Showing papers in "Numerical Algebra, Control and Optimization in 2021"
TL;DR: The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.
Abstract: One of the challenges facing supply chain designers is designing a sustainable and resilient supply chain network. The present study considers a closed-loop supply chain by taking into account sustainability, resilience, robustness, and risk aversion for the first time. The study suggests a two-stage mixed-integer linear programming model for the problem. Further, the robust counterpart model is used to handle uncertainties. Furthermore, conditional value at risk criterion in the model is considered in order to create real-life conditions. The sustainability goals addressed in the present study include minimizing the costs, \begin{document}$ \text{CO}_2 $\end{document} emission, and energy, along with maximizing employment. In addition, effective environmental and social life-cycle evaluations are provided to assess the associated effects of the model on society, environment, and energy consumption. The model aims to answer the questions regarding the establishment of facilities and amount of transported goods between facilities. The model is implemented in a car assembler company in Iran. Based on the results, several managerial insights are offered to the decision-makers. Due to the complexity of the problem, a constraint relaxation is applied to produce quality upper and lower bounds in medium and large-scale models. Moreover, the LP-Metric method is used to merge the objectives to attain an optimal solution. The results revealed that the robust counterpart provides a better estimation of the total cost, pollution, energy consumption, and employment level compared to the basic model.
94 citations
TL;DR: In this article, a modified self-adaptive inertial subgradient extragradient algorithm is presented, in which the two projections are made onto some half spaces and under mild conditions, the sequence generated by the proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space.
Abstract: In this paper, we present a new modified self-adaptive inertial subgradient extragradient algorithm in which the two projections are made onto some half spaces Moreover, under mild conditions, we obtain a strong convergence of the sequence generated by our proposed algorithm for approximating a common solution of variational inequality problem and common fixed point of a finite family of demicontractive mappings in a real Hilbert space The main advantages of our algorithm are: strong convergence result obtained without prior knowledge of the Lipschitz constant of the related monotone operator, the two projections made onto some half-spaces and the inertial technique which speeds up rate of convergence Finally, we present an application and a numerical example to illustrate the usefulness and applicability of our algorithm
34 citations
TL;DR: In this article, conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map, and the relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in higher gaps.
Abstract: In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.
19 citations
TL;DR: In this paper, a modified extragradient algorithm for split pseudo-monotone variational inequality problem in real Hilbert spaces is presented. But the authors do not consider the Lipschitz constant of the pseudo monotone operator.
Abstract: In this paper, we introduce and study a modified extragradient algorithm for approximating solutions of a certain class of split pseudo-monotone variational inequality problem in real Hilbert spaces. Using our proposed algorithm, we established a strong convergent result for approximating solutions of the aforementioned problem. Our strong convergent result is obtained without prior knowledge of the Lipschitz constant of the pseudo-monotone operator used in this paper, and with minimized number of projections per iteration compared to other results on split variational inequality problem in the literature. Furthermore, numerical examples are given to show the performance and advantage of our method as well as comparing it with related methods in the literature.
17 citations
TL;DR: A feature selection approach based on the binary whale optimization algorithm with different kinds of updating techniques for the time-varying transfer functions is proposed and it proved that BWOA-TV2 has consistency in feature selection and it gives rise to the high accuracy of the classification with more congruent in the convergence.
Abstract: Feature selection is a valuable tool in supervised machine learning research fields, such as pattern recognition or classification problems. Feature selection used to eliminate irrelevant and noise features that adversely affect results. Swarm algorithms are usually used in feature selection problem; these algorithms need transfer functions that change search space from continuous to the discrete. However, transfer functions are the backbone of all binary swarm algorithms. Transfer functions in the current formula cannot provide binary swarm algorithms with a fit balance between exploration and exploitation stages. In this work, a feature selection approach based on the binary whale optimization algorithm with different kinds of updating techniques for the time-varying transfer functions is proposed. To evaluate the performance of the proposed method, three of each chemical and biological binary datasets are used. The results proved that BWOA-TV2 has consistency in feature selection and it gives rise to the high accuracy of the classification with more congruent in the convergence. It worth mentioning that the proposed method is proved advance in performance over competitor optimization algorithms, such as particle swarm optimization (PSO) and firefly optimization (FO) that commonly used in this field.
