I have a domain problem formulation in MathProg, where the cost function uses geometrical distances. The data sets contain only X,Y coordinates and not the actual distances. Right now, my formulation calculates the distances directly:
minimize total: sum{(f, c) in S} x[f, c] * sqrt(((facilityXs[f] - customerXs[c])**2) + ((facilityYs[f] - customerYs[c])**2));
And I want to know, whether the MathProg compiler is smart enough to see that the expression inside sqrt is constant and thus the whole thing can be precomputed, or whether it recalculates the expression every time, and how can I write it in a more elegant way.
Yes the MathProg 'compiler' is smart enough. It will precompute all equations containing solely parameters (and then create a computation matrix containing just one numeric value per cell). If you put variables in non linear functions like sqrt() the precomputation will fail.
A more elegant way is to keep your core set of equations linear. I often use separate parameters calculated by 'prequations', to keep the main formulations clean and simple.
param distance{(f,c) in S} := sqrt(((facilityXs[f] - customerXs[c])**2) + ((facilityYs[f] - customerYs[c])**2);
minimize total: sum{(f, c) in S} x[f, c] * distance[f,c]);
If the expression inside sqrt doesn't contain variables, then it will be evaluated at the translation stage and sent to the solver as a constant (coefficient of x[f, c]).
Related
I use the scipy.optimize.minimize ( https://docs.scipy.org/doc/scipy/reference/tutorial/optimize.html ) function with method='L-BFGS-B.
An example of what it returns is here above:
fun: 32.372210618549758
hess_inv: <6x6 LbfgsInvHessProduct with dtype=float64>
jac: array([ -2.14583906e-04, 4.09272616e-04, -2.55795385e-05,
3.76587650e-05, 1.49213975e-04, -8.38440428e-05])
message: 'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
nfev: 420
nit: 51
status: 0
success: True
x: array([ 0.75739412, -0.0927572 , 0.11986434, 1.19911266, 0.27866406,
-0.03825225])
The x value correctly contains the fitted parameters. How do I compute the errors associated to those parameters?
TL;DR: You can actually place an upper bound on how precisely the minimization routine has found the optimal values of your parameters. See the snippet at the end of this answer that shows how to do it directly, without resorting to calling additional minimization routines.
The documentation for this method says
The iteration stops when (f^k - f^{k+1})/max{|f^k|,|f^{k+1}|,1} <= ftol.
Roughly speaking, the minimization stops when the value of the function f that you're minimizing is minimized to within ftol of the optimum. (This is a relative error if f is greater than 1, and absolute otherwise; for simplicity I'll assume it's an absolute error.) In more standard language, you'll probably think of your function f as a chi-squared value. So this roughly suggests that you would expect
Of course, just the fact that you're applying a minimization routine like this assumes that your function is well behaved, in the sense that it's reasonably smooth and the optimum being found is well approximated near the optimum by a quadratic function of the parameters xi:
where Δxi is the difference between the found value of parameter xi and its optimal value, and Hij is the Hessian matrix. A little (surprisingly nontrivial) linear algebra gets you to a pretty standard result for an estimate of the uncertainty in any quantity X that's a function of your parameters xi:
which lets us write
That's the most useful formula in general, but for the specific question here, we just have X = xi, so this simplifies to
Finally, to be totally explicit, let's say you've stored the optimization result in a variable called res. The inverse Hessian is available as res.hess_inv, which is a function that takes a vector and returns the product of the inverse Hessian with that vector. So, for example, we can display the optimized parameters along with the uncertainty estimates with a snippet like this:
ftol = 2.220446049250313e-09
tmp_i = np.zeros(len(res.x))
for i in range(len(res.x)):
tmp_i[i] = 1.0
hess_inv_i = res.hess_inv(tmp_i)[i]
uncertainty_i = np.sqrt(max(1, abs(res.fun)) * ftol * hess_inv_i)
tmp_i[i] = 0.0
print('x^{0} = {1:12.4e} ± {2:.1e}'.format(i, res.x[i], uncertainty_i))
Note that I've incorporated the max behavior from the documentation, assuming that f^k and f^{k+1} are basically just the same as the final output value, res.fun, which really ought to be a good approximation. Also, for small problems, you can just use np.diag(res.hess_inv.todense()) to get the full inverse and extract the diagonal all at once. But for large numbers of variables, I've found that to be a much slower option. Finally, I've added the default value of ftol, but if you change it in an argument to minimize, you would obviously need to change it here.
