Nonlinear Constraint Functions Research Articles

An Improved Optimization Model to Predict the MOR of Glulam Prepared by UF-Oxidized Starch Adhesive: A Hybrid Artificial Neural Network-Modified Genetic Algorithm Optimization Approach.

The purpose of the present article is to study the bending strength of glulam prepared by plane tree (Platanus Orientalis-L) wood layers adhered by UF resin with different formaldehyde to urea molar ratios containing the modified starch adhesive with different NaOCl concentrations. Artificial neural network (ANN) as a modern tool was used to predict this response, too. The multilayer perceptron (MLP) models were used to predict the modulus of rapture (MOR) and the statistics, including the determination coefficient (R2), root mean square error (RMSE), and mean absolute percentage error (MAPE) were used to validate the prediction. Combining the ANN and the genetic algorithm by using the multiple objective and nonlinear constraint functions, the optimum point was determined based on the experimental and estimated data, respectively. The characterization analysis, performed by FTIR and XRD, was used to describe the effect of the inputs on the output. The results indicated that the statistics obtained show excellent MOR predictions by the feed-forward neural network using Levenberg-Marquardt algorithms. The comparison of the optimal output of the actual values obtained by the genetic algorithm resulting from the multi-objective function and the optimal output of the values estimated by the nonlinear constraint function indicates a minimum difference between both functions.

Building a novel multivariate nonlinear MGM(1,m,N|γ) model to forecast carbon emissions.

With the proposal of the carbon neutrality target, China's attention to carbon emissions has been further enhanced. Effective prediction of future carbon emissions is important for the formulation of carbon neutralization target and action plans in the region. Many factors affecting carbon emissions, cause their development trends may be nonlinear. To forecast the carbon emissions of coal and natural gas in the industrial sector more accurately, a new MGM(1,m,N|γ) model considering nonlinear characteristics is proposed in this paper. The new model introduces power function γ as nonlinear parameter, and the γ value is solved by nonlinear constraint function. We further deduce the simulation and prediction formula and then apply the improved model to the carbon emission forecast. The comparisons show that the nonlinear parameters can modify the trend of sequences and improve the prediction accuracy, which verifies the validity of the model. Finally, according to the influencing factors and forecast results, this paper analyzes the causes of high carbon emissions and puts forward reasonable suggestions for China's carbon governance.

Penjadwalan Mata Pelajaran Menggunakan Integer Nonlinear Programming

Timetabling is one of the problems faced by large numbers of institutions, including schools. In this paper, this timetabling problem is mathematically modeled using Integer Nonlinear Programming to optimize the result with the non-linear objective function or constraint function. The model was implemented to solve the timetabling problem in one of Madrasah Tsanawiyah Islamic junior high school in Gorontalo. The result effective solutions in the form of subject and instructor timetabling that overcome the obstacles are obtained. To better the timetabling, supplementary teachers are still required for some subjects.

Stochastic Dynamic Cutting Plane for Multistage Stochastic Convex Programs

We introduce Stochastic Dynamic Cutting Plane (StoDCuP), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors. Finally, numerical experiments are presented on nondifferentiable multistage stochastic programs where Inexact StoDCuP computes a good approximate policy quicker than StoDCuP while SDDP and the previous inexact variant of SDDP combined with Mosek library to solve subproblems were not able to solve the differentiable reformulation of the problem.

Constrained optimization of line-start PM motor based on the gray wolf optimizer

This paper presents the algorithm and computer software for constrained optimization based on the gray wolf algorithm. The gray wolf algorithm was combined with the external penalty function approach. The optimization procedure was developed using Borland Delphi 7.0. The developed procedure was then applied to design of a line-start PM synchronous motor. The motor was described by three design variables which determine the rotor structure. The multiplicative compromise function consisted of three maintenance parameters of designed motor and one non-linear constraint function was proposed. Next, the result obtained for the developed procedure (together with the gray wolf algorithm) was compared with results obtained using: (a) the particle swarm optimization algorithm, (b) the bat algorithm and (c) the genetic algorithm. The developed optimization algorithm is characterized by good convergence, robustness and reliability. Selected results of the computer simulation are presented and discussed.

