Active Set Technique Research Articles

Optimal control of thermoregulation in the human dermal regions investigated through the stochastic integrated techniques

In this research, stochastic computing techniques based on artificial neural networks are applied to the proposed singular nonlinear differential equation to explain thermoregulation in the human dermal region problem to investigate and predict the effect of bioheat temperature on human skin at different atmospheric temperatures with different cases with considering the impact of both air convection thermal sensitivity and temperature, where sequential quadratic programming, interior point technique, and active set technique are included in the designed techniques. Moreover, thermal dynamics in the epidermal layer that is affected by both cooling and heating treatments, and the temperature profile across space and time have been examined. Numerical convergence analysis has been applied to endorse the nature of precision and convergence of the designed stochastic computing techniques. To preserve thermal balance, this leads to modifications in the human body's temperature regulation. The statistical analysis is provided comprehensively in the form of graphs and tables to further enhance the significance in terms of accuracy, efficiency, and convergence of the current study.

Integrated Stochastic Investigation of Singularly Perturbed Delay Differential Equations for the Neuronal Variability Model

The proposed research utilizes a computational approach to attain a numerical solution for the singularly perturbed delay differential equation (SPDDE) problem arising in the neuronal variability model through artificial neural networks (ANNs) with different solvers. The log-sigmoid function is used to construct the fitness function. The implementation of ANN on SPDDE problems is formulated for different solvers and trained with different weights. The optimization solvers such as the genetic algorithm (GA), sequential quadratic programming (SQP), and pattern search (PS) are hybridized with the active set technique (AST) and the interior-point technique (IPT) and is used to check the accuracy and rapid convergence of the numerical results of the SPDDE model. The numerical outcomes demonstrate that the system is easy to handle and efficient to solve with boundary conditions. Moreover, we used the mean residual error for one hundred runs for each solver to validate the accuracy of the proposed scheme.

Evolutionary Techniques for the Solution of Bio-Heat Equation Arising in Human Dermal Region Model

The proposed research work analyzes the bio-inspired problem through artificial neural networks with a feed-forward approach utilized to approximate the numerical results for singular nonlinear bio-heat equation (BHE) with boundary conditions based on four different scenarios created on the variation of environmental temperature to illustrate the effects of temperature on the human dermal region. The log-sigmoid function is used to construct the fitness function, while the optimization solvers: pattern search and genetic algorithm, are then hybridized with the active set technique, interior point technique, sequential quadratic programming for accurate and reliable results of the proposed BHE with various scenarios where the convergence of the numerical results is also analyzed. Moreover, a comparison of the proposed technique is expressed through residual error that reveals the nature of the numerical results and their efficiency. Additionally, a comprehensive statistical analysis is presented for the designed technique to better illustrate the accuracy, reliability, and efficiency of the obtained results.

Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation

We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.

A decomposition method for lasso problems with zero-sum constraint

In this paper, we consider lasso problems with zero-sum constraint, commonly required for the analysis of compositional data in high-dimensional spaces. A novel algorithm is proposed to solve these problems, combining a tailored active-set technique, to identify the zero variables in the optimal solution, with a 2-coordinate descent scheme. At every iteration, the algorithm chooses between two different strategies: the first one requires to compute the whole gradient of the smooth term of the objective function and is more accurate in the active-set estimate, while the second one only uses partial derivatives and is computationally more efficient. Global convergence to optimal solutions is proved and numerical results are provided on synthetic and real datasets, showing the effectiveness of the proposed method. The software is publicly available.

Investigations of non-linear induction motor model using the Gudermannian neural networks

This study aims to solve the non-linear fifth-order induction motor model (FO-IMM) using the Gudermannian neural networks (GNN) along with the optimization procedures of global search as a genetic algorithm together with the quick local search process as active-set technique (GNN-GA-AST). The GNN are executed to discretize the non-linear FO-IMM to prompt the fitness function in the procedure of mean square error. The exactness of the GNN-GA-AST is observed by comparing the obtained results with the reference results. The numerical performances of the stochastic GNN-GA-AST are provided to tackle three different variants based on the non-linear FO-IMM to authenticate the consistency, significance and efficacy of the designed stochastic GNN-GA-AST. Additionally, statistical illustrations are available to authenticate the precision, accuracy and convergence of the designed stochastic GNN-GA-AST.

An active set quasi-Newton method with projection step for monotone nonlinear equations

In this paper, an active set quasi-Newton method for bound constrained nonlinear equation is proposed. By using this active set technique, we only need to solve a reduced dimension linear equation at each iteration to generate the search direction. The algorithm is a combination of the quasi-Newton method and projection method. Firstly we use the quasi-Newton step as the trial step and then use a projection technique to generate the next iteration point. Our key observation is that the algorithm generates a bounded iteration sequence automatically even if the bounds are equal to infinity and the global convergence is obtained in the sense that the whole sequence converges to the stationary point. The numerical tests show the efficiency of the algorithm.

