M6 := 1+(-ln(2)-2*ln(a))*a+(-1+(1/2)*(-ln(2)-2*ln(a))^2)*a^2;
with(gfun):
seriestoratpoly(series(M6, a, 4), ['egf']);

@Markiyan Hirnyk 

First try, i change to 

result1 := Optimization:-Minimize([ans>=0, ans<=0],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

Error, (in Optimization:-Minimize) objective function must be an algebraic expression or procedure

Second try, i change to use ans for >=0, ans2 <=0

ans:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2>=0,i=1..N)
end proc;
ans2:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2<=0,i=1..N)
end proc;
ans(.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003);
result1 := Optimization:-Minimize([ans, ans2],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

Error, (in Optimization:-Minimize) objective function must be an algebraic expression or procedure

x11 := [0.208408965651696e-3, -0.157194487523421e-2, -0.294739401402979e-2, 0.788206708183853e-2, 0.499394753201753e-2, 0.191468321959759e-3, 0.504980449104750e-2, 0.222150494088535e-2, 0.132091821964287e-2, 0.161118434883258e-2, -0.281236534046873e-2, -0.398055875132037e-2, -0.111753680372819e-1, 0.588868146012489e-2, -0.354191562612469e-2, 0.984082837373291e-3, -0.116041186868374e-1, 0.603027845850267e-3, -0.448778128168742e-2, -0.127561485214862e-1, -0.412027655195339e-2, 0.379387381798949e-2, -0.602550446997765e-2, -0.605986284736216e-2, -0.751396992404410e-2, 0.633613424008655e-2, -0.677581832613623e-2]:
y11 := [ -21321.9719565717, 231.709204951251, 1527.92905167191, -32.8508507060675, 54.9408176234139, -99.4222178124229, -675.771433486265, 42.0838668074923, -12559.3183308951, 5.21412214166344*10^5, 1110.50031772203, 3.67149699000155, -108.543878970269, -8.48861069398811, -521.810552387313, 26.4792411876883, -8.32240296737599, -1085.40982521906, -44.1390030597906, -203.891397612798, -56.3746416571417, -218.205643256096, -178.991498697065, -42.2468018350386, .328546922634921, -1883.18308996621, 111.747881085748]:
z11 := [ 1549.88755331800, -329.861725802688, 8.54200301129155, -283.381775745327, -54.5469129127573, 1875.94875597129, -16.2230517860850, 6084.82381954832, 1146.15489803104, -456.460512914647, 104.533252701641, 16.3998365630734, 11.5710907832054, -175.370276462696, 33.8045539958636, 2029.50029336951, 1387.92643570857, 9.54717543291120, -1999.09590358328, 29.7628085078953, 2.58210333216737*10^6, 57.7969622731082, -6.42551196941394, -8549.23677077892, -49.0081775323244, -72.5156360537114, 183.539911458475]: 
u11 := [7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7];
a1 := Diff(x1(t),t) = k1*x1(t)+ k2*y1(t)+ k3*z1(t)+k4*u1(t);
b1 := Diff(y1(t),t) = k5*x1(t)+ k6*y1(t)+ k7*z1(t)+k8*u1(t);
c1 := Diff(z1(t),t) = k9*x1(t)+ k10*y1(t)+ k11*z1(t)+k12*u1(t);
d1 := Diff(u1(t),t) = 0;
ICS:=x1(1)=x11[1],y1(1)=y11[1],z1(1)=z11[1],u1(1)=u11[1];
sol:=dsolve({a1,b1,c1,d1, a2,b2,c2,d2,ICS}, numeric, method=rkf45, parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12],output=listprocedure);
X,Y,Z,U:=op(subs(sol,[x1(t),y1(t),z1(t),u1(t)]));
tim := [seq(n, n=1..27)];
N:=nops(tim):
ans:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2,i=1..N)
end proc;
ans(.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003);
result1 := Optimization:-Minimize([ans>=0, ans<=0],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

@Markiyan Hirnyk 

First try, i change to 

result1 := Optimization:-Minimize([ans>=0, ans<=0],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

