I was able to come up with a creative solution that achieved an objective value of 1861.54 by combining various ideas.

1. Created unordered color clusters of size 8 by utilizing a min-cost matching approach and merging matched subclusters, repeating the process three times. The distance function used for subclusters C1 and C2 is d(C1, C2) = ∑_{c1 in C1} ∑_{c2 in C2} d(c1, c2).

2. Determined the optimal 2 × 5 cluster arrangement based on the aforementioned distance function. This required brute forcing 10! permutations (or 10!/4 considering symmetry, which I opted not to exploit).

3. Optimized the layout of each cluster individually by testing all 4 × 2 arrangement permutations through brute force method (potential for further symmetry breaking, although I did not pursue this).

4. Explored all possible ways (4^10) to flip the clusters using brute force technique. (Additional symmetry breaking opportunities were available but not pursued.)

5. Enhanced the arrangement further through local search. Implemented two types of rounds: a 2-opt round where pairs of positions are considered for swapping, and a large-neighborhood round where a random maximal independent set is selected and optimally reassigned using the Hungarian method (facilitated by the constraint that moved items cannot be adjacent).

The final output has been visualized here:

https://i.sstatic.net/Ki7O9.png

For those interested, the Python implementation can be found at https://github.com/eisenstatdavid/felt-tip-pens

To crack this puzzle, shift your focus away from seeing it as an array for a moment and instead fix your gaze on the corners.

First things first, clearly define the problem you're aiming to solve. Traditional colors operate in three dimensions: hue, saturation, and value (darkness). Attempting to map all three dimensions onto a two-dimensional grid won't work seamlessly. However, there's a workaround.

If your goal is to orderly arrange colors ranging from white to black and red to purple, tweak your distance function to interpret variations in darkness as distance along with differences in hue values (no warping!). This adjustment will yield a color sorting that aligns with a four-corner scheme.

Next, tether each of your colors to the four corners by defining (0:0) as black, (1:1) as white, (0,1) as red (0 hue), and (1:0) as purple-red (350+ hue), simulating that purple-red equals purple for simplicity:

https://i.sstatic.net/ApfWB.png

Now, you have established two extremes: darkness and hue. But here's the twist... if we rotate the box by 45 degrees...

https://i.sstatic.net/T1Aq6.png

Spot the revelation? Not yet? The X and Y axes are now aligned with our two extreme metrics! All that remains is dividing each color's distance from white by the distance of black from white, and each color's distance from purple by the distance of red from purple, thus obtaining our Y and X coordinates, respectively!

Add a few more pens to the mix:

https://i.sstatic.net/JIY6E.png

Proceed to iterate through all the pens at O(n)^2 complexity, pinpointing the nearest distance between any pen and its final position uniformly across the rotated grid. Maintain a record of these distances, replacing them if a pen's slot has been occupied. By doing so, positioning pens in their closest slots becomes achievable in polynomial time O(n)^3.

https://i.sstatic.net/jdzrS.png

But wait, there's more to be done. HSV operates in three dimensions, prompting us to incorporate this aspect into our model as well! Expand the previous algorithm by integrating a third dimension before calculating the closest distances. Insert the 2D plane into a 3D space by intersecting it with the two color extremes and the horizontal line between white and black. This can be accomplished by locating the midpoint of the two color extremes and adjusting darkness slightly. Subsequently, distribute our pen slots uniformly on this plane and position our pens directly in this 3D space based on their HSV values - H corresponding to X, V representing Y, and S for Z.

https://i.sstatic.net/J0y63.png

With the inclusion of saturation in the 3D representation of the pens, resume iterating over the pen positions to determine the nearest one for each within polynomial time.

Tada! Your pens are now neatly organized. Should you desire the outcome in an array format, evenly generate coordinates for each array index and proceed sequentially!

Shift focus from pen sorting and delve into crafting code!

In the comment section, it was brought to your attention that you appear to be interested in locating one of the global minima for a discrete optimization problem. If this is a new concept for you, it might be beneficial to delve deeper into the topic.

Consider an error (objective) function that calculates the sum of distances between all pairs of adjacent pens. An optimal solution (pen arrangement) minimizes this error function. There may exist multiple optimal solutions, and it's important to note that different error functions can yield different results beyond the simplistic example I just provided.

You could utilize a pre-made optimizer tool (like CPLEX or Gurobi) by supplying it with a valid formulation of your problem. It may discover an optimal solution, but even if not, it might present a satisfactory sub-optimal solution.

An alternative approach would be to develop your own heuristic algorithm (e.g., a specialized genetic algorithm) to potentially surpass what the solver could achieve within its time and space constraints. Given your resources including input data, a function for assessing color dissimilarity, and JavaScript skills, crafting a heuristic algorithm is likely the more comfortable route for you.

This response initially lacked code samples as real-world problems like this usually don't have simple one-size-fits-all solutions.

The process of performing these calculations using JavaScript, especially on a web browser, may seem unconventional. Yet, at the request of the author, here's a JavaScript implementation of a basic evolutionary algorithm hosted on CodePen.