One approach to this common problem is to use scipy.optimize.leastsq after using minimize with 'L-BFGS-B' starting from the solution found with 'L-BFGS-B'. That is, leastsq will (normally) include and estimate of the 1-sigma errors as well as the solution.
Of course, that approach makes several assumption, including that leastsq can be used and may be appropriate for solving the problem. From a practical view, this requires the objective function return an array of residual values with at least as many elements as variables, not a cost function.
You may find lmfit (https://lmfit.github.io/lmfit-py/) useful here: It supports both 'L-BFGS-B' and 'leastsq' and gives a uniform wrapper around these and other minimization methods, so that you can use the same objective function for both methods (and specify how to convert the residual array into the cost function). In addition, parameter bounds can be used for both methods. This makes it very easy to first do a fit with 'L-BFGS-B' and then with 'leastsq', using the values from 'L-BFGS-B' as starting values.
Lmfit also provides methods to more explicitly explore confidence limits on parameter values in more detail, in case you suspect the simple but fast approach used by leastsq might be insufficient.
It really depends what you mean by "errors". There is no general answer to your question, because it depends on what you're fitting and what assumptions you're making.
The easiest case is one of the most common: when the function you are minimizing is a negative log-likelihood. In that case the inverse of the hessian matrix returned by the fit (hess_inv) is the covariance matrix describing the Gaussian approximation to the maximum likelihood.The parameter errors are the square root of the diagonal elements of the covariance matrix.
Beware that if you are fitting a different kind of function or are making different assumptions, then that doesn't apply.
I'm trying to use the scipy.optimize.slsqp for an industrial-related constrained optimization. A highly non-linear FE model is used to generate the objective and the constraint functions, and their derivatives/sensitivities.
The objective function is in the form:
obj=a number calculated from the FE model
A series of constraint functions are set, and most of them are in the form:
cons = real number i - real number j (calculated from the FE model)
I would like to try to restrict the design variables to integers as that would be what input into the plant machine.
Another consideration is to have a log file recording what design variable have been tried. if a set of design variable (integer) is already tried for, skip the calculation, perturb the design variable and try again. By limiting the design variable to integers, we are able to limit the number of trials (while leaving the design variable to real, a change in the e.g. 8th decimal point could be regarded as untried values).
I'm using SLSQP as it is one of the SQP method (please correct me if I am wrong), and the it is said to be powerful to deal with nonlinear problems. I understand the SLSQP algorithm is a gradient-based optimizer and there is no way I can implement the restriction of the design variables being integer in the algorithm coded in FORTRAN. So instead, I modified the slsqp.py file to the following (where it calls the python extension built from the FORTRAN algorithm):
slsqp(m, meq, x, xl, xu, fx, c, g, a, acc, majiter, mode, w, jw)
for i in range(len(x)):
x[i]=int(x[i])
The code stops at the 2nd iteration and output the following:
Optimization terminated successfully. (Exit mode 0)
Current function value: -1.286621577077517
Iterations: 7
Function evaluations: 0
Gradient evaluations: 0
However, one of the constraint function is violated (value at about -5.2, while the default convergence criterion of the optimization code = 10^-6).
Questions:
1. Since the FE model is highly nonlinear, I think it's safe to assume the objective and constraint functions will be highly nonlinear too (regardless of their mathematical form). Is that correct?
2. Based on the convergence criterion of the slsqp algorithm(please see below), one of which requires the sum of all constraint violations(absolute values) to be less than a very small value (10^-6), how could the optimization exit with successful termination message?
IF ((ABS(f-f0).LT.acc .OR. dnrm2_(n,s,1).LT.acc).AND. h3.LT.acc)
Any help or advice is appreciated. Thank you.