An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system

We consider the solution of the fully kinetic (including electrons) Vlasov-Ampère system in a one-dimensional physical space and two-dimensional velocity space (1D-2V) for an arbitrary number of species with a time-implicit Eulerian algorithm. The problem of velocity-space meshing for disparate thermal and bulk velocities is dealt with by an adaptive coordinate transformation of the Vlasov equation for each species, which is then discretized, including the resulting inertial terms. Mass, momentum, and energy are conserved, and Gauss's law is enforced to within the nonlinear convergence tolerance of the iterative solver through a set of nonlinear constraint functions while permitting significant flexibility in choosing discretizations in time, configuration, and velocity space. We mitigate the temporal stiffness introduced by, e.g., the plasma frequency through the use of high-order/low-order (HOLO) acceleration of the iterative implicit solver. We present several numerical results for canonical problems of varying degrees of complexity, including the multiscale ion-acoustic shock wave problem, which demonstrate the efficacy, accuracy, and efficiency of the scheme.

Improved population initialization method and its application in bridge optimization

When solving bridge optimization problems by traditional genetic algorithms, a random method is often used to generate the initial population, which can’t ensure that the initial population is evenly distributed in the solution space and reasonably represents the solution space. In order to solve the shortcomings of random method, an improved population initialization method is proposed by combining random method with good point set. Based on it, aimed at the characteristic of implicit non-linear objective function or constraint function in bridge structure optimization, a BP neural network is used to simulate the relations between the objective function or constraint function and design variables. Then this paper takes a steel truss bridge as an example of optimization. Taken the allowable deflection of bridges as limit and aiming at minimizing the amount of steel, optimization model of bridge is established. Finally, the improved population initialization method is applied to solve the optimization model. The results show that the improved population initialization method can not only improve the uniformity of the population, but also be suitable for optimization design of bridges.

An augmented step size adjustment method for the performance measure approach: Toward general structural reliability-based design optimization

Heavy computational burden has been one of the largest barriers to the application of reliability-based design optimization (RBDO) in real-world structures. The key difficulty of RBDO lies in how to perform reliability analysis efficiently and robustly. In performance measure approach (PMA)-based RBDO, the reliability analysis process searches for the minimum performance target point (MPTP) with the target reliability index in standard normal space. Many methods have been proposed to improve the efficiency and robustness of the PMA. However, these methods may face the convergence problem for highly nonlinear constraint functions; or high computational cost for weakly nonlinear ones. More importantly, most existing methods are very sensitive to the selection of algorithm parameters. In this paper, an augmented step size adjustment (ASSA) method is proposed to boost the iterative process in terms of both efficiency and robustness. According to the relative positions of the direction vector and negative gradient direction and the angle between them at each iterative point, a fire-new strategy is established to identify the oscillation during the iterative process and define the iterative step size. Seven inverse reliability analysis problems and three RBDO benchmarks are used to validate the performance of the ASSA method. The results indicate that the ASSA method has wide applicability for nonlinear constraint functions and achieves efficient and robust performance. Furthermore, the results demonstrate that the ASSA method can be regarded as a reliable and effective method for addressing PMA-based RBDO problems.

A heuristic optimization considering probabilistic constraints via an equivalent single variable Pearson distribution system

The computational cost of an optimization considering probabilistic constraints is often expensive due to the reliability analysis performed inside the optimization loop. This study computes reliability using an equivalent single variable Pearson’s distribution system. To retain the accuracy without sacrificing efficiency, dimension reduction method is used to reduce the required number of points in Gaussian–Hermite integration. In addition, four advance machine learning techniques are used to construct a hyper-probability density function to solve the singularity issue in the Pearson distribution system. The ε-constrained is utilized in a metaheuristic optimization algorithm (PSO/SOS) to consider the probabilistic constraints. The proposed algorithm is verified with several literature studies. Results shown that it is able to find an optimal design for problems having linear, highly nonlinear, and implicit probabilistic constraint functions with normal or non-normal variables. It is found that increasing number of variates in dimension reduction can provide a more accurate estimation of moments of the performance functions and enhance the solution accuracy. To demonstrate the applicability of the proposed algorithm to a practical problem, a platform of the SAP2000 Open Application Programming Interface (OAPI) is developed to find an optimal design for a three story steel frame under nonlinear time history analysis.

Uncertainty quantification for constitutive model calibration of brain tissue.

