F[x_] = Integrate[Exp[z]/{1 + Exp[z]}^2, {z, -Infinity, x}];

L[a1_,a2_,b1_,b2_,b3_]=Sum[y1[[i]]*Log[F[a1-(
b1*x1[[i]]+b2*x2[[i]]+b3*
x3[[i]])]]+
y2[[i]]*Log[F[a2-(b1*x1[[i]]+b2*x2[[i]]+b3*x3[[i]])]-
F[a1-(b1*x1[[i]]+b2*x2[[i]]+
b3*x3[[i]])]]+y3[[i]]*Log[1-
F[a2-(b1*x1[[i]]+b2*x2[[i]]+b3*x3[[i]])]],{i,1,100}];

K1[a1_,a2_,b1_,b2_,b3_]=D[L[a1,a2,b1,b2,b3],a1];

K2[a1_,a2_,b1_,b2_,b3_]=D[L[a1,a2,b1,b2,b3],a2];

K3[a1_,a2_,b1_,b2_,b3_]=D[L[a1,a2,b1,b2,b3],b1];

K4[a1_,a2_,b1_,b2_,b3_]=D[L[a1,a2,b1,b2,b3],b2];

K5[a1_,a2_,b1_,b2_,b3_]=D[L[a1,a2,b1,b2,b3],b3];

FindRoot[
{K1[a1,a2,b1,b2,b3]\[Equal]0,
K2[a1,a2,b1,b2,b3]\[Equal]0,
K3[a1,a2,b1,b2,b3]\[Equal]0,
K4[a1,a2,b1,b2,b3]\[Equal]0,
K5[a1,a2,b1,b2,b3]\[Equal]0},
{a1,0},{a2,1},{b1,1},{b2,3},{b3,1}
]