Optimization of a function with possibly complex values

Question

I have the following problem:

I have the following function of 3 parameters A, B, mu: $$ g(A,B,\mu) = \int_{BZ} dk_1 dk_2\sqrt{(\mu+ F (\cos k_1 + \cos k_2 + \cos (k_1+k_2)))^2 - A^2 (\sin k_1 + \sin k_2 - \sin (k_1+k_2))^2} - 1.28 \mu + 3A^2 - 3 F^2 $$ where $BZ$ denotes the regular hexagon

H =  Polygon[ 4 \[Pi]/ 
 3 {{1, 0}, {1/2, Sqrt[3]/2}, {-1/2, Sqrt[3]/2}, {-1, 
  0}, {-1/2, -Sqrt[3]/2}, {1/2, -Sqrt[3]/2}}];

The expression under integral becomes purely imaginary for some choices of $A,F,\mu$. My aim is to find the local maximum of this function (where it has real values). I know, that it should be at the point $(A,F,\mu) = (0.181,-0.0528,0.4025)$. But using

 FindMaximum 

gives an error (becouse the procedure gets gomplex values of the function). Is it possible to tackle this problem?

My code is the following:

Ns = 100;  J = 1 ; kappa = 0.28;
H =  Polygon[ 
   4 \[Pi]/ 
     3 {{1, 0}, {1/2, Sqrt[3]/2}, {-1/2, Sqrt[3]/2}, {-1, 
      0}, {-1/2, -Sqrt[3]/2}, {1/2, -Sqrt[3]/2}}];
pts = RandomPoint[H, Ns^2];
ptsKspace = {#[[1]], -1/2 #[[1]] + (Sqrt[3]/2) #[[2]]} & /@ pts;

f[A_, F_, mu_, k1_, k2_] = 
  Sqrt[(mu + F (Cos[k1] + Cos[k2] + Cos[k1 + k2]))^2 - 
    A^2 (Sin[k1] + Sin[k2] - Sin[k1 + k2])^2];

En3 [A_, F_, mu_] := Module[{re, im},
   re = Sqrt[3]/(8 \[Pi]^2) Integrate[
       Re[f[A, F, mu, k1, -1/2 k1 + Sqrt[3]/2 k2] ], {k1, 
         k2} \[Element] H] - mu (1 + kappa) + 3  Abs[A]^2  - 
     3  Abs[F]^2;
   im = Sqrt[3]/(8 \[Pi]^2) Integrate[
      Im[f[A, F, mu, k1, -1/2 k1 + Sqrt[3]/2 k2] ], {k1, 
        k2} \[Element] H] ;
   Return[re + I im]; 
];

FindMaximum[En3[x, y, z], {x, 0.181}, {y, -0.05}, {z, 0.45}]

And I ge an error. The firs lines with random points from hexagon are for possible Monte carlo calculation of integral.

Answer 1

I redefined so that the numeric functions only evaluate for explicitly numeric input, and the objective returns a negative value well below the max for cases where there is a nontrivial imaginary part.

ns = 100; J = 1; kappa = 0.28;
hh = Polygon[
   4 \[Pi]/3 {{1, 0}, {1/2, Sqrt[3]/2}, {-1/2, Sqrt[3]/2}, {-1, 
      0}, {-1/2, -Sqrt[3]/2}, {1/2, -Sqrt[3]/2}}];
pts = RandomPoint[hh, ns^2];
ptsKspace = {#[[1]], -1/2 #[[1]] + (Sqrt[3]/2) #[[2]]} & /@ pts;

f[a_?NumberQ, f_?NumberQ, mu_?NumberQ, k1_?NumberQ, 
  k2_?NumberQ] := 
 Sqrt[(mu + f (Cos[k1] + Cos[k2] + Cos[k1 + k2]))^2 - 
   a^2 (Sin[k1] + Sin[k2] - Sin[k1 + k2])^2]

en3[aA_?NumberQ, fF_?NumberQ, mu_?NumberQ] := 
 Module[{re, im}, 
  re = Sqrt[3]/(8 \[Pi]^2) NIntegrate[
      Re[f[aA, fF, mu, k1, -1/2 k1 + Sqrt[3]/2 k2]], {k1, 
        k2} \[Element] hh] - mu (1 + kappa) + 3 Abs[aA]^2 - 
    3 Abs[fF]^2;
  im = NIntegrate[
    Im[f[aA, fF, mu, k1, -1/2 k1 + Sqrt[3]/2 k2]], {k1, 
      k2} \[Element] hh];
  If[Abs[im] > .0001, -100, re]
  ]

This is slow and gives warnings but it also delivers a result after several minutes of deep cogitation.

In[21]:= FindMaximum[en3[x, y, z], {x, 0.181}, {y, -0.05}, {z, 0.45}]

(* During evaluation of In[21]:= NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

During evaluation of In[21]:= NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

During evaluation of In[21]:= NIntegrate::izero: Integral and error estimates are 0 on all integration subregions. Try increasing the value of the MinRecursion option. If value of integral may be 0, specify a finite value for the AccuracyGoal option.

During evaluation of In[21]:= General::stop: Further output of NIntegrate::izero will be suppressed during this calculation.

During evaluation of In[21]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

During evaluation of In[21]:= NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 10.077065638905038` and 0.00025052670223458276` for the integral and error estimates.

During evaluation of In[21]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

During evaluation of In[21]:= NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 7.2069039646135185` and 0.0002487749558483572` for the integral and error estimates.

During evaluation of In[21]:= NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small.

During evaluation of In[21]:= General::stop: Further output of NIntegrate::slwcon will be suppressed during this calculation.

During evaluation of In[21]:= NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 2000 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 10.077065638905038` and 0.00025052670223458276` for the integral and error estimates.

During evaluation of In[21]:= General::stop: Further output of NIntegrate::eincr will be suppressed during this calculation.

During evaluation of In[21]:= FindMaximum::lstol: The line search decreased the step size to within the tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient increase in the function. You may need more than MachinePrecision digits of working precision to meet these tolerances.

Out[21]= {-0.0860541069032, {x -> 0.247531713662, 
  y -> -0.103777453975, z -> 0.487104944188}} *)