Maximize[{x+y,100000000000000000000 (x^2)-20000000000 (x^4)+(x^6)+100000000000000000000 (y^2)-40000000000 (x^2) (y^2)+3 (x^4) (y^2)-20000000000 (y^4)+3 (x^2) (y^4)+(y^6)≤1},{x,y}]//N[#,20]&

Maximize[{x+y,100000000000000000001 (x^2)-20000000000 (x^4)+(x^6)+100000000000000000000 (y^2)-40000000000 (x^2) (y^2)+3 (x^4) (y^2)-20000000000 (y^4)+3 (x^2) (y^4)+(y^6)≤1},{x,y}]//N[#,20]&

NMaximize[{x+y,100000000000000000000 (x^2)-20000000000 (x^4)+(x^6)+100000000000000000000 (y^2)-40000000000 (x^2) (y^2)+3 (x^4) (y^2)-20000000000 (y^4)+3 (x^2) (y^4)+(y^6)≤1},{x,y}]

NMaximize[{x+y,100000000000000000001 (x^2)-20000000000 (x^4)+(x^6)+100000000000000000000 (y^2)-40000000000 (x^2) (y^2)+3 (x^4) (y^2)-20000000000 (y^4)+3 (x^2) (y^4)+(y^6)≤1},{x,y}]

Clear[f,x,y,z,t];
f=-Log[x]-2 Log[y] - 3Log[y]-3Log[t];
cons = {200 (x^2)+(y^2)+(z^2)+(t^2)==100,x>0,y>0,z>0,t>0};
vars = {x,y,z,t};
sol=NMinimize[{f,cons},vars]

sol=NMinimize[{f,cons},vars,Method->{NelderMead, PostProcess->KKT}]

sol=NMinimize[{f,cons},vars,Method->{NelderMead, PostProcess->FindMinimum}]

Clear[f,cons,vars,x];
{f,cons,vars}={((20 x[2] x[6])/(((x[1])^2) x[4] ((x[5])^2)))+((15 x[3] x[4])/(x[1] ((x[2])^2) x[5] ((x[7])^(0.5`))))+((10 x[1] ((x[4])^2) ((x[7])^(0.125`)))/(x[2] ((x[6])^3)))+((25 ((x[1])^2) ((x[2])^2) ((x[5])^(0.5`)) x[7])/(x[3] ((x[6])^2))),{0.1`≤x[1]≤10,0.1`≤x[2]≤10,0.1`≤x[3]≤10,0.1`≤x[4]≤10,0.1`≤x[5]≤10,0.1`≤x[6]≤10,0.01`≤x[7]≤10,1-((0.2` x[3] ((x[6])^(2/3)) ((x[7])^(0.25`)))/(x[2] ((x[4])^(0.5`))))-((0.7` ((x[1])^3) x[2] x[6] ((x[7])^(0.5`)))/((x[3])^2))-((0.5` ((x[1])^(0.5`)) x[7])/(x[3] ((x[6])^2)))≥0,1-((3.1` ((x[2])^(0.5`)) ((x[6])^(1/3)))/(x[1] ((x[4])^2) x[5]))-((1.3` x[2] x[6])/(((x[1])^(0.5`)) x[3] x[5]))-((0.8` x[3] ((x[6])^2))/(x[4] x[5]))≥0,1-((x[2] ((x[3])^(0.5`)) x[5])/(x[1]))-((0.1` x[2] x[5])/(((x[3])^(0.5`)) x[6] ((x[7])^(0.5`))))-((2 x[1] x[5] ((x[7])^(1/3)))/(((x[3])^(1.5`)) x[6]))-((0.65` x[3] x[5] x[7])/(((x[2])^2) x[6]))≥0,1-((0.3` ((x[1])^(0.5`)) ((x[2])^2) x[3] ((x[4])^(1/3)) ((x[7])^(0.25`)))/((x[5])^(2/3)))-((0.2` x[2] ((x[5])^(0.5`)) ((x[7])^(1/3)))/(((x[1])^2) x[4]))-((0.5` x[4] ((x[7])^(0.5`)))/((x[3])^2))-((0.4` x[3] x[5] ((x[7])^(0.75`)))/(((x[1])^3) ((x[2])^2)))≥0,((20 x[2] x[6])/(((x[1])^2) x[4] ((x[5])^2)))+((15 x[3] x[4])/(x[1] ((x[2])^2) x[5] ((x[7])^(0.5`))))+((10 x[1] ((x[4])^2) ((x[7])^(0.125`)))/(x[2] ((x[6])^3)))+((25 ((x[1])^2) ((x[2])^2) ((x[5])^(0.5`)) x[7])/(x[3] ((x[6])^2)))≥100,((20 x[2] x[6])/(((x[1])^2) x[4] ((x[5])^2)))+((15 x[3] x[4])/(x[1] ((x[2])^2) x[5] ((x[7])^(0.5`))))+((10 x[1] ((x[4])^2) ((x[7])^(0.125`)))/(x[2] ((x[6])^3)))+((25 ((x[1])^2) ((x[2])^2) ((x[5])^(0.5`)) x[7])/(x[3] ((x[6])^2)))≤3000},{x[1],x[2],x[3],x[4],x[5],x[6],x[7]}};

FindMinimum[{f,cons},vars]//Timing

NMinimize[{f,cons},vars]//Timing