I have a program which must quickly find $x$ and $y$ where $x,y\in\mathbb{N_0}$ which correspond with maximum value of a function:

$$f(x,y)=\frac{\sum_{i=0}^{|b|-1}{|b_i \sin^k{\frac{2xi\pi+y}{44100}}|}}{\sum_{i=0}^{|b|-1}{g(i)}+l}$$

where $g(x)=1$ if $|b_x\sin^k{\frac{2xi\pi+y}{44100}}|>l$, else $0$; $k\gg1$; $l\rightarrow0,l>0$; $b$ is a list of arbitrary values.

$x,y$ are most often in range $[0, 10000\rangle$, so technically, it is possible for a computer program to find global maximum using brute force, but it takes too long. I need to optimize the algorithm for finding global maximum, but don't know how to simplify the expression, nor can I derive it.

Is there a way to calculate global maximum of such a function? If not, can this function be simplified?

  • $\begingroup$ What is $R$ and how can $l^R\to 0$ if $l,R$ are - as it seems - fixed? $\endgroup$ – Hagen von Eitzen May 9 '15 at 12:37
  • $\begingroup$ You may certainly assume that $0\le x<22050$, but it sounds unplausible that $f$ has a global maximum with resopect to $y\in\mathbb N$. Also, where does this problem come from, and for example why does the standard CD sampling rate of 44.1 kHz play a role? $\endgroup$ – Hagen von Eitzen May 9 '15 at 12:40
  • $\begingroup$ @HagenvonEitzen this program processes audio (44100 samples per second); it is possible that global maximum may be greater in domain of real numbers $x,y$; but they are defined as integers, despite $f$ giving real solutions for real $x,y$ $\endgroup$ – Mirac7 May 10 '15 at 11:57

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