How do I multiply two arrays of different rank, element-wise? For example, element-wise multiplying every row of a matrix with a vector.
real :: a(m,n), b(n)
My initial thought was to use spread(b,...), but it is my understanding that this tiles b in memory, which would make it undesirable for large arrays.
In MATLAB I would use bsxfun for this.
If the result of the expression is simply being assigned to another variable (versus being an intermediate in a more complicated expression or being used as an actual argument), then a loop (DO [CONCURRENT]) or FORALL assignment is likely to be best from the point of view of execution speed (though it will be processor dependent).
so I am making a simple simulation of different planets with individual velocity flying around space and orbiting each other.
I plan to simulate their pull on each other by considering each planet as projecting their own "gravity vector field." Each time step I'm going to add the vectors outputted from each planets individual vector field equation (V = -xj + (-yj) or some notation like it) except the one being effected in the calculation, and use the effected planets position as input to the equations.
However this would inaccurate, and does not consider the gravitational pull as continuous and constant. Bow do I calculate the movement of my planets if each is continuously effecting the others?
Thanks!
In addition to what Blender writes about using Newton's equations, you need to consider how you will be integrating over your "acceleration field" (as you call it in the comment to his answer).
The easiest way is to use Euler's Method. The problem with that is it rapidly diverges, but it has the advantage of being easy to code and to be reasonably fast.
If you are looking for better accuracy, and are willing to sacrifice some performance, one of the Runge-Kutta methods (probably RK4) would ordinarily be a good choice. I'll caution you that if your "acceleration field" is dynamic (i.e. it changes over time ... perhaps as a result of planets moving in their orbits) RK4 will be a challenge.
Update (Based on Comment / Question Below):
If you want to calculate the force vector Fi(tn) at some time step tn applied to a specific object i, then you need to compute the force contributed by all of the other objects within your simulation using the equation Blender references. That is for each object, i, you figure out how all of the other objects pull (apply force) and those vectors when summed will be the aggregate force vector applied to i. Algorithmically this looks something like:
for each object i
Fi(tn) = 0
for each object j ≠ i
Fi(tn) = Fi(tn) + G * mi * mj / |pi(tn)-pj(tn)|2
Where pi(tn) and pj(tn) are the positions of objects i and j at time tn respectively and the | | is the standard Euclidean (l2) normal ... i.e. the Euclidean distance between the two objects. Also, G is the gravitational constant.
Euler's Method breaks the simulation into discrete time slices. It looks at the current state and in the case of your example, considers all of the forces applied in aggregate to all of the objects within your simulation and then applies those forces as a constant over the period of the time slice. When using
ai(tn) = Fi(tn)/mi
(ai(tn) = acceleration vector at time tn applied to object i, Fi(tn) is the force vector applied to object i at time tn, and mi is the mass of object i), the force vector (and therefore the acceleration vector) is held constant for the duration of the time slice. In your case, if you really have another method of computing the acceleration, you won't need to compute the force, and can instead directly compute the acceleration. In either event, with the acceleration being held as constant, the position at time tn+1, p(tn+1) and velocity at time tn+1, v(tn+1), of the object will be given by:
pi(tn+1) = 0.5*ai(tn)*(tn+1-tn)2 + vi(tn)*(tn+1-tn)+pi(tn)
vi(tn+1) = ai(tn+1)*(tn+1-tn) + vi(tn)
The RK4 method fits the driver of your system to a 2nd degree polynomial which better approximates its behavior. The details are at the wikipedia site I referenced above, and there are a number of other resources you should be able to locate on the web. The basic idea is that instead of picking a single force value for a particular timeslice, you compute four force vectors at specific times and then fit the force vector to the 2nd degree polynomial. That's fine if your field of force vectors doesn't change between time slices. If you're using gravity to derive the vector field, and the objects which are the gravitational sources move, then you need to compute their positions at each of the four sub-intervals in order compute the force vectors. It can be done, but your performance is going to be quite a bit poorer than using Euler's method. On the plus side, you get more accurate motion of the objects relative to each other. So, it's a challenge in the sense that it's computationally expensive, and it's a bit of a pain to figure out where all the objects are supposed to be for your four samples during the time slice of your iteration.
