I am outlining a method for solving Non Linear optimization problems.

Consider the system of equations:--------------------------------- 1
f1(a0, a1, a2, a3 ......... an) = 0

f2(a0, a1, a2, a3 ......... an) = 0

fk(a0, a1, a2, a3 ...........an) = 0

Here f1, f2 ... fk are non linear functions in variables a0, a1, a2 ......... an.

Our goal is to find out the values of a0, a1, a2 ....an which satisfy the above system of equations. We will assume these variables can take values 0 and 1 only.
This would be similar to combinatorial optimization.

Now model each variable as a square wave with slightly different periods.

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I have used the Fourier series representation of a square wave for this purpose. $$ a0 = \frac{4}{\pi }\sum_{n=1,3,5...}^{\infty} \frac{1}{n}sin(\frac{n\pi x}{L0}) $$

$$ a1 = \frac{4}{\pi }\sum_{n=1,3,5...}^{\infty} \frac{1}{n}sin(\frac{n\pi x}{L1}) $$

$$ a2 = \frac{4}{\pi }\sum_{n=1,3,5...}^{\infty} \frac{1}{n}sin(\frac{n\pi x}{L2}) $$


$$ an = \frac{4}{\pi }\sum_{n=1,3,5...}^{\infty} \frac{1}{n}sin(\frac{n\pi x}{Ln}) $$

where L0, L1, L2.......Ln are the half peroids of the associated square waves.

We can further simplify the sine terms using Taylor series. $$ \sin (\frac{n\pi x}{L}) = \sin (\frac{n\pi x}{P}) + (P-L)f'(P) + \frac{(P-L)^{2}}{2!}f''(P) + .... $$ where f' , f'' ... are the derievaties of the function $\sin (\frac{n\pi x}{L})$ evaluated at L=P.

The above equation will converge provided that L and P are closely spaced.

We can use the above equation to express $\sin (\frac{n\pi x}{L0})$ ,$\sin (\frac{n\pi x}{L1})$,$\sin (\frac{n\pi x}{L2})$........$\sin (\frac{n\pi x}{Ln})$ in terms of a single term $\sin (\frac{n\pi x}{P})$.

Now we can substitute this in the original system of equations above (1), to arrive at a polynomial equation in x and $\sin (\frac{n\pi x}{P})$.

We can assume $\sin (\frac{n\pi x}{P})$ = 1 to obtain a polynomial equation in x. Solving this polynomial equation can yield the values of a0,a1,a2.....an.

Please let me know of any similar approach used by anyone previously. I did some amount of research but could not find any on the internet.

  • $\begingroup$ This may be better suited to math.stackexchange. $\endgroup$ – kodlu Oct 8 '17 at 21:38
  • $\begingroup$ posted in math.stackexchange as well but no responses yet. $\endgroup$ – rajeesh Oct 21 '17 at 12:41

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