16 citations
TL;DR: Li et al. as discussed by the authors developed a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system, which is obtained by combining its search direction with the projection method.
Abstract: By exploiting the idea employed in the spectral Dai-Yuan method by Xue et al. [IEICE Trans. Inf. Syst. 101 (12)2984-2990 (2018)] and the approach applied in the modified Hager-Zhang scheme for nonsmooth optimization [PLos ONE 11(10): e0164289 (2016)], we develop a Dai-Yuan type iterative scheme for convex constrained nonlinear monotone system. The scheme's algorithm is obtained by combining its search direction with the projection method [Kluwer Academic Publishers, pp. 355-369(1998)]. One of the new scheme's attribute is that it is derivative-free, which makes it ideal for solving non-smooth problems. Furthermore, we demonstrate the method's application in image de-blurring problems by comparing its performance with a recent effective method. By employing mild assumptions, global convergence of the scheme is determined and results of some numerical experiments show the method to be favorable compared to some recent iterative methods.
7 citations
TL;DR: It is shown that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach and that the method can solve problems of this type much faster in comparison with (very few) available options.
Abstract: We consider some important computational aspects of the long-step path-following algorithm developed in our previous work and show that a broad class of complicated optimization problems arising in quantum information theory can be solved using this approach. In particular, we consider one difficult optimization problem involving the quantum relative entropy in quantum key distribution and show that our method can solve problems of this type much faster in comparison with (very few) available options.
6 citations
TL;DR: An observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible and guarantees that theclosed-loop system is regular, impulse-free and stable in the event of actuator failures.
Abstract: A method of designing observer-based feedback controller against actuator failures for uncertain singular fractional order systems (SFOS) is presented in this paper. By establishing actuator fault model and state observer, an observer-based fault-tolerant state feedback controller is developed such that the closed-loop SFOS is admissible. The controller designed by the proposed method guarantees that the closed-loop system is regular, impulse-free and stable in the event of actuator failures. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.
6 citations
TL;DR: A methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM) and results correspond that the primal aim of chaos control in the given system have been attained computationally.
Abstract: In this manuscript, we design a methodology to investigate the anti-synchronization scheme in chaotic chemical reactor system using adaptive control method (ACM). Initially, an ACM has been proposed and analysed systematically for controlling the microscopic chaos found in the discussed system which is essentially described by employing Lyapunov stability theory (LST). The required asymptotic stability criterion of the state variables of the discussed system having unknown parameters is derived by designing appropriate control functions and parameter updating laws. In addition, numerical simulation results in MATLAB software are performed to illustrate the effective presentation of the considered strategy. Simulations outcomes correspond that the primal aim of chaos control in the given system have been attained computationally.
6 citations
TL;DR: Yang et al. as discussed by the authors presented an iterative method for solving the convex constraint nonlinear equation problem, which incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method.
Abstract: We present an iterative method for solving the convex constraint nonlinear equation problem. The method incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method by Yang et al. for solving the unconstrained optimization problem. The proposed method does not require the Jacobian information, nor does it require to store any matrix at each iteration. Thus, it has the potential to solve large-scale non-smooth problems. Under some standard assumptions, the convergence analysis of the method is established. Finally, to show the applicability of the proposed method, the proposed method is used to solve the \begin{document}$ \ell_1 $\end{document} -norm regularized problems to restore blurred and noisy images. The numerical experiment indicates that our result is a significant improvement compared with the related methods for solving the convex constraint nonlinear equation problem.
6 citations
TL;DR: A rigorous analysis in damping of oscillations in a power network utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics.