There is no such thing as "continuous" when dealing with computers, so you'll have to approximate continuity with very small intervals of time.
That being said, why are you using a vector field? What's wrong with Newton?
And the sum of the forces on an object is that above equation. Equate the two and solve for a
So you'll just have to loop over all the objects one by one and find the acceleration on it.
Greetings. I'm trying to approximate the function
Log10[x^k0 + k1], where .21 < k0 < 21, 0 < k1 < ~2000, and x is integer < 2^14.
k0 & k1 are constant. For practical purposes, you can assume k0 = 2.12, k1 = 2660. The desired accuracy is 5*10^-4 relative error.
This function is virtually identical to Log[x], except near 0, where it differs a lot.
I already have came up with a SIMD implementation that is ~1.15x faster than a simple lookup table, but would like to improve it if possible, which I think is very hard due to lack of efficient instructions.
My SIMD implementation uses 16bit fixed point arithmetic to evaluate a 3rd degree polynomial (I use least squares fit). The polynomial uses different coefficients for different input ranges. There are 8 ranges, and range i spans (64)2^i to (64)2^(i + 1).
The rational behind this is the derivatives of Log[x] drop rapidly with x, meaning a polynomial will fit it more accurately since polynomials are an exact fit for functions that have a derivative of 0 beyond a certain order.
SIMD table lookups are done very efficiently with a single _mm_shuffle_epi8(). I use SSE's float to int conversion to get the exponent and significand used for the fixed point approximation. I also software pipelined the loop to get ~1.25x speedup, so further code optimizations are probably unlikely.
What I'm asking is if there's a more efficient approximation at a higher level?
For example:
Can this function be decomposed into functions with a limited domain like
log2((2^x) * significand) = x + log2(significand)
hence eliminating the need to deal with different ranges (table lookups). The main problem I think is adding the k1 term kills all those nice log properties that we know and love, making it not possible. Or is it?
Iterative method? don't think so because the Newton method for log[x] is already a complicated expression
Exploiting locality of neighboring pixels? - if the range of the 8 inputs fall in the same approximation range, then I can look up a single coefficient, instead of looking up separate coefficients for each element. Thus, I can use this as a fast common case, and use a slower, general code path when it isn't. But for my data, the range needs to be ~2000 before this property hold 70% of the time, which doesn't seem to make this method competitive.
Please, give me some opinion, especially if you're an applied mathematician, even if you say it can't be done. Thanks.
You should be able to improve on least-squares fitting by using Chebyshev approximation. (The idea is, you're looking for the approximation whose worst-case deviation in a range is least; least-squares instead looks for the one whose summed squared difference is least.) I would guess this doesn't make a huge difference for your problem, but I'm not sure -- hopefully it could reduce the number of ranges you need to split into, somewhat.
If there's already a fast implementation of log(x), maybe compute P(x) * log(x) where P(x) is a polynomial chosen by Chebyshev approximation. (Instead of trying to do the whole function as a polynomial approx -- to need less range-reduction.)
I'm an amateur here -- just dipping my toe in as there aren't a lot of answers already.
One observation:
You can find an expression for how large x needs to be as a function of k0 and k1, such that the term x^k0 dominates k1 enough for the approximation:
x^k0 +k1 ~= x^k0, allowing you to approximately evaluate the function as
k0*Log(x).
This would take care of all x's above some value.
I recently read how the sRGB model compresses physical tri stimulus values into stored RGB values.
It basically is very similar to the function I try to approximate, except that it's defined piece wise:
k0 x, x < 0.0031308
k1 x^0.417 - k2 otherwise
I was told the constant addition in Log[x^k0 + k1] was to make the beginning of the function more linear. But that can easily be achieved with a piece wise approximation. That would make the approximation a lot more "uniform" - with only 2 approximation ranges. This should be cheaper to compute due to no longer needing to compute an approximation range index (integer log) and doing SIMD coefficient lookup.
For now, I conclude this will be the best approach, even though it doesn't approximate the function precisely. The hard part will be proposing this change and convincing people to use it.