Which Functions Have Multiple Local Minima?

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SUMMARY

The discussion centers on identifying functions that exhibit multiple local minima, specifically within the context of decentralized algorithms. Participants suggest using the function F:R→R, with examples including the sum of trigonometric functions such as ##\sin x + \cos( \sqrt 2 x) + \sin( \sqrt3 x)##, which lacks a common period, and the function ##\sin(1/x)##, known for its clustering of local minima. These functions serve as effective test cases for understanding local minima in single-variable scenarios.

PREREQUISITES
  • Understanding of single-variable calculus and local minima
  • Familiarity with trigonometric functions and their properties
  • Basic knowledge of decentralized algorithms
  • Experience with function mapping from R to R
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  • Research the properties of trigonometric functions and their combinations
  • Explore the concept of local minima in multi-variable functions
  • Study decentralized algorithms for optimization problems
  • Investigate the behavior of the function ##\sin(1/x)## and its implications in optimization
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Mathematicians, computer scientists, and algorithm developers interested in optimization techniques and the behavior of functions with multiple local minima.

Old Monk
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I've been working on a decentralized algorithm for finding local minima. Can anyone give me a few examples of mappings of the form F:R→R that have multiple local minima. I'm having problems defining neighbourhood on mappings from R2→R, so I thought I'll test it out on single variable functions first.

Thanks.
 
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Try the sum of some trig functions with no common periouds, for example
##\sin x + \cos( \sqrt 2 x) + \sin( \sqrt3 x)##.

Or if you want a cluster of local minima, throw in something like ##\sin(1/x)##
 

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