843 research outputs found

    A Collection of Challenging Optimization Problems in Science, Engineering and Economics

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    Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.

    Solving Polynomial Equations using Modified Super Ostrowski Homotopy Continuation Method

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    Homotopy continuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations.  HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions.   The starting value problem has been solved using Super Ostrowski homotopy continuation method for the initial guesses between . Nevertheless, Super Ostrowski homotopy continuation method was only used to find out real roots of nonlinear equations.  In this paper, we employ the Modified Super Ostrowski-HCM to solve several real life applications which involves polynomial equations by expanding the range of starting values. The results indicate that the Modified Super Ostrowski-HCM performs better than the standard Super Ostrowski-HCM. In other words, the complex roots of polynomial equations can be found even the starting value is real with this proposed scheme

    Estimating Static Models of Strategic Interaction

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    We propose a method for estimating static games of incomplete information. A static game is a generalization of a discrete choice model, such as a multinomial logit or probit, which allows the actions of a group of agents to be interdependent. Unlike most earlier work, the method we propose is semiparametric and does not require the covariates to lie in a discrete set. While the estimator we propose is quite flexible, we demonstrate that in most cases it can be easily implemented using standard statistical packages such as STATA. We also propose an algorithm for simulating the model which finds all equilibria to the game. As an application of our estimator, we study recommendations for high technology stocks between 1998-2003. We find that strategic motives, typically ignored in the empirical literature, appear to be an important consideration in the recommendations submitted by equity analysts.

    Newton Homotopy Continuation Method for Solving Nonlinear Equations using Mathematica

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    In this paper, we solve the nonlinear equations by using a classical method and a powerful method.  A powerful method known as homotopy continuation method (HCM) is used to solve the problem of classical method.  We use Newton-HCM to solve the divergence problem that the classical Newton’s method always faces. The divergence problem occurs when a bad initial guess is used. The problem with Newton’s method happens when the derivative of given function at initial point equal to zero. The division by zero makes the scheme become nonsense. Thus, an approach used to solve this mathematical problem by using Newton-HCM. The results are implemented by mathematical software known as Mathematica 7.0. The results obtained indicate the ability of Newton-HCM to solve this mathematical problem

    Order-of-magnitude speedup for steady states and traveling waves via Stokes preconditioning in Channelflow and Openpipeflow

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    Steady states and traveling waves play a fundamental role in understanding hydrodynamic problems. Even when unstable, these states provide the bifurcation-theoretic explanation for the origin of the observed states. In turbulent wall-bounded shear flows, these states have been hypothesized to be saddle points organizing the trajectories within a chaotic attractor. These states must be computed with Newton's method or one of its generalizations, since time-integration cannot converge to unstable equilibria. The bottleneck is the solution of linear systems involving the Jacobian of the Navier-Stokes or Boussinesq equations. Originally such computations were carried out by constructing and directly inverting the Jacobian, but this is unfeasible for the matrices arising from three-dimensional hydrodynamic configurations in large domains. A popular method is to seek states that are invariant under numerical time integration. Surprisingly, equilibria may also be found by seeking flows that are invariant under a single very large Backwards-Euler Forwards-Euler timestep. We show that this method, called Stokes preconditioning, is 10 to 50 times faster at computing steady states in plane Couette flow and traveling waves in pipe flow. Moreover, it can be carried out using Channelflow (by Gibson) and Openpipeflow (by Willis) without any changes to these popular spectral codes. We explain the convergence rate as a function of the integration period and Reynolds number by computing the full spectra of the operators corresponding to the Jacobians of both methods.Comment: in Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics, ed. Alexander Gelfgat (Springer, 2018

    Homotopy continuation methods for coupled-cluster theory in quantum chemistry

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    Homotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by the coupled-cluster polynomial equations. This endeavor has been pioneered by quantum chemists that have undertaken both elaborate numerical as well as mathematical investigations. Recently, from the perspective of applied mathematics, new interest in these approaches has emerged using both topological degree theory and algebraically oriented tools. This article provides an overview of describing the latter development
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