# Something about smoothing parameters with ipopt

This is my model, which is not smooth because it contains symbolic functions, absolute values, and if-else statements. I rewrote him into the following ampl code, where beta is the smoothing argument of the symbolic function, Z[2,j,k] is the speed v, and delta is the smoothing parameter of the absolute value.This is my model, which is not smooth because it contains symbolic functions, absolute values, and if-else statements. I rewrote him into the following ampl code, where beta is the smoothing argument of the symbolic function, Z[2,j,k] is the speed v, and delta is the smoothing parameter of the absolute value. Why can’t the optimal solution be obtained by changing the model when the beta becomes 1e-4 or less?

var z1{j in Sfe, k in Scp} = sqrt(Z[2,j,k]^2+seta)-delta;
var z3{j in Sfe, k in Scp} = SaP[j,k]-ka Z[1,j,k]-Finit;
var s1{j in Sfe, k in Scp} = 0.5*(z1[j,k]/sqrt(z1[j,k]^2+4beta^2)+1);
var f1{j in Sfe, k in Scp} = -((Sa
P[j,k]-kaZ[1,j,k]-Finit-delta)/((Sa P[j,k]-kaZ[1,j,k]-Finit-delta)^2+4 beta^2)^(1/2))0.5 sqrt((z3[j,k]+Fs-sqrt((z3[j,k]-Fs)^2+seta))^2+seta);
var f2{j in Sfe, k in Scp} = -(Fc+(Fs-Fc)exp(-(Z[2,j,k]/Vs)^2)) (Z[2,j,k]-delta)/((Z[2,j,k]-delta)^2+4beta^2)^0.5-Fv Z[2,j,k];
var Ff{j in Sfe, k in Scp} = f1[j,k]*(1-s1[j,k])+f2[j,k]*s1[j,k];