Integrated neuro-evolution-based computing solver for dynamics of nonlinear corneal shape model numerically

In this study, bio-inspired computational techniques have been exploited to get the numerical solution of a nonlinear two-point boundary value problem arising in the modelling of the corneal shape. The computational process of modelling and optimization makes enormously straightforward to obtain accurate approximate solutions of the corneal shape models through artificial neural networks, pattern search (PS), genetic algorithms (GAs), simulated annealing (SA), active-set technique (AST), interior-point technique, sequential quadratic programming and their hybrid forms based on GA–AST, PS–AST and SA–AST. Numerical results show that the designed solvers provide a reasonable precision and efficiency with minimal computational cost. The efficacy of the proposed computing strategies is also investigated through a descriptive statistical analysis by means of histogram illustrations, probability plots and one-way analysis of variance.

Limited memory BFGS algorithm for the matrix approximation problem in Frobenius norm

This paper proposes an L-BFGS algorithm based on the active set technique to solve the matrix approximation problem: given a symmetric matrix, find a nearest approximation matrix in the sense of Frobenius norm to make it satisfy some linear equalities, inequalities and a positive semidefinite constraint. The problem is a convex optimization problem whose dual problem is a nonlinear convex optimization problem with non-negative constraints. Under the Slater constraint qualification, it has zero duality gap with the dual problem. To handle large-scale dual problem, we make use of the active set technique to estimate the active constraints, and then the L-BFGS method is used to accelerate free variables. The global convergence of the proposed algorithm is established under certain conditions. Finally, we conduct some preliminary numerical experiments, and compare the L-BFGS method with the inexact smoothing Newton method, the projected BFGS method, the alternating direction method and the two-metric projection method based on the L-BFGS. The numerical results show that our algorithm has some advantages in terms of CPU time when a large number of linear constraints are involved.

Neuro-Heuristic Computational Intelligence for nonlinear Thomas-Fermi equation using trigonometric and hyperbolic approximation

We propose a computational intelligence technique to solve the nonlinear Thomas-Fermi equation for a finite interval. This technique uses an optimized approximation to reduce the error between the reference solution and optimized numerical solution with trigonometric and hyperbolic functions. Computational intelligence technique is based on optimization tools like Active Set Technique (AST), Interior Point Technique (IPT) with Sequential Quadratic Programming (SQP) through artificial neural networks (ANN). Furthermore, the proposed model results can be applied in atomic models phenomena. Numerical results showed that the optimizers provided a high level of accuracy, as well as the efficiency with marginal computational requirements infinite domain. Statistical analysis is presented, particularly fitting of normal distribution based on absolute errors (AEs) of proposed solvers and Chi-square distribution data curve fitting. It provides evidence that the present solution is highly accurate in the context of confidence bands.

Novel applications of intelligent computing paradigms for the analysis of nonlinear reactive transport model of the fluid in soft tissues and microvessels

This article presents a methodology to solve a one-dimensional steady-state nonlinear reactive transport model (RTM) that is meant for fluid and solute transport model of soft tissues and microvessels. The methodology integrates the artificial neural network (ANN), genetic algorithms (GAs), and pattern search (PS) aided by active-set technique (AST) and interior-point technique (IPT). The RTM is represented with nonlinear second-order system based on the boundary value problem of ordinary differential equation. The ANN modeling is used for governing expression of RTM to form a fitness function in mean square sense, and optimization solvers based on the GA, PS, GA-AST, GA-IPT, PS-AST, PS-IPT are used for viable learning of weights. Proposed techniques are applied to different nonlinear RTMs based on variation in the characteristic reaction rate and half-saturation concentration. The proposed stochastic numerical solutions are compared with state-of-the-art solvers in order to check the accuracy and convergence based on sufficient large multiple runs of the algorithms.

Stochastic technique for solutions of non-linear fin equation arising in thermal equilibrium model

In this study, a stochastic numerical technique is used to investigate the numerical solution of heat transfer temperature distribution system using feed forward artificial neural networks. Mathematical model of fin equation is formulated with the help of artificial neural networks. The effect of the heat on a rectangular fin with thermal conductivity and temperature dependent internal heat generation is calculated through neural networks optimization with optimizers like active set technique, interior point technique, pattern search, genetic algorithm and a hybrid approach of pattern search - interior point technique, genetic algorithm - active set technique, genetic algorithm - interior point technique, and genetic algorithm - sequential quadratic programming with different selections of weights. The governing fin equation is transformed into an equivalent non-linear second order ODE. For this transformed ODE model we have performed several simulations to provide the justification of better convergence of results. Moreover, the effectiveness of the designed models is validated through a complete statistical analysis. This study reveals the importance of rectangular fins during the heat transformation through the system.

A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head

Bio-inspired computing approaches are effective to solve a variety of dynamical problems. The strength of these stochastic solvers is exploited for the numerical treatment of a nonlinear heat conduction model of the human head using artificial neural networks (ANNs), genetic algorithms (GAs), active-set technique (AST), and their hybrids. The universal function approximation competencies of unsupervised ANNs are utilized in constructing the mathematical model of the problem by defining an error function in the mean squared sense. The training of the design parameters of ANN models is made with global search brilliance of GAs, viably local search with AST and hybrid approach GA-AST. The results of the proposed schemes are determined in terms of temperature profiles by considering variants of the problem with different Biot numbers, metabolic thermogenesis slope parameters and thermogenesis heat production factors. The correctness, effectiveness and convergence of the proposed approaches are also ascertained through statistics.