The results of a study comparing model calibration techniques for Ogden's constitutive model that describes the hyperelastic behavior of brain tissue are presented. One and two-term Ogden models are fit to two different sets of stress-strain experimental data for brain tissue using both least squares optimization and Bayesian estimation. For the Bayesian estimation, the joint posterior distribution of the constitutive parameters is calculated by employing Hamiltonian Monte Carlo (HMC) sampling, a type of Markov Chain Monte Carlo method. The HMC method is enriched in this work to intrinsically enforce the Drucker stability criterion by formulating a nonlinear parameter constraint function, which ensures the constitutive model produces physically meaningful results. Through application of the nested sampling technique, 95% confidence bounds on the constitutive model parameters are identified, and these bounds are then propagated through the constitutive model to produce the resultant bounds on the stress-strain response. The behavior of the model calibration procedures and the effect of the characteristics of the experimental data are extensively evaluated. It is demonstrated that increasing model complexity (i.e., adding an additional term in the Ogden model) improves the accuracy of the best-fit set of parameters while also increasing the uncertainty via the widening of the confidence bounds of the calibrated parameters. Despite some similarity between the two data sets, the resulting distributions are noticeably different, highlighting the sensitivity of the calibration procedures to the characteristics of the data. For example, the amount of uncertainty reported on the experimental data plays an essential role in how data points are weighted during the calibration, and this significantly affects how the parameters are calibrated when combining experimental data sets from disparate sources.

A power balancing method of distributed generation and electric vehicle charging for minimizing operation cost of distribution systems with uncertainties

AbstractA power balancing method of distributed generation (DG) and electric vehicle charging is presented for minimizing operation costs of distribution systems with uncertainties, which includes the uncertainties in output power of DG and the randomness of charging power of electric vehicles (EV). A probability model is established for the uncertain characteristics of DG and electric vehicle charging. A multi‐state optimization coordination method for DG of renewable energy systems and electric vehicle charging in distribution systems based on quadratic rotation cone programming is presented to minimize the expected generation cost of generators in the main power grid, the expected operation cost of DG systems in distribution systems, and the expected social outage loss. An objective function maximizing the operation efficiency of distribution systems with DGs and EVs is proposed. Using quadratic rotated conic programming, the nonlinear objective function and constraint functions are transformed into a linear form. An IEEE 14‐node distribution system is used as a study example to illustrate adaptability of the proposed model and the feasibility of the proposed method. The simulation results show that the proposed method simplifies the original problem of the optimization problem and makes its solution faster, more stable, and better.

Derived heuristics-based consistent optimization of material flow in a gold processing plant

ABSTRACTMaterial flow in a chemical processing plant often follows complicated control laws and involves plant capacity constraints. Importantly, the process involves discrete scenarios which when modelled in a programming format involves if–then–else statements. Therefore, a formulation of an optimization problem of such processes becomes complicated with nonlinear and non-differentiable objective and constraint functions. In handling such problems using classical point-based approaches, users often have to resort to modifications and indirect ways of representing the problem to suit the restrictions associated with classical methods. In a particular gold processing plant optimization problem, these facts are demonstrated by showing results from MATLAB®'s well-known fmincon routine. Thereafter, a customized evolutionary optimization procedure which is capable of handling all complexities offered by the problem is developed. Although the evolutionary approach produced results with comparatively less variance over multiple runs, the performance has been enhanced by introducing derived heuristics associated with the problem. In this article, the development and usage of derived heuristics in a practical problem are presented and their importance in a quick convergence of the overall algorithm is demonstrated.

A Hybrid Loop Approach Using the Sufficient Descent Condition for Accurate, Robust, and Efficient Reliability-Based Design Optimization

For reliability-based design optimization (RBDO) problems, single loop approaches (SLA) are very efficient but prone to converge to inappropriate point for highly nonlinear constraint functions, and double loop approaches (DLA) are proven to be accurate but require more iterations to achieve stable results. In this paper, an adjusted advanced mean value (AAMV) method is firstly proposed to improve the robustness and efficiency of performance measure approach. The global convergence of the AAMV is guaranteed using sufficient descent condition for the reliability loop in RBDO. Then, a hybrid RBDO method is developed to improve the efficiency of DLA and accuracy of SLA, on the basis of sufficient descent condition and AAMV method, named as hybrid single and double loops (HSD) method. Three nonlinear concave and convex performance functions are used to illustrate the efficiency and robustness of the AAMV method; then the accuracy, robustness, and efficiency of the proposed HSD method are compared to current SLA and DLA through another three benchmark nonlinear RBDO examples. Results show that the AAMV is more robust and efficient than the existing reliability analysis methods. The HSD is more accurate than the SLA for highly nonlinear problems, and also exhibits a better performance than the DLA from the point of view of both robustness and efficiency.

A Derivative-Free Approach to Constrained Multiobjective Nonsmooth Optimization

In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. We define a linesearch-based solution method, and we show that it converges to a set of Pareto stationary points. To this aim, we carry out a theoretical analysis of the problem by only assuming Lipschitz continuity of the functions; more specifically, we give new optimality conditions that take explicitly into account the bound constraints, and prove that the original problem is equivalent to a bound constrained problem obtained by penalizing the nonlinear constraints with an exact merit function. Finally, we present the results of some numerical experiments on bound constrained and nonlinearly constrained problems, showing that our approach is promising when compared to a state-of-the-art method from the literature.

