Which Functions Have Multiple Local Minima?

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The discussion focuses on finding examples of single-variable functions with multiple local minima for a decentralized algorithm. Participants suggest using combinations of trigonometric functions with no common periods, such as sin(x) + cos(√2 x) + sin(√3 x). Additionally, the function sin(1/x) is recommended for generating a cluster of local minima. The aim is to explore these functions before tackling more complex mappings from R² to R. The conversation highlights the challenges of defining neighborhoods in higher dimensions.
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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)##
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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