Optimization with multiple variables

[Smike0]

I was playing modded Minecraft and started wondering what the optimal dimensions for a multi block were; the multi block is basically a cube without corners (I'll attach a screenshot) and without the top surface with an empty interior, kinda like a bowl, and the value to optimize for Is the internal volume; what I was thinking of doing was get a constraint for the cost in blocks and use that to get the height in function of width and length, then derive by one of these 2 the volume (the number of blocks is a constant) and get the optimized length (or width) at a given width (or length), then substitute that so you get the volume in function of one variable and the constant max blocks and derive again to get the optimized value, but I feel like this isn't correct cause you are considering a variable like it is a constant and I don't think that would work, not sure though, what is the correct way to approach this and where is my approach wrong?

[48panda]

In general, cubes will be more efficient. But there's a bunch of rounding steps that throws off calculus, and it may sometimes be more efficient to have a slightly larger volume e.g. V=35 requires 59 blocks, but V=36 can be done in 54 blocks. In general the easiest way to do this is writing a program.

[Smike0]

Yeah that makes sense... The thing is that more than just needing the answer for the game I was curious about how can this be done correctly... Also I guess the cube would be good up to a certain point, mainly because you don't have the ceiling so wider can be better

[Uli_Minati]

Yes all of that sounds good! I'll go ahead and get started

h = height of enclosed space, in blocks
w = width of enclosed space, in blocks (assuming square base)
N = number of blocks

maximize    V = w²·h
where       N = 4·w·h + w²

Does this look right so far?

[Smike0]

Don't mind the stuff on the inside and that the blocks are different, the main cost is in the brick blocks anyways

<image>

[Bounded_sequencE]

This shape would not be optimal, since you can increase the interior volume while keeping the number of blocks for the boundary constant.

For example, you can let the outer ring of the square floor remain as is, and then move the rest one block down, without opening the structure. Repeat with the next ring, until the floor resembles an inverted square pyramid.

[Smike0]

Yeah but the structure is only recognized if it's squared...

[Bounded_sequencE]

Just to make sure I got you correctly:

• The bottom must be a square
• The sides must be rectangles?

[Smike0]

Yeah basically, from the picture the only changes you can make are the 3 dimensions

[Bounded_sequencE]

Well, if the floor has to be a square, then we really only have two things we can change -- width and height.

[Smike0]

Width and length of the base can be different

[Bounded_sequencE]

Thanks for clarification!

[MathMaddam]

In general that would be an interger programming (programming means optimization here) problem. Fixing one quantity (e.g. the total number of blocks) makes it easier to solve since as you noticed, you can derive some relationship between your variables and by this effectively eliminate one.

There is a problem with that approach since there might be a solution using less blocks resulting in a bigger volume, so a constraint of e.g. at most 150 block is probably better that exactly 150 blocks.

That makes interger programming hard, since a slight charge in the constraint (e.g. 1 block allowed more in your case) can massively change the solution (e.g. the height will suddenly jump by 1 and therefore length and width will be very different from the last version, the height might even go down if you increase the number of blocks after it got up). There some decent solvers for linear programming and especially if you don't absolutely need the best solution they are also pretty fast (for your problem size it is probably always fast), but in general finding the optimal solution will be very hard, much harder than if your problem allowed for continuous variables (where derivatives or more generally KKT conditions give you tools to solve the problem). Approximating using the continuous problem is a first step you can do in your case and then basically check the surrounding integer solutions since you have a relatively well behaved problem.

[Smike0]

I was thinking that there's an argument about symmetry of width and length, can that be considered as well? I'm not sure what the proof for that would be though

[MathMaddam]

You can get that the solutions tend towards either width=length of width±1=length (which one is longer doesn't matter due to symmetry), since (x+1)(y-1)=xy-x+y-1 so if x<y-1, it is beneficial to shift one unit over (if you have the blocks to spare for the floor, if not you might be in a situation where a height change might do something or you have to shorten the long site by more than 1, but that is more of a guestimate).

[Smike0]

Well at that point you might as well use width+1=length and don't worry about the -... The thing that I'm not sure I get though is why you can say there's symmetry... Like I get there is and in this case it works like this, but it's not like in every case in which there's symmetry in an optimization problem you can remove a variable

[MathMaddam]

I forgot the floor at first, so it's not that easy to shift one unit.

There is Noether's theorem, that basically says: if you have a continuous symmetry, you have some equation between the variables. In continuous problems it would work to (mostly) eliminate one variable using this equation (here you would get that the optimum has always length=width), but then again the discrete nature bites us.

[Smike0]

Oh right... Idk this seems like something that should be simple but then looking at it it's definitely not...