Optimization of MA parameters

The sum of squared residuals (1.11) is a nonlinear non-convex function of MA parameters $b$. Thus we have to consider the global optimization algorithms. Denote

$$f(x)=\log S(a(x),x),$$ (16)

where $x=b$ and $S(a,b)$ is from (1.11) at optimal parameter $a=a(b)$. Denote

$$b^0 =x^0 =arg \min_x f(x).$$ (17)