why isn't xtable the first argument of ff and cff in FindMinimum:

In[1]:= xmax = 2.;

In[2]:= n = 10; nk = 15;

In[4]:= ktable = Table[Pi (i - 1)/nk; 2 Pi Cos[phi] // N, {i, 1, nk}];

In[9]:= xtable := RandomReal[{-xmax, xmax}, {n}];

In[6]:= ff[xtable_, ktable_, nk_, n_] := 
  Module[{is, ik}, 
   Min[Abs[Table[
      Sum[Exp[I ktable[[ik]] xtable[[is]]], {is, 1, n}], {ik, 1, 
       nk}]]]];

In[7]:= cff = 
  Compile[{{xtable, _Real, 1}, {ktable, _Real, 
     1}, {nk, _Integer}, {n, _Integer}}, 
   Module[{is, ik}, 
    Min[Abs[Table[
       Sum[Exp[I ktable[[ik]] xtable[[is]]], {is, 1, n}], {ik, 1, 
        nk}]]]], 
   "RuntimeOptions" -> {"EvaluateSymbolically" -> False}, 
   CompilationTarget -> "C"];

In[10]:= FindMinimum[{ff[xtable, ktable, nk, n]}, varsp, 
 Compiled -> True]

During evaluation of In[10]:= FindMinimum::fmgz: Encountered a gradient that is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point.

Out[10]= {1.74504, {varsp -> 1.}}

In[11]:= FindMinimum[{cff[xtable, ktable, nk, n]}, varsp, 
 Compiled -> True]

During evaluation of In[11]:= FindMinimum::fmgz: Encountered a gradient that is effectively zero. The result returned may not be a minimum; it may be a maximum or a saddle point.

Out[11]= {1.15655, {varsp -> 1.}}

yes, that was the intention, as it allowed me to test cff in a loop for many different arguments. The issue remains that FindMinimum and NMinimize appear to evaluate the objective function symbolically as implied by the error message

During evaluation of In[1157]:= CompiledFunction::cfta: Argument {x[1],x[2],x[3],x[4],x[5],x[6],x[7],x[8],x[9],x[10]} at position 1 should be a rank 1 tensor of machine-size real numbers.

the error messages are saying that your compiled function is not working correctly.