeneous ensemble learning method (GSH-EL). Firstly, the geographically weighted regression model, geographically optimal similarity model, and random forest model are used as three base learners to express local spatial heterogeneity, global feature correlation, and nonlinear relationship of geographic elements, respectively. Then, a spatially weighted ensemble neural network module (SWENN) of GSH-EL is proposed to express spatial heterogeneity by exploring the complex nonlinear relationship between the spatial proximity and ensemble weights. Finally, the outputs of the three base learners are combined with the spatial heterogeneous ensemble weights from SWENN to obtain the spatial prediction results. The proposed method is validated on the PM2.5 air quality and landslide dataset in China, both of which obtain more accurate prediction results than the existing ensemble learning strategies. The results confirm the need to accurately express spatial heterogeneity in the model ensemble process.

Asymptotic tracking control for a class of uncertain nonlinear systems with output constraint

In this paper, we investigate the asymptotic tracking control for a class of strict-feedback nonlinear systems with unknown nonlinear functions and output constraint. The nonlinear transformation function-based system transformation approach is employed to convert the initial constrained system into an unconstrained one. To achieve the objective of asymptotic tracking, the tracking error is scaled by an exponential proportional function. Furthermore, to address the challenge of repetitive derivation of virtual control laws that occurs in the traditional design process of high-order backstepping methods, a first-order filter is applied. The designed control scheme guarantees that the system output will asymptotically track a predefined reference tracking trajectory while ensuring that the output constraint is not violated. Moreover, it is shown that the filter error will converge asymptotically to the origin. The effectiveness of the developed control strategy is verified through simulation results.

Master–slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this paper, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

A Stochastic Inexact Sequential Quadratic Optimization Algorithm for Nonlinear Equality-Constrained Optimization

A stochastic algorithm is proposed, analyzed, and tested experimentally for solving continuous optimization problems with nonlinear equality constraints. It is assumed that constraint function and derivative values can be computed but that only stochastic approximations are available for the objective function and its derivatives. The algorithm is of the sequential quadratic optimization variety. Distinguishing features of the algorithm are that it only employs stochastic objective gradient estimates that satisfy a relatively weak set of assumptions (while using neither objective function values nor estimates of them) and that it allows inexact subproblem solutions to be employed, the latter of which is particularly useful in large-scale settings when the matrices defining the subproblems are too large to form and/or factorize. Conditions are imposed on the inexact subproblem solutions that account for the fact that only stochastic objective gradient estimates are employed. Convergence results are established for the method. Numerical experiments show that the proposed method vastly outperforms a stochastic subgradient method and can outperform an alternative sequential quadratic programming algorithm that employs highly accurate subproblem solutions in every iteration. Funding: This material is based upon work supported by the National Science Foundation [Awards CCF-1740796 and CCF-2139735] and the Office of Naval Research [Award N00014-21-1-2532].

Event-triggered consensus tracking control of flexible manipulators with nonlinear time-varying fault-tolerant actuators and input constraint

This paper introduces a novel event-triggered consensus control strategy for a multi-agent system composed of underactuated flexible manipulators. Most existing research on consensus tracking control assumes that the model targeted is fully actuated, the control signal is continuous, and that both time-varying actuator failures and input constraints are managed separately. These challenges were tackled in this study within the context of multi-agent systems, addressing input constraints and nonlinear time-varying actuator failures simultaneously. Moreover, an event-triggered mechanism is proposed to reduce signal communication. A double closed-loop consensus tracking method is designed based on the multi-agent topology, enabling all flexible manipulators to track the angle and angular velocity of the leader and an adaptive control law is designed to solve actuator faults. Finally, the simulation results demonstrate the effectiveness of the proposed control scheme.

Threshold Effects between Ecosystem Services and Natural and Social Drivers in Karst Landscapes

It has been shown that there are thresholds of influence on the response of ecosystem services to their drivers, and the range of drivers that provide high levels of ecosystem services can be delineated through thresholds. However, due to the spatial heterogeneity of landscapes in karst regions, the results of ecosystem service threshold studies in non-karst regions may not be applicable to karst regions. This study explores the threshold effects between ecosystem services in karst landscapes and their natural and social drivers. It is shown that there are nonlinear constraints between them, and different critical thresholds exist for different kinds of ecosystem services. The main thresholds for water supply services include the slope (43.64°) and relief amplitude (331.60 m); for water purification services, they include relief amplitude (147.05 m) and distance to urban land (DTUL) (32.30 km); for soil conservation services, they include the normalized difference vegetation index (NDVI) (0.80) and nighttime light intensity (43.58 nW∙cm−2∙sr−1); the main thresholds for biodiversity maintenance services include population density (1481.06 person∙km−2) and distance to urban land (DTUL) (32.80 km). This enables regional ecological conservation planning based on different threshold ranges corresponding to different ecosystem services to meet the different needs of different decision makers.

In-silico optimal operating policies of a batch or a fed-batch bioreactor for mAb production using a hybridoma cell culture

Production of monoclonal antibodies (mAb) is a well-known method to synthesize a large number of identical antibodies, of huge importance in medicine. In thus context, huge efforts have been spent to maximize the mAb production in industrial bioreactors by using hybridoma cell cultures. However, the optimal operation of these bioreactors is an engineering problem difficult to solve due to the highly nonlinear bioprocess dynamics, and a bioreactor involving a large number of decision (control) variables, subjected to multiple nonlinear process constraints, which often translates into a non-convex optimization problem. Based on an adequate kinetic model adopted from literature, this paper is aiming at in-silico, off-line deriving and comparing the optimal operating policies of a batch bioreactor (BR), and a fed-batch bioreactor (FBR) operated in several feeding alternatives (including substrates and the viable biomass) with using a hybridoma culture immobilized on a porous support (alginate) for mAb production. FBR with a variable time stepwise optimal feeding policy proved to reach better performances in terms of mAb production maximization with a minimal raw-material consumption.

Learning to Optimize Contextually Constrained Problems for Real-Time Decision Generation

The topic of learning to solve optimization problems has received interest from both the operations research and machine learning communities. In this paper, we combine ideas from both fields to address the problem of learning to generate decisions to instances of optimization problems with potentially nonlinear or nonconvex constraints where the feasible set varies with contextual features. We propose a novel framework for training a generative model to produce provably optimal decisions by combining interior point methods and adversarial learning, which we further embed within an iterative data generation algorithm. To this end, we first train a classifier to learn feasibility and then train the generative model to produce optimal decisions to an optimization problem using the classifier as a regularizer. We prove that decisions generated by our model satisfy in-sample and out-of-sample optimality guarantees. Furthermore, the learning models are embedded in an active learning loop in which synthetic instances are iteratively added to the training data; this allows us to progressively generate provably tighter optimal decisions. We investigate case studies in portfolio optimization and personalized treatment design, demonstrating that our approach yields advantages over predict-then-optimize and supervised deep learning techniques, respectively. In particular, our framework is more robust to parameter estimation error compared with the predict-then-optimize paradigm and can better adapt to domain shift as compared with supervised learning models. This paper was accepted by Chung Piaw Teo, optimization. Funding: This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2020.03565 .