A new homotopy for seeking all real roots of a nonlinear equation

Rahimian, Saeed Khaleghi; Jalali, Farhang; Seader, J. D.; White, Ralph E.

doi:10.1016/j.compchemeng.2010.04.007

Cited by 47 publications

(33 citation statements)

References 25 publications

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“…It is also due to the absence of real space connections between separate homotopy path branches, thus preventing the attainment of solutions on isolated branches. Recently, Rahimian et al [8][9] combined the features of fixed-point (FP) and Newton (N) homotopies to form a new homotopy called as the FPN homotopy. Rahimian et al [8][9] applied FPN homotopy in the solving of a multitude of different chemical engineering related problems to reach all the feasible solutions of the problems successfully.…”

Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

“…Recently, Rahimian et al [8][9] combined the features of fixed-point (FP) and Newton (N) homotopies to form a new homotopy called as the FPN homotopy. Rahimian et al [8][9] applied FPN homotopy in the solving of a multitude of different chemical engineering related problems to reach all the feasible solutions of the problems successfully. However, the formulation of FPN homotopy does not include variables bounding, and hence the path may travel outside the feasible variable domain.…”

Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

“…Mjalli et al [2]; Švandová et al [3]; Kiss et al [4]; Brooks [5]; Yermakova and Anikeev [6]; Waschler et al [7]). However, the emphasis in the current studies in the literature has shifted towards the investigation of CSTR systems from the perspective of numerical solving methods [8][9][10][11][12]. The reason for this is that CSTR models exhibit characteristics that pose challenges to practically all the available general-purpose numerical solving methods.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

Malinen¹,

Kangas

Ahola

et al. 2016

Period. Polytech. Chem. Eng.

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Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

Section: Characteristics Of Some Present Solving Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

Malinen¹,

Kangas

Ahola

et al. 2016

Period. Polytech. Chem. Eng.

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“…As pointed out in [6] and [21], the classical methods have some weaknesses. For example, the performance of the Newton and quasi-Newton type methods is highly sensitive to the initial guess; the embedding methods cannot directly deal with variable bounds and inequality constraints; the interval-Newton methods are computationally expensive; and the deterministic branch-and-bound methods need the specification of bounded intervals.…”

Section: B Motivationmentioning

confidence: 99%

“…As reviewed in [6], [20], and [21], several different classical methods have been proposed to solve NESs, which can be classified as follows.…”

Section: A Related Workmentioning

confidence: 99%

Finding Multiple Roots of Nonlinear Equation Systems via a Repulsion-Based Adaptive Differential Evolution

Gong

Wang

Cai

et al. 2020

IEEE Trans. Syst. Man Cybern, Syst.

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Abstract-Finding multiple roots of nonlinear equation systems (NESs) in a single run is one of the most important challenges in numerical computation. We tackle this challenging task by combining the strengths of the repulsion technique, diversity preservation mechanism, and adaptive parameter control. First, the repulsion technique motivates the population to find new roots by repulsing the regions surrounding the previously found roots. However, to find as many roots as possible, algorithm designers need to address a key issue: how to maintain the diversity of the population. To this end, the diversity preservation mechanism is integrated into our approach, which consists of the neighborhood mutation and the crowding selection. In addition, we further improve the performance by incorporating the adaptive parameter control. The purpose is to enhance the search ability and remedy the trial-and-error tuning of the parameters of differential evolution (DE) for different problems. By assembling the above three aspects together, we propose a repulsion-based adaptive DE, called RADE, for finding multiple roots of NESs in a single run. To evaluate the performance of RADE, 30 NESs with diverse features are chosen from the literature as the test suite. Experimental results reveal that RADE is able to find multiple roots simultaneously in a single run on all the test problems. Moreover, RADE is capable of providing better results than the compared methods in terms of both root rate and success rate.

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Equation‐oriented flowsheet simulation and optimization using pseudo‐transient models

2014

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Tight integration through material and energy recycling is essential to the energy efficiency and economic viability of process and energy systems. Equation-oriented (EO) steady-state process simulation and optimization are key enablers in the optimal design of integrated processes. A new process modeling and simulation concept based on pseudotransient continuation is introduced. An algorithm for reformulating the steady-state models of process unit operations as differential-algebraic equation systems that are statically equivalent with the original model is presented. These pseudo-transient models improve the convergence of EO process flowsheet simulations by expanding the convergence basin. This concept is used to build a library of pseudo-transient models for common process unit operations, and this modeling concept seamlessly integrates with a previously developed time-relaxation optimization algorithm. Two design case studies are presented to validate the proposed framework.The framework developed in this article consists of converting the algebraic model into a pseudo-transient process model with an equivalent steady-state solution, which can be obtained by a DAE time integration.The pseudo-transient models described above were implemented in gPROMS. 57 In the following sections, we 4116

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A new homotopy for seeking all real roots of a nonlinear equation

Cited by 47 publications

References 25 publications

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

A New Homotopy-Based Strategy for the Robust Determination of All the Feasible Solutions for CSTR Systems

Finding Multiple Roots of Nonlinear Equation Systems via a Repulsion-Based Adaptive Differential Evolution

Equation‐oriented flowsheet simulation and optimization using pseudo‐transient models

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