estimating spectral optimization
If you know the number of sinusoidal components, parametric estimators such as MUSIC may be suitable. ESPRIT might be another.  Both seem similar to you proposed optimization/minimization search.  

But parametric estimators are reported to produce bogus decompositions if given the wrong number of components.  Possible reason: For a decomposition into an arbitrary number of arbitrary frequency sinusoids, there can be a near infinite number of solutions that might be arbitrarily close (e.g. you can get a bunch of cosines to converge to one sine wave of yet another frequency.)