Optimal Magnet Design for Lorentz Force Eddy-Current Testing

We propose a procedure to determine optimal magnet systems in the framework of the nondestructive evaluation technique Lorentz force eddy-current testing (LET). The underlying optimization problem is clearly defined considering the problem specificity of nondestructive testing scenarios. The quantities involved are classified as design variables, and system and scaling parameters to provide a high level of generality. The objective function is defined as the absolute defect response signal (ADS) of the Lorentz force resulting from an inclusion inside the object under test. Associated constraints are defined according to the applied force sensor technology. A numerical procedure based on the finite-element method is proposed to evaluate the nonlinear objective and constraint functions, and the method of sequential quadratic programming is applied to determine unconstrained and constrained optimal magnet designs. Consequently, we propose a new magnet design based on the Halbach principle in combination with high saturation magnetization iron–cobalt alloys. The proposed magnet system outperforms currently available cylindrical magnets in terms of weight and performance. The corresponding defect response signal is increased up to 180% in the case of small defects located close to the surface of the specimen. The combination of active and passive magnetic materials provides an increase of the ADS by 15% compared with the magnet designs that are built solely from permanent magnet material. The proposed procedure provides a highly adaptive optimization strategy in the framework of LET and proposes new magnet systems with inherently improved characteristics.

Hybrid Methodology of Nonlinear Goal Programming

What we demonstrate here is a nonlinear goal-programming (NGP) algorithm based on hybrid connection of the modified simplex method of goal programming, gradient method of feasible directions and method of optimal displacement size finding-called HNGPM. Iterative methodology is given in five steps: (1) linearization the set of nonlinear constraints at particular point, (2) solving the problem of normalized linear goal programming, (3) feasible direction computation, (4) calculating optimal step length displacement, and (5) testing out convergence problem. Our idea was to apply Euler’s theorem for the “total” linearization of the nonlinear constraints (in the space) around particular point. According to Euler’s theorem, it is possible to apply this methodology to solve the problems of NGP whether the nonlinear constraint functions are linearly or positively homogeneous.

A novel acc-jerk-limited NURBS interpolation enhanced with an optimized S-shaped quintic feedrate scheduling scheme

This paper presents a new adaptive acc-jerk-limited nonuniform rational B-spline (NURBS) interpolation method based on an optimized S-shaped C2 quintic feedrate planning scheme. At first, the modified quintic feedrate profile for each sharp corner during the acceleration/deceleration (acc/dec) stage is constructed. To this end, two feedrate slope correction coefficients (FSCC) are introduced for zero end point acceleration and jerk conception in acc/dec stage of each sharp corner. Also, a new algorithm is recommended to compute the acc/dec stage traverse time with respect to the deceleration starting time in this paper. Then, the modified quintic feedrate scheduling scheme equipped with the FSCC is improved for the tool path containing several sharp corners. The FSCC and the deceleration starting times corresponding to all sharp corners are evaluated using an optimization method such that the total machining time to be minimized. In this paper, the pattern search algorithm equipped with the nonlinear constraint function including the acceleration and jerk limitations Matlab code is used for obtaining the optimized parameters to accomplish the acc-jerk-limited feedrate scheduling scheme along the tool path. The proposed interpolation method is performed for several case studies and compared with the previously published methods to evaluate the effectiveness of the designed adaptive acc-jerk-limited feedrate scheduling scheme. The simulation results demonstrate that the proposed interpolation algorithm is capable for providing a smooth feedrate transition for all stages of motion along the tool path and yields satisfactory performances such as total machining time and the interpolation steps.

Deriving robust counterparts of nonlinear uncertain inequalities

In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.

Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187---1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The ?BB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.

Deriving Robust Counterparts of Nonlinear Uncertain Inequalities

In this paper we provide a systematic way to construct the robust counterpart of a nonlinear uncertain inequality that is concave in the uncertain parameters. We use convex analysis (support functions, conjugate functions, Fenchel duality) and conic duality in order to convert the robust counterpart into an explicit and computationally tractable set of constraints. It turns out that to do so one has to calculate the support function of the uncertainty set and the concave conjugate of the nonlinear constraint function. Conveniently, these two computations are completely independent. This approach has several advantages. First, it provides an easy structured way to construct the robust counterpart both for linear and nonlinear inequalities. Second, it shows that for new classes of uncertainty regions and for new classes of nonlinear optimization problems tractable counterparts can be derived. We also study some cases where the inequality is nonconcave in the uncertain parameters.