Abstract: The assimilation of flexible AC transmission (FACTS) controllers to the existing power network outweigh the numerous alternatives in enhancing the damping behavior for the inter-area /intra-area system oscillations of a power network. This paper provides a rigorous analysis in damping of oscillations in a power network. It utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics. A comprehensive system mathematical modelling has been developed for demonstrating the system behavior under different loading conditions. A novel hybrid augmented grey wolf optimization-particle swarm optimization (AGWO-PSO) is proposed for the coordinated design of controllers static synchronous compensator (STATCOM) and power system stabilizers (PSSs). A multi-objective function, comprising damping ratio improvement and drifting the real part to the left-hand side of S-plane of the system poles, has been developed to achieve the objective and the effectiveness of the proposed algorithms have been analyzed by monitoring the system performance under different loading conditions. Eigenvalue analysis and damping nature of the system states under perturbation have been presented for the proposed algorithms under different loading conditions, and the performance evaluation of the proposed algorithms have been done by means of time of execution and the convergence characteristics.
TL;DR: In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered, and stability and convergence of the proposed scheme are discussed using mathematical induction.
Abstract: In this paper, a numerical approximation solution of a space-time fractional diffusion equation (FDE), involving Caputo-Katugampola fractional derivative is considered. Stability and convergence of the proposed scheme are discussed using mathematical induction. Finally, the proposed method is validated through numerical simulation results of different examples.
TL;DR: The so-called vector E $\ end{document} -dual problem in the sense of Mond-Weir is defined for the considered E $\end{ document} -differentiable multiobjective programming problem and several theorems are derived also under appropriate assumptions.
Abstract: In this paper, a new concept of generalized convexity is introduced for not necessarily differentiable vector optimization problems with \begin{document}$ E $\end{document} -differentiable functions. Namely, for an \begin{document}$ E $\end{document} -differentiable vector-valued function, the concept of \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity is defined as a generalization of the \begin{document}$ E $\end{document} -differentiable \begin{document}$ E $\end{document} -invexity notion and the concept of \begin{document}$ V $\end{document} -invexity. Further, the sufficiency of the so-called \begin{document}$ E $\end{document} -Karush-Kuhn-Tucker optimality conditions are established for the considered \begin{document}$ E $\end{document} -differentiable vector optimization problems with both inequality and equality constraints under \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity hypotheses. Furthermore, the so-called vector \begin{document}$ E $\end{document} -dual problem in the sense of Mond-Weir is defined for the considered \begin{document}$ E $\end{document} -differentiable multiobjective programming problem and several \begin{document}$ E $\end{document} -duality theorems are derived also under appropriate \begin{document}$ V $\end{document} - \begin{document}$ E $\end{document} -invexity assumptions.
TL;DR: In this article, the authors discuss conditions for exact (approximate) controllability and exact observability of stochastic implicit systems in Banach spaces, in terms of the GE-evolution operator and the dual principle.
Abstract: This paper discusses exact (approximate) controllability and exact (approximate) observability of stochastic implicit systems in Banach spaces. Firstly, we introduce the stochastic GE-evolution operator in Banach space and discuss existence and uniqueness of the mild solution to stochastic implicit systems by stochastic GE-evolution operator in Banach space. Secondly, we discuss conditions for exact (approximate) controllability and exact (approximate) observability of the systems considered in terms of stochastic GE-evolution operator and the dual principle. Finally, an illustrative example is given.
TL;DR: This approach proposes to formulate any piecewise smooth function as the expectation of a random variable and proposes to use the Boltzmann distribution as a smoothing approximation for this probability distribution.
Abstract: In this article, we present a new approach to construct smoothing approximations for piecewise smooth functions. This approach proposes to formulate any piecewise smooth function as the expectation of a random variable. Based on this formulation, we show that smoothing all elements of a defined space of piecewise smooth functions is equivalent to smooth a single probability distribution. Furthermore, we propose to use the Boltzmann distribution as a smoothing approximation for this probability distribution. Moreover, we present the theoretical results, error estimates, and some numerical examples for this new smoothing method in both one-dimensional and multiple-dimensional cases.
