Numerical Optimization

The numerical optimization-integration methods considered in this paper use the function

$$h(t) = P(t)-(1-\alpha ).$$

These methods involve finding $t_\alpha$, the point where $h(t_\alpha) = 0$, using a numerical optimization method. But $h(t)$ is often expensive to compute using numerical integration, particularly for large $m$, so a numerical optimization method that requires only a few iterations is needed. This need can be satisfied if we combine a method for getting good starting points for the optimization method, with an optimization method that converges rapidly. An additional complication that arises with the combined numerical optimization-integration method is the presence of numerical integration errors, which must be controlled along with with the numerical optimization errors. These issues are discussed in the next three subsections.