“Giambattista Vico was a practical roundheaded Neapolitan. It pleases Croce to consider him as a mystic, essentially speculative, “disdegnoso dell’ empirismo.” It is a surprising interpretation, seeing that more than three-fifths of his Scienza Nuova is concerned with empirical investigation.Who's Vico? Giambattista Vico - the philosopher and social scientist?
And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?A great example of how to motivate Cauchy convergence is the following:
Consider using Differential Evolution (DE) to minimize a function in n space, Choose an initial population of NP individuals (vectors) all of which ultimately converge to the unknown optimum. Convergence is reached when they all converge to each other.
The distance between the population members determines convergence. There are several ways to measure, e.g. maximum distance between trial vectors and best vector or difference between best and worst objective values.
Each generation is one iteration in the evolution scheme.
In what way does this Gedankenexperiment motivate Cauchy convergence?A great example of how to motivate Cauchy convergence is the following:
Consider using Differential Evolution (DE) to minimize a function in n space, Choose an initial population of NP individuals (vectors) all of which ultimately converge to the unknown optimum. Convergence is reached when they all converge to each other.
The distance between the population members determines convergence. There are several ways to measure, e.g. maximum distance between trial vectors and best vector or difference between best and worst objective values.
Each generation is one iteration in the evolution scheme.
Are there physical systems that embody DE? Or is DE another thing from the minds of mathematicians?And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?
This is a mathematical issue (necessary and sufficient conditions) to be addressed. This has been addressed in the literature. In general, DE converges to the global optimum in contrast to gradient methods.
Think of a population as a vector of 'haploid' bitsets, essentially.
I'm not interested in biological evolution here. Don't want 2 simultaneous discussions. It adds no precision.
But a harmonic oscillator never actually converges below some threshold epsilon that is a function of some noise floor (thermal, quantum fluctuations, etc.) and system design.A harmonic oscillator? I'm not sure why you think the Cauchy convergence is not present in physical system - I gave you several examples, and if you read some more on mathematical physics, you'll see see that it's ubiquitous.
That's the main point of this topic.It's only a model and that is a product of the mind.
Do you mean damping? BTW, everything in physics in a harmonic oscillator!But a harmonic oscillator never actually converges below some threshold epsilon that is a function of some noise floor (thermal, quantum fluctuations, etc.) and system design.A harmonic oscillator? I'm not sure why you think the Cauchy convergence is not present in physical system - I gave you several examples, and if you read some more on mathematical physics, you'll see see that it's ubiquitous.
This is "theory-laden" too. What distance metric, and what space are we going to position both "the concept" and "reality"?The distance from reality to concept >> distance from concept to realty.
Instead of always asking questions, maybe try to answer the question yourself.Are there physical systems that embody DE? Or is DE another thing from the minds of mathematicians?And what if convergence is suboptimal? What if the superior solution is to randomly flip between the multiple optima?
This is a mathematical issue (necessary and sufficient conditions) to be addressed. This has been addressed in the literature. In general, DE converges to the global optimum in contrast to gradient methods.
Think of a population as a vector of 'haploid' bitsets, essentially.
I'm not interested in biological evolution here. Don't want 2 simultaneous discussions. It adds no precision.