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I'm working on optimizing the structure of an optical meta device. I have a randomly generated 2D matrix, where 0,1 represents the presence/absence of a hole. Each structure manipulates light differently, thus giving rise to a unique spectrum.

The problem I wish to solve is to maximize the efficiency of this structure. Given the large size of the solution space ($2^{100}$ in this case), it isn't possible to simulate each structure. Is there any search method I could use to complete this optimization?

A general workflow would be:

  1. Generate a random hole structure
  2. Flip one or some bits (based on the optimization algorithm)
  3. Compute the spectrum
  4. Go back to step 2 and make a decision based on the previously computed spectrum.

Here's a link to a sample hole array.

Apologies for the vague statement of the problem.

The objective function to score a structure: Thank you D.W. for pointing this out. The problem is of course meaningless without an objective function. I'd initially omitted it in the question, so as to not make it irrelevant for this forum by adding too much physics.

The system has a transmission $(T)$ and a reflection $(R)$, so a good objective function would be:

$\mathcal{L} = min(T) - \alpha*max(R)$

Maximizing $\mathcal{L}$ would Maximize the Transmission while minimizing the reflection.

Based on the suggestions in the comments, I did try out a Genetic Algorithm Approach (basic algorithm given below)

for iterations:
    for batch_size:
        flip n bits in the structure randomly
        compute objective function

    structure = structure with the highest objective function value in the batch

However, though there is a slight increase in the Objective function initially, it saturates quickly and doesn't reach the expected values (Image below X-Axis - Iterations, Y-Axis - Objective Function Value).

X-Axis - Iterations, Y-Axis - Objective Function Value

I have tried multiple values of batch size, bitflips (n). Is there any alternate algorithm or any possible modification to the current algorithm?

Thanks in advance!

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    $\begingroup$ how will you use each structure? Some usages will be totally different when flipping 1 or 2 specific bits, while other uses would not change at all $\endgroup$ – nir shahar Jun 28 at 17:41
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    $\begingroup$ Do you have an objective function? i.e., a function that given a candidate structure, can compute how "good" it is? Maybe you can compute the spectrum and then compute the efficiency from the spectrum? Can you please edit the question to describe that function in as much detail as possible? What properties does it have, what form does it take, etc.? The answer will likely depend on the nature of that function. In general, for a completely arbitrary function, the problem is not solvable, so any solution will have to exploit some regularities or properties of that function. $\endgroup$ – D.W. Jun 28 at 18:17
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    $\begingroup$ Im not really sure how $T$ and $R$ depend on your hole matrix $M$, but you can try to use a genetic algorithms-like method: Generate a starting (or a few starting matricies) matrix $M$. Generate new matricies from $M$ by flipping random bits in it. Take the one that maximizes $L$ from the group you just created, and iterate this process a few times. This should give a pretty good approximation for the maximum if $L$ is a "nice" function $\endgroup$ – nir shahar Jun 29 at 13:19
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    $\begingroup$ Do you want to find one solution that maximizes the objective function? Or do you want to find a set of solutions that are somehow as "different" as possible or "cover the space of solutions" as well as possible? If the latter, how do you propose to measure "different" or "coverage"? $\endgroup$ – D.W. Jun 29 at 18:16
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    $\begingroup$ If you can't give specifics of your particular situation, I wonder if you would do better to ask a new question that is more general and more likely to be useful to others in the future? e.g., something like "how do I choose among the various optimization algorithms for combinatorial optimization?" $\endgroup$ – D.W. Sep 20 at 19:27

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