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I am trying to solve an equation that can include truncations in Python with a numerical approach. I am wondering what the best library and approach would be? Following is more detail about the problem:

The equation changes every time. From a human perspective, the equations should be pretty simple; they include common operators such as +,-,*,/, and they also sometimes have truncation functions (truncate to integer) or limit functions (limit the value in parenthesis between two provided bounds) or (rarely) multiple variables. A couple of examples (with these being separate examples and not a system of equations) would be:

  1. TRUNCATE(VAR_1 + 300) - 50.4 = 200
  2. (VAR_2 + VAR_3)*3 = 35
  3. LIMIT(3,5)(VAR_4) = 8
  4. VAR_5 = 34

(This is not exactly what the equations look like, because I am writing them in postfix notation, but I have a calculator to determine their value with provided input values.)

All I need for these equations is some value for each variable that would solve each equation; I do not need to know every solution.

Some additional things to note about this is a) these variables all have maximum and minimum values, b) while perfection would be nice, occasional errors are acceptable, and c) some of the variables are integers, which I expect really complicates things. Right now, I'm handling this very sloppily but also mostly acceptably for my case by rounding the integer values to the nearest int.

In an attempt to solve this problem, I tried solving analytically with Sympy (which as you might expect didn't work on truncations and was difficult to implement), and I also tried using Scipy minimize as follows:

minimize(minimization, x0, method = 'SLSQP', constraints = cons, tol = 1e-3, options={'ftol': 1e-3, 'disp':True, 'maxiter': 100, "eps":.1}, args = (x_vals, postfix, const_values, value))

This one got stuck on truncations, presumably because it didn't know what direction to move, unless I set the step to 1, which decreased accuracy. For some reason, it also didn't seem to follow the ftol, because it would give acceptable answers within the tolerance but would just keep going to the iteration limit.

I am considering using something that does random walks like the "Markov Chain Monte Carlo" method, but I really don't know much about this and was eager to hear other thoughts.

Hattie35
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    Note that a truncation or ceil function is not differentiable, so you can't use the SLSQP method. Instead, you can try a derivative-free method like Powell. According to the docs: *the function need not be differentiable, and no derivatives are taken*. – joni Jul 15 '21 at 15:14

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I ended up solving the problem two slightly different ways. Both of them used the Powell solver as suggested by joni in the comments on the original question, and for both of them I had to multiply the output of the function that gets passed to the "fun" parameter (a function that I named minimize) by 100, because I could never get the tolerance adjusted in the solver function call.

  1. When the equation had only one variable, I removed the truncation from the minimize function. This worked for my purposes because the reason the equations I was using was being truncated was so they would equal an integer value (generally). So, when the equation output is an integer and there is only one variable, I believe the correct solution will be obtained by just pretending the truncation function does not exist in the solver (though remember to be wary of floating point math). (And if any numbers outside of the truncation are integers, the equation may not have a solution anyways.)

  2. In cases with multiple variables, my solution was to a) include the truncation function in the minimize function and b) round the x values suggested by the solver as I planned to round them in the end (ex. round them to an integer if they were an integer value).

Anyways, this solution worked for the problem defined above, but it otherwise has some limitations. It is not guaranteed to always find the correct output, especially the second part. Another approach people with this problem may wish to consider would be some sort of integer programming, if they have linear equations.

Hattie35
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