Due to the increased input size compared to the 5x5 demo originally presented, along with prolonged algorithmic iterations and slow code execution, the runtime is extensive. Although adjustments were made to avoid excessive re-calculation due to mutations, running the program still requires substantial time. The provided solution required approximately 45 minutes to execute via CodePen's debug mode.

https://i.sstatic.net/lyZ3v.png

This solution yielded an objective function just below 2060 and was generated utilizing specific parameters:

const SelectionSize = 100;
const MutationsFromSolution = 50;
const MutationCount = 5;
const MaximumGenerationsWithoutImprovement = 5;

It should be noted that slight modifications to parameters can lead to significant changes in the algorithm's outcomes. Increasing mutation count or selection size enhances program duration but can also result in superior results. Experimenting with various parameters is advised to seek better solutions, though they'll likely demand additional computational time.

Oftentimes, optimal enhancements derive from algorithmic alterations rather than sheer computing power. Innovative strategies concerning mutations and recombinations are often key to achieving improved results when employing a genetic algorithm.

Utilizing an explicitly seeded and reproducible pseudo-random number generator (PRNG) instead of Math.random() enables repeated program runs for debugging and reproducibility proof.

Considering setting up a visualization mechanism for monitoring the algorithm's progress (beyond mere console.log outputs) offers insight into the process rather than solely focusing on the final result.

Allowing human intervention (to propose mutations and guide the search based on personal color perception) can aid in attainments desired. This introduces an Interactive Genetic Algorithm (IGA). For further exploration, refer to the publication J. C. Quiroz, S. J. Louis, A. Shankar and S. M. Dascalu, "Interactive Genetic Algorithms for User Interface Design," 2007 IEEE Congress on Evolutionary Computation, Singapore, 2007, pp. 1366-1373, doi: 10.1109/CEC.2007.4424630..

If a total ordering function could be created to determine the 'darker' color between two colors, it would be possible to sort an array of colors from dark to light (or vice versa) using this function.

Starting at the top left with the first color in the sorted array, continue diagonally across the grid while filling in subsequent elements. This method results in a gradient-filled rectangular grid where neighboring colors are similar.

https://i.sstatic.net/hiYBe.png

Do you believe this approach aligns with your goal?

You have the option to alter the appearance by adjusting the behavior of the total ordering function. For instance, when arranging colors based on similarity using a color map depicted below, the total ordering can be defined as a traversal from one cell to the next. Through modifying which cell is selected next in the traversal process, various color-similar gradient grid fills can be achieved.

https://i.sstatic.net/5nmn3.png

There appears to be a possible simple solution for this issue by calculating the average color surrounding each color and placing it in that approximate position. For example:

C[j] ~ sum_{i=1...8}(C[i])/8

This is known as the discrete Laplace operator, where solving this equation means defining a discrete harmonic function over the color vector space. Harmonic functions adhere to the mean-value property, which asserts that the average value of the function in a neighborhood is equal to its center value.

To find a specific solution, we must establish boundary conditions by fixing at least two colors on the grid. Selecting 4 extreme colors and placing them in the corners of the grid seems suitable in this scenario.

One method to solve Laplace's equation effectively is through the relaxation technique, solving one linear equation at a time. While not directly applicable due to the combinatorial nature of the problem, utilizing relaxation may still yield results.

The process involves fixing the corner colors, filling the grid with their bilinear interpolation, then iteratively replacing each color C_j with the closest Laplacian color L_j from the neighboring colors until convergence criteria are met.

Guidelines for finding the closest color include proximity based on Euclidean metric and distinctiveness from neighbor colors. This operation acts as a projection into the input set of colors.

Convergence might not be achieved strictly; thus, limiting iterations to around 10 times the number of cells in the grid is reasonable. Singularities like repeated or unplaced colors should be addressed individually for optimal results.

Selecting initial corner colors could involve identifying the most diverse colors using Euclidean metrics or performing PCA on the point cloud to determine extremal points for boundary conditions.

Preliminary implementation shows promising visual results, albeit with notable singularities. Further refinement and optimization may lead to a more accurate approximation, especially when dealing with smoothly varying input colors.

UPDATE:

I have begun implementing these ideas and provided a visual comparison below. Despite some issues with singularities, the overall approach shows potential. Feel free to reach out if you find the results beneficial, and I can consider adapting the code to JavaScript.

https://i.sstatic.net/uqOky.png

In order to effectively organize a grid of colors, I propose implementing a system of color regions. These regions would consist of groups of colors where the distance between points is within a certain tolerance. Within each region, the central point would be determined as the closest to all other points in terms of average distance.

Initially, my algorithm would analyze the grid of colors and identify clusters that could form cohesive color regions. The challenge then lies in determining how these regions interact with one another. Due to the structured nature of a region and the sharp contrast at its edges, there must be a consideration for interregion compatibility. It may be beneficial for two regions to align along specific sides rather than others, which could necessitate rotating one or both regions to achieve optimal positioning.

The final step involves assessing the boundaries between regions after any necessary rotations have been made. Adjustments may need to be made to ensure cohesion between neighboring regions.