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In the following code, RHS of NL-constraints should change. but the error happens.
ERROR: UndefVarError: setRHS not defined. could you please learn me why this error happens?. thanks for your helps
using JuMP,CPLEX, Ipopt
#parameters--------------------------------------------------------
sig=0.86;
#---------------------------------------------------------------------------
ALT= Model(optimizer_with_attributes(Juniper.Optimizer, "nl_solver"=>optimizer_with_attributes(Ipopt.Optimizer, "print_level" => 0),
"mip_solver"=>optimizer_with_attributes(CPLEx.Optimizer, "logLevel" => 0),"registered_functions" =>[Juniper.register( :f, 1, f; autodiff = true)])
)
# variables-----------------------------------------------------------------
f(x) = cdf(Normal(0, 1), x);
JuMP.register(ALT, :f, 1, f; autodiff = true);
#variable(ALT, h >= 0.1);
#variable(ALT, L >= 0.0001);
#variable(ALT, n>=2, Int);
#-------------------------------------------------------------------
#NLexpression(ALT,k7,1-f(L-sig*sqrt(n))+f(-L-sig*sqrt(n)));
#NLexpression(ALT,f2,1/k7)
#constraints--------------------------------------------------------
#NLconstraint(ALT, f(-L) <= 1/400);
#NLconstraint(ALT,rf2,f2<=10000);
#-------------------------------------------------------------------
#NLobjective(ALT, Min, f2)
optimize!(ALT)
JuMP.setRHS(rf2,getvalueNLobjective(1/k7))
You are using outdated JuMP version examples. As of today you should use set_normalized_rhs:
set_normalized_rhs(con_ref, some_rhs_value)
Note that this sets the normalized RHS that is after it has been pre-computed by JuMP. For example for #constraint(model, 2x - 5 <= 2) the normalized value is 7.
See also https://jump.dev/JuMP.jl/v0.21/constraints/#Constraint-modifications-1 for more details.
I have an array Z in Julia which represents an image of a 2D Gaussian function. I.e. Z[i,j] is the height of the Gaussian at pixel i,j. I would like to determine the parameters of the Gaussian (mean and covariance), presumably by some sort of curve fitting.
I've looked into various methods for fitting Z: I first tried the Distributions package, but it is designed for a somewhat different situation (randomly selected points). Then I tried the LsqFit package, but it seems to be tailored for 1D fitting, as it is throwing errors when I try to fit 2D data, and there is no documentation I can find to lead me to a solution.
How can I fit a Gaussian to a 2D array in Julia?
The simplest approach is to use Optim.jl. Here is an example code (it was not optimized for speed, but it should show you how you can handle the problem):
using Distributions, Optim
# generate some sample data
true_d = MvNormal([1.0, 0.0], [2.0 1.0; 1.0 3.0])
const xr = -3:0.1:3
const yr = -3:0.1:3
const s = 5.0
const m = [s * pdf(true_d, [x, y]) for x in xr, y in yr]
decode(x) = (mu=x[1:2], sig=[x[3] x[4]; x[4] x[5]], s=x[6])
function objective(x)
mu, sig, s = decode(x)
try # sig might be infeasible so we have to handle this case
est_d = MvNormal(mu, sig)
ref_m = [s * pdf(est_d, [x, y]) for x in xr, y in yr]
sum((a-b)^2 for (a,b) in zip(ref_m, m))
catch
sum(m)
end
end
# test for an example starting point
result = optimize(objective, [1.0, 0.0, 1.0, 0.0, 1.0, 1.0])
decode(result.minimizer)
Alternatively you could use constrained optimization e.g. like this:
using Distributions, JuMP, NLopt
true_d = MvNormal([1.0, 0.0], [2.0 1.0; 1.0 3.0])
const xr = -3:0.1:3
const yr = -3:0.1:3
const s = 5.0
const Z = [s * pdf(true_d, [x, y]) for x in xr, y in yr]
m = Model(solver=NLoptSolver(algorithm=:LD_MMA))
#variable(m, m1)
#variable(m, m2)
#variable(m, sig11 >= 0.001)
#variable(m, sig12)
#variable(m, sig22 >= 0.001)
#variable(m, sc >= 0.001)
function obj(m1, m2, sig11, sig12, sig22, sc)
est_d = MvNormal([m1, m2], [sig11 sig12; sig12 sig22])
ref_Z = [sc * pdf(est_d, [x, y]) for x in xr, y in yr]
sum((a-b)^2 for (a,b) in zip(ref_Z, Z))
end
JuMP.register(m, :obj, 6, obj, autodiff=true)
#NLobjective(m, Min, obj(m1, m2, sig11, sig12, sig22, sc))
#NLconstraint(m, sig12*sig12 + 0.001 <= sig11*sig22)
setvalue(m1, 0.0)
setvalue(m2, 0.0)
setvalue(sig11, 1.0)
setvalue(sig12, 0.0)
setvalue(sig22, 1.0)
setvalue(sc, 1.0)
status = solve(m)
getvalue.([m1, m2, sig11, sig12, sig22, sc])
In principle, you have a loss function
loss(μ, Σ) = sum(dist(Z[i,j], N([x(i), y(j)], μ, Σ)) for i in Ri, j in Rj)
where x and y convert your indices to points on the axes (for which you need to know the grid distance and offset positions), and Ri and Rj the ranges of the indices. dist is the distance measure you use, eg. squared difference.
