Minimization of a function with non-numeric parameters?

Question

I have the following problem (in Mathematica) I want to calculate the following expressions:

Assuming[{t > 0, t \[Element] Integers}, 
 Minimize[Sqrt[(Sin[t*Pi/6] - x)^2 + (Pi/6 - x)^2], x]]

How can I get a unique result?

The other question: what is if my function has 2 parameters, like t and s which do not get any numerical values however restrict to be from N (Integer). What is the key name, may I say "multivariate minimisation"?

Thanks

Answer 1

I suggest removing the Sqrt. Here the minima are derived in different ways:

f[u_, x_, t_] := u[(Sin[t*Pi/6] - x)^2 + (Pi/6 - x)^2]
p1 = Plot[Evaluate[f[Identity, x, #] & /@ Range[-5, 5]], {x, -4, 4}];
exp = x /. First@Solve[D[f[Identity, x, t], x] == 0, x];
pts = Table[{exp, f[Identity, exp, t]} /. t -> j, {j, -5, 5}];
p2 = ListPlot[List /@ pts, PlotMarkers -> {Automatic, 20}];
min = Minimize[f[Identity, x, t], x];
gm = t /. Last@Minimize[min[[1]], t];
cv = (x /. min[[2]]) /. t -> gm;
pi = Show[p1, p2, 
   Graphics[{Text[Style["x", 20], {cv, f[Identity, cv, gm]}]}], 
   PlotRange -> {{-1, 1}, {0, 5}}, ImageSize -> 300];
pc = Plot[Evaluate[f[Sqrt, x, #] & /@ Range[-5, 5]], {x, -4, 4}, 
   Epilog -> {PointSize[0.03], Point[{cv, f[Sqrt, cv, gm]}]}, 
   ImageSize -> 300];
Row[{pi, pc}]
Row[{"t=", gm, ", x=", cv}]

The family of curves is shown for illustration. The minimum of family of curves is black cross in no sqrt and black point in sqrt plot. The values of t and xare shown below.

Answer 2

Apparently, there is no analytic solution, so just use numerical minimization over your two variables:

NMinimize[{Sqrt[(Sin[t π/6] - x)^2 + (π/6 - x)^2], 
  t > 0, 
  t \[Element] Integers}, 
  {x, t}]

$\{0.0166869,\{x\to 0.511799,t\to 1\}\}$

And try this, to visualize:

Plot[
 Evaluate@
  Table[Sqrt[(Sin[t*Pi/6] - x)^2 + (Pi/6 - x)^2], {t, 7}], {x, 0, 1},
 PlotStyle -> Table[Hue[i]/10, {i, 7}],
 PlotLegends -> Automatic]

If your question is instead: "Given a $t \in \mathbb{Z}^+$, find the $x$ that minimizes the function", then:

Minimize[Sqrt[(Sin[t π/6] - x)^2 + (π/6 - x)^2], x]

An even better way is to find the value of $x$ (as a function of $t$) for the problem without the square root (avoiding multiple roots), then substituting the found $x$ into your full function:

Sqrt[(Sin[t π/6] - x)^2 + (π/6 - x)^2] /.
 Minimize[(Sin[t π/6] - x)^2 + (π/6 - x)^2, x][[2]]

$\sqrt{\left(\frac{1}{12} \left(-6 \sin \left(\frac{\pi t}{6}\right)-\pi \right)+\frac{\pi }{6}\right)^2+\left(\frac{1}{12} \left(-6 \sin \left(\frac{\pi t}{6}\right)-\pi \right)+\sin \left(\frac{\pi t}{6}\right)\right)^2}$