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I have n number of 2d vectors (all have the same magnitude),

for each vector I can choose whether or not to multiply it by -1 (spinning it through 180 degrees),

all of the vectors are then added together,

The objective is to maximize the magnitude of the final vector, the only variable being whether or not to multiply the vector by -1.

Here is an example,

I have 2 vectors, both magnitude 1, one at 30° and one at -80° from the horizontal axis. Adding both of these together I will have a new vector at -37.6° with magnitude 1.31. However if I flip the -80° vector (spin it through 180), it becomes 100°. The new resultant vector is 64.7° with a magnitude of 1.64.

It is easy to check all combinations for a small number of vectors but it soon becomes unfeasible.

The main observation that I keep coming back to is that the objective should be to minimize the sum of the differences between the angles (i.e. the maximum magnitude will be achieved when the vectors change direction the least amount - trying to make them as similar to a straight line as possible). I am having trouble figuring out how to write the problem down mathematically so it makes sense.

Does anyone have any ideas on this? I feel like it should be really quite simple to solve and just requires some good math common sense but I have been fiddling with it for days now and I am going around in circles (literally).

I should point out that I do have a solution that seems to work but is very messy. I pick a single point far away from the origin (further than if all the vectors made a straight line) and then find the arrangement of vectors that gives the minimum distance to that point (this is a convex linear programming problem). I then move the point through several degrees relative to the origin and repeat the process. I keep going until the point has made a full circle. I then find which solution gave the largest magnitude (there are always at least 2 because for every solution there is a symmetrical one). Although I am pretty certain this approach finds the maximum magnitude it seems very messy to me and I cannot prove that it works mathematically.

Any help greatly appreciated,

Jeff

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# Vector optimization problem

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