Finding all solutions of systems of nonlinear equations with free variables

Tsai, Jung‐Fa; Lin, Chia-Chen

doi:10.1080/03052150701243853

Cited by 11 publications

(3 citation statements)

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“…There is a danger in doing this, since it is possible that the values estimated hold little value and misleading assumptions can impact the kinetic model developed . One of the challenges with nonlinear regression is finding a global solution where often the method will converge to a local minimum. , …”

Section: Analytical Experimental and Numerical Considerations For Dev...mentioning

confidence: 99%

“…141 One of the challenges with nonlinear regression is finding a global solution where often the method will converge to a local minimum. 142,143 Estimation of nonlinear parameter coefficients is a nontrivial task. Robust methods that include Newton, Gauss−Newton, Levenberg−Marquardt, and trust-region reflective methods have been developed to find solutions to the nonlinear objective function.…”

Section: Acs Sustainable Chemistry and Engineeringmentioning

confidence: 99%

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Review and Analysis of CO₂ Photoreduction Kinetics

Thompson

Fernández

Maroto‐Valer

2020

ACS Sustainable Chem. Eng.

112

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Microkinetic, in situ analytics, and empirical modeling approaches for developing intrinsic CO2 photoreduction kinetics are presented in this Perspective. Intrinsic kinetic models that are independent of photoreactor geometry are critical for scaling CO2 photoreduction photoreactors. Successfully scaling CO2 photoreduction is limited using the current extrinsic CO2 photoreduction kinetic models described in this Perspective, because they are dependent on the photoreactor geometry and scale used. The impact of different photoreactor geometries and light transport that lead to extrinsic kinetic models is reviewed. The impacts of temperature and pressure on surface diffusion is highlighted as additional important process parameters. The current Langmuir–Hinshelwood-based kinetic models are discussed, and their limitations are highlighted, with respect to modeling the possible deactivation of the photocatalyst. With a view on developing an intrinsic kinetic model, the challenges for developing CO2 photoreduction kinetics are highlighted and discussed with reference to the current extrinsic CO2 photoreduction kinetic model examples found in the literature. Robust analytical methods for collecting CO2 photoreduction kinetic data and for confirming the carbon source are discussed. The false positive production from adventitious carbon and organic impurities introduced during the synthesis and/or coating of photocatalysts with solvents and degradation of photoreactor components is highlighted as a challenge to collecting CO2 photoreduction kinetic data. It is shown that a wide range of the kinetic model coefficient values are possible when using a multistart, genetic algorithm, or particle swarm approach for estimating nonlinear model coefficients. Finally, an easy to test and implement approach using a mean median multistart and trust-region reflective algorithm method is presented for the estimation of nonlinear CO2 photoreduction kinetic model coefficients.

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Section: Analytical Experimental and Numerical Considerations For Dev...mentioning

confidence: 99%

Section: Acs Sustainable Chemistry and Engineeringmentioning

confidence: 99%

Review and Analysis of CO₂ Photoreduction Kinetics

Thompson

Fernández

Maroto‐Valer

2020

ACS Sustainable Chem. Eng.

112

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“…Convexification of posynomial functions is carried out through the variable transformation x → f (y), where f (y) : IR → IR is a suitable mapping carrying the one-to-one relation between the original variable x and the transformed variable y. For example, Maranas and Floudas (1997) use an exponential transformation, x → e y ; Tsai and Lin (2007) use x → y −1 , which is a special case of the power transformation x → y β . Note that, in general, not all variables need to be transformed.…”

Section: Appendix 1 a Convex Reformulation Of The Modelmentioning

confidence: 99%

A parsimonious model for generating arbitrage-free scenario trees

Consiglio

Carollo

Zenios

2016

Quantitative Finance

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Simulation models of economic, financial and business risk factors are widely used to assess risks and support decision-making. Extensive literature on scenario generation methods aims at describing some underlying stochastic processes with the least number of scenarios to overcome the 'curse of dimensionality'. There is, however, an important requirement that is usually overlooked when one departs from the application domain of security pricing: the no-arbitrage condition. We formulate a moment matching model to generate multi-factor scenario trees for stochastic optimization satisfying no-arbitrage restrictions with a minimal number of scenarios and without any distributional assumptions. The resulting global optimization problem is quite general. However, it is non-convex and can grow significantly with the number of risk factors, and we develop convex lower bounding techniques for its solution exploiting the special structure of the problem. Applications to some standard problems from the literature show that this is a robust approach for tree generation. We use it to price a European basket option in complete and incomplete markets.

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