TL;DR: In this article, the authors analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space.
Abstract: This paper analyzed the new extragradient type algorithm with inertial extrapolation step for solving self adaptive split null point problem and pseudomonotone variational inequality in real Hilbert space. Furthermore, in this study, a strong convergence result is obtained without assuming Lipschitz continuity of the associated mapping and the operator norm is self adaptive. Additionally, the proposed algorithm only uses one projections onto the feasible set in each iteration. More so, the strong convergence results are obtained under some relaxed conditions on the initial factor and the iterative parameters. Numerical results are presented to illustrate the performance of the proposed algorithm.The results obtained in this study improved and extended related studies in the literature.
TL;DR: A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point.
Abstract: A bang-bang iteration method equipped with a component-wise line search strategy is introduced to solve unconstrained optimization problems. The main idea of this method is to formulate an unconstrained optimization problem as an optimal control problem to obtain an optimal trajectory. However, the optimal trajectory can only be generated by impulsive control variables and it is a straight line joining a guessed initial point to a minimum point. Thus, a priori bounds are imposed on the control variables in order to obtain a feasible solution. As a result, the optimal trajectory is made up of bang-bang control sub-arcs, which form an iterative model based on the Lyapunov function's theorem. This is to ensure monotonic decrease of the objective function value and convergence to a desirable minimum point. However, a chattering behavior may occur near the solution. To avoid this behavior, the Newton iterations are then applied to the proposed method via a two-phase approach to achieve fast convergence. Numerical experiments show that this new approach is efficient and cost-effective to solve the unconstrained optimization problems.
TL;DR: A two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractions-order plant model and the presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain.
Abstract: The approximation of the fractional-order controller (FOC) has already been recognized as a distinguished field of research in the literature of system and control. In this paper, a two-step design approach is presented to realize a fractional-order proportional integral controller (FOPI) for a class of fractional-order plant model. The design goals are based on some frequency domain specifications. The first stage of the work is focused on developing the pure continuous-time FOC, while the second stage actually realizes the FOPI controller in discrete-time representation. The presented approach is fundamentally dissimilar with respect to the conventional approaches of z -domain. In the process of realizing the FOC, the delta operator has been involved as a generating function due to its exclusive competency to unify the discrete-time system and its continuous-time counterpart at low sampling time limit. The well-known continued fraction expansion (CFE) method has been employed to approximate the FOPI controller in delta-domain. Simulation outcomes exhibit that the discrete-time FOPI controller merges to its continuous-time counterpart at the low sampling time limit. The robustness of the overall system is also investigated in delta-domain.
TL;DR: It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal controlProblem when use as a delayed boundary value solver of the necessary conditions.
Abstract: There are a limited number of user-friendly, publicly available optimal control software packages that are designed to accommodate problems with delays. GPOPS-Ⅱ is a well developed MATLAB based optimal control code that was not originally designed to accommodate problems with delays. The use of GPOPS-Ⅱ on optimal control problems with delays is examined for the first time. The use of various formulations of delayed optimal control problems is also discussed. It is seen that GPOPS-Ⅱ finds a suboptimal solution when used as a direct transcription delayed optimal control problem solver but that it is often able to produce a good solution of the optimal control problem when used as a delayed boundary value solver of the necessary conditions.
TL;DR: In this paper, a novel quadrature rule is formed combining Lobatto six point transformed rule and Gauss-Legendre five point transformed rules each having precision nine, and the mixed rule so formed is of precision eleven.
Abstract: A novel quadrature rule is formed combining Lobatto six point transformed rule and Gauss-Legendre five point transformed rule each having precision nine. The mixed rule so formed is of precision eleven. Through asymptotic error estimation the novelty of the quadrature rule is justified. Some test integrals have been evaluated using the mixed rule and its constituents both in non-adaptive and adaptive modes. The results are found to be quite encouraging for the mixed rule which is in conformation with the theoretical prediction.
TL;DR: In this article, the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor are discussed.
