Ψ -transformations

[John Raymaker]

Journal of Physics: Conference Series

Journal of Physics: Conference Series

DAvid, your comments on Ψ -transformations are quite technical and interesting. I am trying to understand what you wrote. Could you elaborate a bit more as to how Ψ -transformations can be part of an overal solution, John

DPaper • The following article isOpen access

Application of Ψ -transformation to the search for continuous function’s global extremum on simplex

A Sizikov

Published under licence by IOP Publishing Ltd
Journal of Physics: Conference Series, Volume 1203, International Conference "Applied Mathematics, Computational Science and Mechanics: Current Problems" 17–19 December 2018, Voronezh State University, Voronezh, Russian FederationCitation A Sizikov 2019 J. Phys.: Conf. Ser. 1203 012073DOI 10.1088/1742-6596/1203/1/012073

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A nonconvex problem of mathematical programming, the acceptance region of which is simplex. A two-stage algorthm is suggested to solve the problem. At the first stage, the global optimum region is determined; at the second stage, local clean-up of the solution is carried out. The first stage is realized by the Ψ -transformation method, which is an alternative to direct random search techniques. The method is to build and use Ψ -function. Ψ -function is built emperically based on statistic tests. To perform the tests, the generator of random points evenly distributed in simplex is used. Even distribution in simplex is achieved through affine and linear transformations of points evenly distributed in a unit hypercube. For refinement of the approximate solution obtained at the first stage, the method of regular simplex reflection is used. Examples are discussed. The example of algorthm usage to optimize the hydrocarbon mixture make-up is presented.

 • The following article isOpen access

Application of

Journal of Physics: Conference Series

Paper • The following article isOpen access

Application of Ψ -transformation to the search for continuous function’s global extremum on simplex

A Sizikov

Published under licence by IOP Publishing Ltd
Journal of Physics: Conference Series, Volume 1203, International Conference "Applied Mathematics, Computational Science and Mechanics: Current Problems" 17–19 December 2018, Voronezh State University, Voronezh, Russian FederationCitation A Sizikov 2019 J. Phys.: Conf. Ser. 1203 012073DOI 10.1088/1742-6596/1203/1/012073

DownloadArticle PDF

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Abstract

A nonconvex problem of mathematical programming, the acceptance region of which is simplex. A two-stage algorthm is suggested to solve the problem. At the first stage, the global optimum region is determined; at the second stage, local clean-up of the solution is carried out. The first stage is realized by the Ψ -transformation method, which is an alternative to direct random search techniques. The method is to build and use Ψ -function. Ψ -function is built emperically based on statistic tests. To perform the tests, the generator of random points evenly distributed in simplex is used. Even distribution in simplex is achieved through affine and linear transformations of points evenly distributed in a unit hypercube. For refinement of the approximate solution obtained at the first stage, the method of regular simplex reflection is used. Examples are discussed. The example of algorthm usage to optimize the hydrocarbon mixture make-up is presented.

  to the search for continuous function’s global extremum on simplex

A Sizikov

Published under licence by IOP Publishing Ltd
Journal of Physics: Conference Series, Volume 1203, International Conference "Applied Mathematics, Computational Science and Mechanics: Current Problems" 17–19 December 2018, Voronezh State University, Voronezh, Russian FederationCitation A Sizikov 2019 J. Phys.: Conf. Ser. 1203 012073DOI 10.1088/1742-6596/1203/1/012073

DownloadArticle PDF

Authors

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Article metrics

97 Total downloads

Share this article

Article information

Abstract

A nonconvex problem of mathematical programming, the acceptance region of which is simplex. A two-stage algorthm is suggested to solve the problem. At the first stage, the global optimum region is determined; at the second stage, local clean-up of the solution is carried out. The first stage is realized by the Ψ -transformation method, which is an alternative to direct random search techniques. The method is to build and use Ψ -function. Ψ -function is built emperically based on statistic tests. To perform the tests, the generator of random points evenly distributed in simplex is used. Even distribution in simplex is achieved through affine and linear transformations of points evenly distributed in a unit hypercube. For refinement of the approximate solution obtained at the first stage, the method of regular simplex reflection is used. Examples are discussed. The example of algorthm usage to optimize the hydrocarbon mixture make-up is presented.

[David Bibby]

Dear John,

Thanks for your question and the reference to Sizikov’s article.

The first thing to say is that the ψ_transformations discussed on this site are entirely distinct in origin and purpose from the Ψ-transformations employed by Sizikov and Chichinadze. The latter arise in computational optimisation, where the aim is to locate the global maximum of a continuous function. Their constructed Ψ-function is a heuristic device, obtained statistically, that serves as a guide for where the global optimum might lie.

By contrast, the ψ_terminology I am employing comes from Lonerganian philosophy and mathematics, where the focus is on modelling insight, personhood, and proof. Here the goal is to illuminate the structure of consciousness itself, with ψ representing the dynamism of consciousness, sublation, and the integration of intelligibility.

In this context, ψ_transformation is a movement in “insight space.” A helpful analogy might be a unitary transformation in quantum mechanics: such a transformation changes the form of the wave function (also using the symbol Ψ) while preserving the probability density. This has no physical, measurable impact, but it does alter the mathematical expression, which is its meaning or intelligibility.

That said, there are some resonances between the computational and philosophical contexts. In each case, the Ψ symbol concerns moving from local traps toward a higher integration. In optimisation, Ψ-transformation avoids being stuck in local minima; in ψ_proof, ψ_transformation avoids being confined to partial or lower-level intelligibility. Both point to a structural need to move from the local to the global through an auxiliary operation that cannot be reduced to stepwise deduction.

I am still exploring how ψ_transformations could contribute to an overall solution, but I hope this helps to clarify the distinction and remove any misconceptions.

Best wishes,

David

On 1 Sep 2025, at 07:34, 'John Raymaker' via Lonergan_L <loner...@googlegroups.com> wrote:



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