Contour Pics and Constrained Optimization

f[x_,y_]:= (((x-y))^2)+ 4((( y+ 2 x))^2) + 2 x
ContourPlot[f[x,y],{x,-0.6,0.6},{y,-0.6,0.6},
AxesLabel->{"x","y"},
Contours->30,
Epilog->{Red,Circle[{0,0},0.5]}
]

f[x_,y_]:= (((x-y))^2)- 4((( y+ 2 x))^2) + 3 x
ContourPlot[f[x,y],{x,-0.6,0.6},{y,-0.6,0.6},
AxesLabel->{"x","y"},
Contours->30,
Epilog->{Red,Circle[{0,0},0.5]}
]

Clear[x,y,λ]
f[x_,y_]:= (((x-5y))^2)- 0.01((( y+ 2 x))^2) +12 x
{x0,y0} = {0.2,0.1};
Δ=0.7;
γ=2.6;
Hess=D[f[x,y],{{x,y},2}];
Gradf[x_,y_]=D[f[x,y],{{x,y}}];
g=-Gradf[x0,y0];
Temp=Simplify[LinearSolve[ Hess-2 λ(({{1, 0}, {0, 1}})), g]]
αSol[λ_]=Temp.Temp;
ConstrainedPointsData={{x0,y0}+Temp,λ}/.Solve[αSol[λ]==(Δ^2),λ];
ConstrainedPointsData =Select[ConstrainedPointsData,Im[#[[2]]]==0.0&];
MinPoints=Select[ConstrainedPointsData,#[[2]]<=0 &];
NewtonPoint={x0,y0}+LinearSolve[Hess, g];

ContourPlot[ f[x,y],
{x, x0-γ Δ, x0+γ Δ},
{y,y0-γ Δ,y0+γ Δ} ,
Contours->20,
Epilog->{Red, 
Thickness[0.01],Circle[{x0,y0},Δ],
PointSize[0.02],Point[{x0,y0}],
Blue, Point[NewtonPoint],
Green, Map[Point, ConstrainedPointsData[[All,1]]],
Cyan, Map[Point, MinPoints[[All,1]]],
Black, Table[
{{x,y},λ}=ConstrainedPointsData[[i]];
Text[{λ,f[x,y]},{x,y},{-1,-1},
Background->White],{i,Length[ConstrainedPointsData]} ]
}
]

More dimensions get more solutions.

Clear[x,y,z,f,λ]
f[x_,y_,z_]:= (((x-y))^2)- 4((( y+ 2 x))^2) + 3 x + (((z-x))^2)
{x0,y0,z0} = {0.2,0.1,0.2};
Δ=0.4;
γ=1.6;
Hess=D[f[x,y,z],{{x,y,z},2}];
Gradf[x_,y_,z_]=D[f[x,y,z],{{x,y,z}}];
g=-Gradf[x0,y0,z0];
Temp=LinearSolve[ Hess-2 λ(({{1, 0, 0}, {0, 1, 0}, {0, 0, 1}})), g];
αSol[λ_]=Simplify[Temp.Temp]
ConstrainedPointsData={{x0,y0,z0}+Temp,λ}/.Solve[αSol[λ]==(Δ^2),λ]
NewtonPoint={x0,y0,z0}+LinearSolve[Hess,g]

MinData= Select[ConstrainedPointsData,(z=#[[2]]; And[Im[z]==0,z<0])&]

TableForm[MinData]
NewtonPoint={x0,y0,z0}+LinearSolve[Hess, g]