We wish to bound the single sinusoid (i.e. a one dimensional function) with a parabola as in Equation 6.17. We shall satisfy the 3 requirements (contact, tangentiality and tightness) to derive the optimization algorithm.

Next consider the contact point *x*^{*}=*z* where the parabola and the sinusoid
actually touch in Equation 6.18.

Subsequently, we also enforce the tangentiality requirement that the derivatives are equal at contact to avoid crossover (see Equation 6.19.

Inserting the results of Equation 6.19 and
Equation 6.18 into the inequality in
Equation 6.17 we get an expression for the shape parameter,
as in Equation 6.20. The equation is
manipulated to isolate *w* onto one side of the inequality. *Any*
*w* can be selected now, so long as it satisfies this inequality for
the given *z* and for all *x* in the domain of the function. It is
subsequently straightforward to algebraically compute *y* and *k* and
get a valid parabolic bound on the sinusoid for the given contact
point *z*.

Thus, the critical step still remains: how do we compute *w*? A few ad
hoc ways of finding such a *w* directly exist although these do not
give an optimal value of *w*. Recall, now, that we also have the
tightness requirement in Equation 6.7 which seeks
to compute the smallest |*w*| or largest .
Since *z* is given,
we wish to compute the smallest possible *w* that will satisfy this
inequality in Equation 6.20 for all *x* in the domain of
the function. Ultimately, we wish to recover
*w*_{min}(*z*) which is
the minimum value *w* can have for a given *z* before violating the
inequality (crossing our function *f*(*x*) in some locations,
and invalidating the bound).

It is difficult to find the function
*w*_{min}(*z*) analytically for a
sinusoid. Thus, for each *z*, we solve for *w*_{min} numerically by
searching for the maximum of the right hand side expression in
Equation 6.21 over all *x* (an efficient 1D Brent's search
method is used [48]). The function is then stored as a lookup table (LUT)
and is plotted in Figure 6.6 for visualization.

Given a sinusoid
*f*(*x*)=*sin*(*x*) and a contact point *z*, we can
immediately compute *w*_{min} from this LUT and then the corresponding
*y* (which only depends on *z* and *w*) and finally the *k*. A few
examples of the bounding are shown in Figure 6.7.

Evidently, maximizing these parabolas is trivial (the maximum is *y*)
and if *z* is set to *y*, we can repeat the calculation for a new
contact point and converge to the local maximum as shown in
Figure 6.8 (in fact, in this case, we
converge after only one iteration).

At this point, optimizing the sinusoid is by no means challenging and the answer could have been solved for analytically without any of the above. However, since we have solved for the bound as opposed to a direct solution, the properties outlined earlier allow us to reapply the above derivations and solve a broader class of problems.