Generation of Pareto optimal solutions for multi-objective optimization problems via a reduced interior-point algorithm

ABSTRACTIn this paper, a reduced interior-point (RIP) algorithm is introduced to generate a Pareto optimal front for multi-objective constrained optimization (MOCP) problem. A weighted Tchebychev metric approach is used together with achievement scalarizing function approach to convert MOCP problem to a single-objective constrained optimization (SOCO) problem. An active-set technique is used together with a Coleman–Li scaling matrix and a decrease interior-point method to solve SOCO problem. A Matlab implementation of RIP algorithm was used to solve three cases and application. The results showed that the RIP algorithm is promising when compared with well-known algorithms and the computations may be superior relevant for comprehending real-world application problems.

Intelligent computing to solve fifth-order boundary value problem arising in induction motor models

In this study, biologically inspired intelligent computing approached based on artificial neural networks (ANN) models optimized with efficient local search methods like sequential quadratic programming (SQP), interior point technique (IPT) and active set technique (AST) is designed to solve the higher order nonlinear boundary value problems arise in studies of induction motor. The mathematical modeling of the problem is formulated in an unsupervised manner with ANNs by using transfer function based on log-sigmoid, and the learning of parameters of ANNs is carried out with SQP, IPT and ASTs. The solutions obtained by proposed methods are compared with the reference state-of-the-art numerical results. Simulation studies show that the proposed methods are useful and effective for solving higher order stiff problem with boundary conditions. The strong motivation of this research work is to find the reliable approximate solution of fifth-order differential equation problems which are validated through strong statistical analysis.

Stochastic numerical treatment for solving Falkner–Skan equations using feedforward neural networks

In this article, the artificial intelligence techniques have been used for the solution of Falkner–Skan (FS) equations based on neural networks optimized with three methods including active set technique, sequential quadratic programming and genetic algorithms (GA) hybridization. Log-sigmoid activation function is used in artificial neural network architecture. The proposed techniques are applied to a number of cases for Falkner–Skan problems, and results were compared with GA hybrid results in all cases and were found accurate. The level of accuracy is examined through statistical analyses based on a sufficiently large number of independent runs.

A spline smoothing homotopy method for nonconvex nonlinear programming

Homotopy methods are globally convergent under weak conditions and robust; however, the efficiency of a homotopy method is closely related with the construction of the homotopy map and the path tracing algorithm. Different homotopies may behave very different in performance even though they are all theoretically convergent. In this paper, a spline smoothing homotopy method for nonconvex nonlinear programming is developed using cubic spline to smooth the max function of the constraints of nonlinear programming. Some properties of spline smoothing function are discussed and the global convergence of spline smoothing homotopy under the weak normal cone condition is proven. The spline smoothing technique uses a smooth constraint instead of m constraints and acts also as an active set technique. So the spline smoothing homotopy method is more efficient than previous homotopy methods like combined homotopy interior point method, aggregate constraint homotopy method and other probability one homotopy methods. Numerical tests with the comparisons to some other methods show that the new method is very efficient for nonlinear programming with large number of complicated constraints.

Stochastic approach for the solution of multi-pantograph differential equation arising in cell-growth model

In this paper, a computational technique is introduced for the solution of the first order multi-pantograph differential equation (MPDE) through some well-known optimization algorithms like sequential quadratic programming (SQP) and Active Set Technique (AST). Furthermore, artificial neural network (ANN) is used for networking of the first order multi-pantograph differential equation in used to provide mathematical model based on unsupervised error for equation. Moreover, mathematical modeling has been performed perfectly through multi-runs for simulation to justify the better convergence of the solutions. Also, two examples are presented to exhibit the aptitude of the method SQP and AST. The comparative study will be made with reported techniques such as variational iteration technique (VIT) [6] and collocation based on Bernstein polynomial method (BCM) [6].

A spline smoothing Newton method for finite minimax problems

A spline smoothing stabilized Newton method for finite minimax problems is developed. The spline smoothing technique uses a smooth cubic spline instead of a max function, and, at any fixed point, only a few components of the max function are involved; i.e., it introduces an active set technique, so the proposed method is more efficient for minimizing the maximum function of a large number of complicated functions. Some numerical results comparisons with other methods are also given to show the efficiency of the new method.

An infeasible QP-free algorithm without a penalty function or a filter for nonlinear inequality-constrained optimization

In this paper, we present an infeasible quadratic programming-free method without using a penalty function or a filter for inequality-constrained optimization. At each iteration, three linear equations with the same coefficient matrix are solved. Furthermore, the nearly active set technique is used to reduce the amount of computational work. Under mild conditions, we prove that the proposed method has global and superlinear local convergence. At last, preliminary numerical results and comparison are reported.