Abstract: This article focuses on the hidden insights about the Rayleigh-Taylor instability of two superimposed horizontal layers of nanofluids having different densities in the presence of rotation factor. Conservation equations are subjected to linear perturbations and further analyzed by using the Normal Mode technique. A dispersion relation incorporating the effects of surface tension, Atwood number, rotation factor and volume fraction of nanoparticles is obtained. Using Routh-Hurtwitz criterion the stable and unstable modes of Rayleigh-Taylor instability are discussed in the presence/absence of nanoparticles and presented through graphs. It is observed that in the absence/presence of nanoparticles, surface tension helps to stabilize the system and Atwood number has a destabilizing impact without the consideration of rotation factor. But if rotation parameter is considered (in the absence/presence of nanoparticles) then surface tension destabilizes the system while Atwood number has a stabilization effect (for a particular range of wave number). The volume fraction of nanoparticles destabilizes the system in the absence of rotation but in the presence of rotation the stability of the system is significantly stimulated by the nanoparticles.
TL;DR: A PID control method which combined optimal control strategy is proposed in this paper, and through the numerical simulation, the effectiveness of the proposed method is vertified.
Abstract: A PID control method which combined optimal control strategy is proposed in this paper. The posterior unmodeled dynamics measurement data information are made full use to compensate the unknown nonlinearity of the system, and the unknown increment of the unmodeled dynamics is estimated. Then, a nonlinear PID controller with compensation of the posterior unmodeled dynamics measurement data and the estimation of the increment of the unmodeled dynamics is designed. Finally, through the numerical simulation, the effectiveness of the proposed method is vertified.
TL;DR: A new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex, and it was shown that the method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.
Abstract: In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.
TL;DR: In this article, the authors used the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method to solve the nonlinear fractional type Volterra integro-differential equation.
Abstract: The current work aims at finding the approximate solution to solve the nonlinear fractional type Volterra integro-differential equation \begin{document}$ \begin{equation*} \sum\limits_{k = 1}^{m}F_{k}(x)D^{(k\alpha )}y(x)+\lambda \int_{0}^{x}K(x, t)D^{(\alpha )}y(t)dt = g(x)y^{2}(x)+h(x)y(x)+P(x). \end{equation*} $\end{document} In order to solve the aforementioned equation, the researchers relied on the Bernstein polynomials besides the fractional Caputo derivatives through applying the collocation method. So, the equation becomes nonlinear system of equations. By solving the former nonlinear system equation, we get the approximate solution in form of Bernstein's fractional series. Besides, we will present some examples with the estimate of the error.
TL;DR: A metaheuristic algorithm based on ant colony optimization is applied to obtain the optimal solution of the shortest Hamiltonian path problem, and the rooted shortest path tree structure is applied since in the ideal solution the paths between the restricted nodes are the shortest paths.
Abstract: In a grid network, the nodes could be traversed either horizontally or vertically. The constrained shortest Hamiltonian path goes over the nodes between a source node and a destination node, and it is constrained to traverse some nodes at least once while others could be traversed several times. There are various applications of the problem, especially in routing problems. It is an NP-complete problem, and the well-known Bellman-Held-Karp algorithm could solve the shortest Hamiltonian circuit problem within \begin{document}$ {\rm O(}{{\rm 2}}^{{\rm n}}{{\rm n}}^{{\rm 2}}{\rm )} $\end{document} time complexity; however, the shortest Hamiltonian path problem is more complicated. So, a metaheuristic algorithm based on ant colony optimization is applied to obtain the optimal solution. The proposed method applies the rooted shortest path tree structure since in the optimal solution the paths between the restricted nodes are the shortest paths. Then, the shortest path tree is obtained by at most \begin{document}$ {\rm O(}{{\rm n}}^{{\rm 3}}{\rm )} $\end{document} time complexity at any iteration and the ants begin to improve the solution and the optimal solution is constructed in a reasonable time. The algorithm is verified by some numerical examples and the ant colony parameters are tuned by design of experiment method, and the optimal setting for different size of networks are determined.
