Let's say we have 2 equations and 3 variables (N equations and M variables, where M > N), for example:
0.3a+0.1b+0.2*c=10 + E
0.1a+0.2b+0.3*c=11 + E
We want to find a, b, c that are in the range [0.0, 1.0] and we want to minimize the error E.
We want a smooth solutions that in this context means that if we chart a, b, c in a chart, the chart is smooth.
How this kind of problems can be approached with sagemath?
Here is an example, in which we use x[0], x[1], x[2] for a, b, c, and x[-1] for the (absolute) bound for errors.
milp = MixedIntegerLinearProgram(maximization=False) # we do minimization
x = milp.new_variable(real=True)
milp.set_min(x[0],0)
milp.set_min(x[1],0)
milp.set_min(x[2],0)
milp.set_max(x[0],1)
milp.set_max(x[1],1)
milp.set_max(x[2],1)
eq1 = 0.3*x[0] + 0.1*x[1] + 0.2*x[2] - 10
milp.add_constraint( eq1 <= x[-1] )
milp.add_constraint( eq1 >= -x[-1] )
eq2 = 0.1*x[0] + 0.2*x[1] + 0.3*x[2] - 11
milp.add_constraint( eq2 <= x[-1] )
milp.add_constraint( eq2 >= -x[-1] )
milp.set_objective(x[-1]) # minimize the error bound
milp.solve()
print( milp.get_values(x) )The code produces the following result: {0: 1.0, 1: 1.0, 2: 1.0, -1: 10.4}, that is a=b=c=1.0, and the maximum error of 10.4.
Thanks a lot!
I can see that f = |E|
is translated as f <= E f >= -E
but let's suppose that E=5 and f=4, the translated system of equations will be true in this case while 4 = |5| is false
Errors may differ across the equations. x[-1] bounds their absolute values from above.
Asked: 1 year ago
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Last updated: Apr 20 '24
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This kind of problems can be solved with MILP - see details and examples at https://doc.sagemath.org/html/en/them...
Hello, in my case all variables are non-integer (can have decimals)
That's fine. MILP supports non-integer variables as well.
In fact it's not E which should be minimized but |E|. So the problem should receive a special formalization which can transform a minimization of an absolute value (which is not linear) in the minimization of an affine objective. This can be correctly be done as Max underline with
MixedIntegerLinearPrograminSageMath. Elle if you need some help I can write the program for ypu.Good catch! E is actually |E|. If you have any example of how to code the system would be great!