How can I find solutions for this equation?

I have a function cxyz[s1, e1, s2, e2], and I want to find at least one set of {s1, e1, s2, e2} for which the function's output is {2, 2, 2}. Both NSolve and Solve give me that there are no solutions, but by manually adjusting the variables I can get close to {2, 2, 2}.

The definition of the function is:

ChromaticityPlot["RGB"];
{cx, cy, cz} = 
  Interpolation[#, Method -> "Spline", InterpolationOrder -> 1] & /@ 
   Map[Function[u, 
     Map[{#1[[1]], 

        Total[Select[u, Function[t, t[[1]] <= #1[[1]]]]][[2]]} &, u]],
     Thread[{Image`ColorOperationsDump`$wavelengths, #}] & /@ 
     Transpose[Image`ColorOperationsDump`tris]];
cxyz[s1_, e1_, s2_, e2_] := {cx[e1], cy[e1], cz[e1]} - {cx[s1], cy[s1], 
   cz[s1]} + {cx[e2], cy[e2], cz[e2]} - {cx[s2], cy[s2], cz[s2]}

Answer

You have an underdetermined system (more unknowns than equations). One possibility would be to use NMinimize[] on the sum of squares of differences (actually, NArgMin[] suffices, but you'll see why I didn't use it).

Most of the methods available to NMinimize[] take a "RandomSeed" option, which can be tweaked to produce different results if the objective function has multiple minima. Let's try this on the OP's system, using differential evolution as the optimization method:

Table[NMinimize[{SquaredEuclideanDistance[cxyz[s1, e1, s2, e2], {2, 2, 2}],
                 385 <= {s1, e1, s2, e2} <= 745}, {s1, e1, s2, e2}, 

                Method -> {"DifferentialEvolution", "RandomSeed" -> k, 
                           "ScalingFactor" -> 0.9}],
      {k, 5}]
{{1.53211*10^-12, {s1 -> 479.195, e1 -> 743.045, s2 -> 635.343, e2 -> 503.826}},
 {2.2567*10^-10, {s1 -> 385.967, e1 -> 572.506, s2 -> 562.054, e2 -> 422.602}},
 {2.18925*10^-10, {s1 -> 386.46, e1 -> 422.611, s2 -> 562.054, e2 -> 572.507}},
 {2.24271*10^-10, {s1 -> 562.053, e1 -> 422.587, s2 -> 385.185, e2 -> 572.506}},
 {6.67845*10^-11, {s1 -> 563.374, e1 -> 448.807, s2 -> 443.187, e2 -> 573.769}}}

Note that five rather different minima were obtained, and all resulting in a relatively tiny value of the objective function. Which of these is the one you want/need is now entirely your decision. Using a different "RandomSeed" setting or even a different optimization method will in all likelihood give results different from what you see here.

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