You should be able to pass this into an optimizer by packing μ and Σ into a single vector:
pack(μ, Σ) = [μ; vec(Σ)]
unpack(v) = #views v[1:N], reshape(v[N+1:end], N, N)
loss_packed(v) = loss(unpack(v)...)
where in your case N = 2. (Maybe the unpacking deserves some optimization to get rid of unnecessary copying.)
Another thing is that we have to ensure that Σ is positive semidifinite (and hence also symmetric). One way to do that is to parametrize the packed loss function differently, and optimize over some lower triangular matrix L, such that Σ = L * L'. In the case N = 2, we can write this as
unpack(v) = v[1:2], LowerTriangular([v[3] zero(v[3]); v[4] v[5]])
loss_packed(v) = let (μ, L) = unpack(v)
loss(μ, L * L')
end
(This is of course prone to further optimization, such as expanding the multiplication directly in to loss). A different way is to specify the condition as constraints into the optimizer.
For the optimzer to work you probably have to get the derivative of loss_packed. Either have to find the manually calculate it (by a good choice of dist), or maybe more easily by using a log transformation (if you're lucky, you find a way to reduce it to a linear problem...). Alternatively you could try to find an optimizer that does automatic differentiation.
Excuse me I am new to Wolfram. I have seen people asking questions about how to do convolution of a function with itself in Wolfram. However, I wonder how to do it multiple times in a loop. That is to say I want to do f20* i.e. f*f*f*f*....f totaling 20 f. How to implement it?
Here is my thinking. Of course do not work....
f[x_] := Piecewise[{{0.1`, x >= 0 && x <= 10}, {0, x < 0}, {0, x > 10}}];
g = f;
n = 19;
For[i = 1, i <= n, i++, g = Convolve[f[x], g, x, y]]; Plot[
g[x], {x, -10, n*10 + 10}, PlotRange -> All]
Could anybody help me?
My new code after revising agentp's code
f[x_] := Piecewise[{{0.1, x >= 0 && x <= 10}, {0, x < 0}, {0,x > 10}}];
n = 19;
res = NestList[Convolve[#, f[x], x, y] /. y -> x &, f[x], n];
Plot[res, {x, -10, (n + 1)*10 + 10}, PlotRange -> All,PlotPoints -> 1000]
My buggy image
maybe this?
Nest[ Convolve[#, f[x], x, y] /. y -> x &, f[x] , 3]
If that's not right maybe show what you get by hand for n=2 or 3.
res = NestList[ Convolve[#, f[x], x, y] /. y -> x &, f[x] , 10];
Plot[res, {x, 0, 100}, PlotRange -> All]
this gets very slow with n, I don't have the patience to run it out to 20.
Your approach is nearly working. You just have to
make sure to copy f by value before entering the loop, because otherwise you face infinite recursion.
Assign the result of Convolve to a function which takes a parameter.
This is the code with the mentioned changes:
f[x_] := Piecewise[{{0.1, x >= 0 && x <= 10}, {0, x < 0}, {0, x > 10}}];
g[x_] = f[x];
n = 20;
For[i = 1, i <= n, i++, g[y_] = Convolve[f[x], g[x], x, y]];
Plot[g[x], {x, -10, n*10 + 10}, PlotRange -> All]
Edit: While this works, agentp's answer is more consise and i suspect also faster.
