# How to maximize just with respect two parameters out of four?

There are some posts explaining how do we maximize a function of two parameters with respect to one of those and how to plot the resulting function: Plot a function after taking the supremum with respect to one variable.

Now I have a quite complicated function which has four parameters: $$(n,t,\theta,\phi)$$ where $$n\in \mathbb{N}$$ is natural, $$t\in (-1,1)$$ and $$(\theta,\phi)\in S^2$$ are angles on the sphere.

The function is highly non trivial. It is the output of computations involving matrices, eigenvalues, and a bunch of things like that. The $$n$$ is the dimension of the matrix, which acts as a cuttoff to get an approximation to the real computation with operators on a Hilbert space (so one is actually truncating to dimension $$n$$).

Now I need to maximize with respect to $$(\theta,\phi)$$ and get a function of $$(n,t)$$ that for each $$n$$ fixed can be plotted.

I believe that numerically there sould be no issue in doing that. The method of the question didn't work however. Calling preClassicalCorrelationsAB[nmax_Integer, t_, th_, phi_] the four-argument function, I tried:

f[nmax_Integer, t_?NumericQ] := NMaximize[{preClassicalCorrelationsAB[nmax, t, th, phi], 0 <= th <= Pi, 0 <= phi <= 2 Pi}, t]


And tried plotting f for fixed nmax = 100. It didn't work. In particular I got the error:

0.05102712269051777 is not a valid variable.

The same happend with MaxValue.

I mean, I know the function is highly complicated and so on, but numerically it should be possible to handle it.

So, how can I proceed to maximize this function with respect to $$(\theta,\phi)$$ and get a function of $$(n,t)$$ which I can plot for any value of $$n$$?

Edit: The comment showed I was using NMaximize in the wrong way. Still, I have a problem due to the kind of function I'm considering. The code defining is as follows. First two helper functions are defined:

cs = Compile[{{x, _Real}}, If[0. < x < 1., x Log[2, x], 0.], CompilationTarget -> "C", RuntimeAttributes -> {Listable}, Parallelization -> True]
myEigenvalues[mat_SparseArray?SquareMatrixQ] := If[mat["NonzeroPositions"] === {} && mat["Background"] == 0., ConstantArray[0., Length[mat]], Eigenvalues[N[mat], Method -> "Banded"]]


Next one defines the matrices rhoBPostAPlus and rhoBPostAMinus. Finally one defines:

probPlus[nmax_Integer, t_, th_, phi_] :=Tr[rhoBPostAPlus[nmax, t, th, phi]]
probMinus[nmax_Integer, t_, th_, phi_] := Tr[rhoBPostAMinus[nmax, t, th, phi]]
rhoBPostAPlusNormalized[nmax_Integer, t_, th_, phi_] := rhoBPostAPlus[nmax, t, th, phi]/probPlus[nmax, t, th, phi]
rhoBPostAMinusNormalized[nmax_Integer, t_, th_, phi_] := rhoBPostAMinus[nmax, t, th, phi]/probMinus[nmax, t, th, phi]

entropyBPostAPlus[nmax_Integer, t_, th_, phi_] :=Total[cs@myEigenvalues[rhoBPostAPlusNormalized[nmax, t, th, phi]]]
entropyBPostAMinus[nmax_Integer, t_, th_, phi_] := Total[cs@myEigenvalues[rhoBPostAMinusNormalized[nmax, t, th, phi]]]

preClassicalCorrelationsAB[nmax_Integer, t_, th_, phi_] := entropyB[nmax, t] + probPlus[nmax, t, th, phi]*entropyBPostAPlus[nmax, t, th, phi] + probMinus[nmax, t, th, phi]*entropyBPostAMinus[nmax, t, th, phi]


This preClassicalCorrelationsAB works perfectly if I try to evaluate it. We can even plot it without error.

Now when we evaluate:

f[nmax_Integer, t_?NumericQ] := NMaximize[{preClassicalCorrelationsAB[nmax, t, th, phi],0 <= th <= Pi, 0 <= phi <= 2 Pi}, {th, phi}]


we get the error:

NMaxValue::nnum: The function value -1.90526-0.5
((0<Eigenvalues[SparseArray[Automatic,<<3>>],Method->Banded]<1.)+1.4427
Eigenvalues[SparseArray[<<1>>],<<1>>]
Log[Eigenvalues[<<1>>,Method->Banded]])-0.5
((0<Eigenvalues[SparseArray[Automatic,{11,11},0,
{1,{{0,2,5,8,11,14,17,20,23,26,29,31},{{2},{1},{1},{3},{2},{2},{4},{3},{3},{5},{4},{4},{6},{5},{5},{7},{6},{6},{8},{7},{7},{9},{8},{8},{10},{9},{9},{11},{10},{10},{11}}},
{Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<4>>],Times[<<2>>],Times[<<4>>],Times[<<2>>]}}],Method->Banded]<1.)+1.4427
Eigenvalues[SparseArray[<<1>>],Method->Banded]
Log[Eigenvalues[SparseArray[Automatic,{11,11},0,{1,{<<2>>},
{<<31>>}}],Method->Banded]]) is not a number at {phi,th} =
{6.20181,2.61304}.


Why is that? NMaximize can't handle a function like that?

• I would think that your call to NMaximize within the definition of f should read: NMaximize[{func, cons}, {th, phi}], i.e. you should NOT be trying to maximize with respect to t, but with respect to $\theta$ and $\phi$. t is a numerical value by the time the NMaximize call is evaluated. – MarcoB Mar 12 '19 at 17:55
• What is rhoBPostAPlus [ ] in your code? The function is undefined. Same for entropyB and rhoBPostAMinus. – MikeY Mar 13 '19 at 16:56
• rhoBPostAPlus is a matrix which depends on the four parameters. I didn't add it to avoid making the post even lengthier, since its definition is rather complicated. The same for rhoBPostAMinus. Intuitively I thought the problem couldn't be with them, since the preClassicalCorrelationsAB function works on its own, being even possible to plot it. – user1620696 Mar 13 '19 at 17:03
• Try forcing the preClassicalCorrelationsAB function to not run until it has numeric values, so use preClassicalCorrelationsAB[nmax_Integer, t_?NumericQ, th_?NumericQ, phi_?NumericQ] – MikeY Mar 13 '19 at 17:12
• If you can simulate the behavior with a simplified rhoBPostAPlus we can probably solve this in minutes. – MikeY Mar 13 '19 at 17:47

Define a function to play with...

preClassicalCorrelationsAB[nmax_, t_, th_, phi_] := -(nmax^2 + t^2 + th^2 + phi^2)


Then define

f[nmax_Integer, t_?NumericQ] := NMaximize[preClassicalCorrelationsAB[nmax, t, th, phi], {th, phi}] // First


We are optimizing over th and phi with n and t fixed, then picking off the value of the function with the First call.

Try it.

f[2, 2]


-8

EDIT: just saw @MarcoB's comment, which is the answer too.

preClassicalCorrelationsAB[nmax_Integer, t_?NumericQ, th_?NumericQ, phi_?NumericQ]

If you are having trouble using a Plot command on the function, and know that the function is reasonably smooth, you can try ListPlot3D@Table[etc.,{ }]` as a workaround that lets you control the sampling interval.