TL;DR: A generalization of the successive overrelaxation (GSOR) iteration method is presented to find the solution of the image restoration problem and an improved version of the GSOR (IGSOR) method is also given to solve the proposed problem.
Abstract: In this study, we present a generalization of the successive overrelaxation (GSOR) iteration method to find the solution of the image restoration problem. Moreover, an improved version of the GSOR (IGSOR) method is also given to solve the proposed problem. Convergence of the GSOR and IGSOR methods are investigated. Three numerical examples are given to illustrate the effectiveness and accuracy of the methods.
TL;DR: In this paper, the authors investigated the controllability and observability of second-order linear systems with discrete/continuous time varying and linear time-invariant continuous systems in matrix form.
Abstract: The paper deals with the controllability and observability of second order discrete linear time varying and linear time-invariant continuous systems in matrix form. To this case, we generalize the classical conditions for linear systems of the first order, without reducing them to systems of the first order. Within the framework of Kalman-type criteria, we investigate these concepts for second-order linear systems with discrete / continuous time; we define the initial values and input functions uniquely if and only if the observability and controllability matrices have full rank, respectively. Also a conceptual partner of controllability, that is, reachability of second order discrete time-varying systems is formulated and a necessary and sufficient condition for complete reachability is derived. Also the transfer function of the second order continuous-time linear state-space system is constructed. We have given numerical examples to illustrate the feasibility and effectiveness of the theoretical results obtained.
TL;DR: The purpose of this article is to introduce a methodology to select the list of stocks for investment purpose, and to employ a stochastic fractional programming model to assign money into selected stocks.
Abstract: This article proposes an efficient approach for solving portfolio type problems. It is highly suitable to help fund allocators and decision makers to set up appropriate portfolios for investors. Stock selection is based upon the risk benefits analysis using MADM approach in fuzzy environment. This sort of analysis allows decision makers to identify the list of acceptable portfolios where they can assign some portions of their asset to them. The purpose of this article is two folds; first, to introduce a methodology to select the list of stocks for investment purpose, and second, to employ a stochastic fractional programming model to assign money into selected stocks. This article proposes a hybrid methodology for finding an optimal or new optimal solution of the problem. This hybrid approach considers risks and benefits at the time of stocks prioritization. This is followed by solving a fractional programming to determine the percentages of the budget to be allocated to stocks while dealing with two sets of suitable and non-suitable stocks. For clarification purposes, a sample example problem is solved.
TL;DR: In this paper, sufficient conditions of complete Euclidean space controllability for a singularly perturbed linear time-dependent controlled system with a point-wise nonsmall (of order of \begin{document}$ 1 $\end{document}) delay in the input (the control variable) are derived.
Abstract: A singularly perturbed linear time-dependent controlled system with a point-wise nonsmall (of order of \begin{document}$ 1 $\end{document} ) delay in the input (the control variable) is considered. Sufficient conditions of the complete Euclidean space controllability for this system, robust with respect to the parameter of singular perturbation, are derived. This derivation is based on an asymptotic analysis of the controllability matrix for the considered system and on such an analysis of the determinant of this matrix. However, this derivation does not use a slow-fast decomposition of the considered system. The theoretical result is illustrated by an example.
TL;DR: A new algorithm to obtain roots of the real polynomial represented by f(x) is constructed and it is shown that this algorithm is more useful than others.
Abstract: In this paper, our main interest is to create/ construct a new useful and outstanding algorithm to obtain roots of the real polynomial represented by \begin{document}$ f(x) = c_{0}+c_{1}x+...+c_{i}x^{i}+...+c_{n}x^{n} $\end{document} where coefficients of the polynomials are real numbers and \begin{document}$ x $\end{document} is a real number in the closed interval of \begin{document}$ \mathbb{R} $\end{document} . Also, our results are supported by numerical examples. Then, a new algorithm is compared with the others (writer classical methods) and this algorithm is more useful than others.