Primal problem : minimize x3^2 subject to x1^2+x2^2-x3<=0 and x1+x2+2x3=0

Show[RegionPlot3D[x^2+y^2-z≤0,{x,-2,2},{y,-2,2},{z,-1,2},Mesh->None,LabelStyle->(FontSize->14),PlotPoints->20,PlotStyle->Directive[GrayLevel[4/10,4/10]],BoundaryStyle->None],ContourPlot3D[-x-y-2z+2==0,{x,-2,2},{y,-2,2},{z,-1,2},Mesh->None]]

Lagrangian function

L=x3^2+lambda(x1^2+x2^2-x3)+nu(x1+x2+2x3-2)

Minimize Lagrangian function with respect to x

sol=Solve[{D[L,x1]==0,D[L,x2]==0,D[L,x3]==0},{x1,x2,x3}]

Simplify[L/.sol]

Dual function

g=-((lambda^2)/4)+lambda nu-((nu^2)/(2 lambda))-nu (2+nu)

Maximize dual function with respect to lambda and nu

Solve[{D[g,lambda]==0,D[g,nu]==0},{lambda,nu}]

dstar=g/.{lambda->(1/3),nu->-(1/3)}