Error, (in Optimization:-Minimize) objective function must be an algebraic expression or procedure

Second try, i change to use ans for >=0, ans2 <=0

ans:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2>=0,i=1..N)
end proc;
ans2:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2<=0,i=1..N)
end proc;
ans(.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003);
result1 := Optimization:-Minimize([ans, ans2],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

Error, (in Optimization:-Minimize) objective function must be an algebraic expression or procedure

x11 := [0.208408965651696e-3, -0.157194487523421e-2, -0.294739401402979e-2, 0.788206708183853e-2, 0.499394753201753e-2, 0.191468321959759e-3, 0.504980449104750e-2, 0.222150494088535e-2, 0.132091821964287e-2, 0.161118434883258e-2, -0.281236534046873e-2, -0.398055875132037e-2, -0.111753680372819e-1, 0.588868146012489e-2, -0.354191562612469e-2, 0.984082837373291e-3, -0.116041186868374e-1, 0.603027845850267e-3, -0.448778128168742e-2, -0.127561485214862e-1, -0.412027655195339e-2, 0.379387381798949e-2, -0.602550446997765e-2, -0.605986284736216e-2, -0.751396992404410e-2, 0.633613424008655e-2, -0.677581832613623e-2]:
y11 := [ -21321.9719565717, 231.709204951251, 1527.92905167191, -32.8508507060675, 54.9408176234139, -99.4222178124229, -675.771433486265, 42.0838668074923, -12559.3183308951, 5.21412214166344*10^5, 1110.50031772203, 3.67149699000155, -108.543878970269, -8.48861069398811, -521.810552387313, 26.4792411876883, -8.32240296737599, -1085.40982521906, -44.1390030597906, -203.891397612798, -56.3746416571417, -218.205643256096, -178.991498697065, -42.2468018350386, .328546922634921, -1883.18308996621, 111.747881085748]:
z11 := [ 1549.88755331800, -329.861725802688, 8.54200301129155, -283.381775745327, -54.5469129127573, 1875.94875597129, -16.2230517860850, 6084.82381954832, 1146.15489803104, -456.460512914647, 104.533252701641, 16.3998365630734, 11.5710907832054, -175.370276462696, 33.8045539958636, 2029.50029336951, 1387.92643570857, 9.54717543291120, -1999.09590358328, 29.7628085078953, 2.58210333216737*10^6, 57.7969622731082, -6.42551196941394, -8549.23677077892, -49.0081775323244, -72.5156360537114, 183.539911458475]:
u11 := [7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7,8,7];
a1 := Diff(x1(t),t) = k1*x1(t)+ k2*y1(t)+ k3*z1(t)+k4*u1(t);
b1 := Diff(y1(t),t) = k5*x1(t)+ k6*y1(t)+ k7*z1(t)+k8*u1(t);
c1 := Diff(z1(t),t) = k9*x1(t)+ k10*y1(t)+ k11*z1(t)+k12*u1(t);
d1 := Diff(u1(t),t) = 0;
ICS:=x1(1)=x11[1],y1(1)=y11[1],z1(1)=z11[1],u1(1)=u11[1];
sol:=dsolve({a1,b1,c1,d1, a2,b2,c2,d2,ICS}, numeric, method=rkf45, parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12],output=listprocedure);
X,Y,Z,U:=op(subs(sol,[x1(t),y1(t),z1(t),u1(t)]));
tim := [seq(n, n=1..27)];
N:=nops(tim):
ans:=proc(k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12) sol(parameters=[k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,k11,k12]);
add((X(tim[i])-x11[i])^2,i=1..N)+add((Y(tim[i])-y11[i])^2,i=1..N)+add((Z(tim[i])-z11[i])^2,i=1..N)+add((U(tim[i])-u11[i])^2,i=1..N)
end proc;
ans(.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003);
result1 := Optimization:-Minimize([ans>=0, ans<=0],initialpoint=[.001,.002,.003,.001,.002,.003,.001,.002,.003,.003,.003,.003], feasibilitytolerance=0.01);

How to set feasibility tolerance for optimization command?

Any example to show some syntax.

How to group action on vector space in maple