Hello and thanks in advance,
I came upon an issue trying to plot a piecewise function in Mathematica. When entered directly into the Plot function, the piecewise did NOT plot. However, when I defined the piecewise as a variable, it did plot. Why is this issue occurring and can I still join my plot? (I would like to join it by setting exclusions to none)
The following is my code: (I define a Maxwell strain function and am trying to model the plastic deformation of a polymer over multiple stress cycles)
z = 2*10^10;
h = 10^12;
MaxwellStrain[s_, t_] := s/z + (s*t)/h;
stress = {0, 10^7, -10^7, 5*10^6, 10^7, -5*10^6};
time = {0, 100, 200, 300, 400, 500};
strainList = Join[{0}, Table[MaxwellStrain[stress[[i + 1]], t - time[[i]]], {i, 1, 5}]];
Plot[
Piecewise[
Table[{
Total[strainList[[1 ;; i + 1]]], time[[i]] < t < time[[i + 1]]},
{i, 1, 5}
],
Exclusions -> none]
,
{t, 0, 500}
]
x = Piecewise[
Table[{
Total[strainList[[1 ;; i + 1]]], time[[i]] < t < time[[i + 1]]},
{i, 1, 5}
],
Exclusions -> none]
Plot[x, {t, 0, 500}]
The following is my output: (first plot doesn't show, the second does)
output:
Thank you for the help,
SV
I programmed a Euler function but misread the instructions, so now I have to make a new one, but I can't figure it out.
I have made the following automatic Euler function.
f[x_, y_] := -x y^2;
x0 = 0;
y0 = 2;
xend = 2;
steps = 20;
h = (xend - x0)/steps // N;
x = x0;
y = y0;
eulerlist = {{x, y}};
For[i = 1, i <= steps, y = f[x, y]*h + y;
x = x + h;
eulerlist = Append[eulerlist, {x, y}];
i++
]
Print[eulerlist]
But it just generates the list I have specified.
I would like to have a Euler function which is able to generate this form:
Euler[y, 2, -x y^2, {x, 0, 2}, 20]
I don't seem to get any further.
It is not clear what you are asking, but if what you want is to be able to input
Euler[y, 2, -x y^2, {x, 0, 2}, 20]
and get
{{0,2},{0.1,2.},{0.2,1.96},{0.3,1.88317},{0.4,1.77678},{0.5,1.6505},{0.6,1.51429},{0.7,1.37671},{0.8,1.24404},{0.9,1.12023},{1.,1.00728},{1.1,0.905822},{1.2,0.815565},{1.3,0.735748},{1.4,0.665376},{1.5,0.603394},{1.6,0.548781},{1.7,0.500596},{1.8,0.457994},{1.9,0.420238},{2.,0.386684}}
Then you need to write a function definition like this:
Euler[y0_, f_, {x0_, xend_}, steps_Integer?Positive] := (* body *)
Notice the underscores to denote patterns, the := to denote delayed evaluation and the pattern specification Integer?Positive.
As for the body of the function -- oh my goodness could you have picked a less Mathematica-style approach? Perhaps not. Procedural loops and Append are almost never the best way to do anything in Mathematica.
Here is a better solution.
Euler[y_, y0_, f_, {x_, x0_, xend_}, steps_Integer?Positive] :=
With[{h = N[(xend - x0)/steps], ff = Function[{x, y}, f]},
NestList[{#[[1]] + h, ff[#[[1]], #[[2]]]*h + #[[2]]} &, {x0, y0},
steps]]
Euler[y, 2, -x y^2, {x, 0, 2}, 20]
{{0, 2}, {0.1, 2.}, {0.2, 1.96}, {0.3, 1.88317}, {0.4,
1.77678}, {0.5, 1.6505}, {0.6, 1.51429}, {0.7, 1.37671}, {0.8,
1.24404}, {0.9, 1.12023}, {1., 1.00728}, {1.1, 0.905822}, {1.2,
0.815565}, {1.3, 0.735748}, {1.4, 0.665376}, {1.5, 0.603394}, {1.6,
0.548781}, {1.7, 0.500596}, {1.8, 0.457994}, {1.9, 0.420238}, {2.,
0.386684}}
If you want something that outputs Euler[y, 2, -x y^2, {x, 0, 2}, 20], then typing it into the notebook is the quickest method.