Constrained optimization exact optimization by cylindrical algebraic decomposition

Minimize[{x^2 + y^2, x^3 + y^2 - x == (1/4)}, {x, y}]

Minimize[{x^2 + y^2, -x + x^3 + y^2 == 1/4}, {x, y}]

min = Minimize[{x^2 + y^2, x^3 + y^2 - x == a}, {x, y}]

{0, {x -> 0, y -> 0}}

plot[a_] := ContourPlot[x^3 + y^2 - x == a, {x, -3, 3}, {y, -3, 3}, PlotRange -> {{-3, 3}, {-3, 3}}];
minval[a_] := Evaluate[min[[1]]]
minpt[a_] := Evaluate[min[[2]]]
mmark = Graphics[Text[Style["m=", 10], {1.25, 2.5}]];
mvalue[a_] := Graphics[Text[Style[PaddedForm[minval[a], {5, 3}], 10], {2, 2.5}]];
amark = Graphics[Text[Style["a=", 10], {1.25, 2.8}]];
avalue[a_] := Graphics[Text[Style[PaddedForm[a, {5, 3}], 10], {2, 2.8}]];
mpoint[a_] := Graphics[{PointSize[0.03], Red, Point[Re[{x, y} /. minpt[a]]]}];
origin = Graphics[{PointSize[0.03], Green, Point[{0, 0}]}];
Manipulate[Show[{plot[a], amark, origin, avalue[a], mmark, mvalue[a], mpoint[a]}], {{a, 4.5}, -5, 5}, SaveDefinitions -> True]

Manipulate[Show[{plot[a], amark, origin, avalue[a], mmark, mvalue[a], mpoint[a]}], {{a, 4.5}, -5, 5}, SaveDefinitions -> True]