  One common approach to detecting change-points is minimizing a cost function
over possible numbers and locations of change-points. The framework includes
several well-established procedures, such as the penalized likelihood and
minimum description length. Such an approach requires finding the cost value
repeatedly over different segments of the data set, which can be time-consuming
when (i) the data sequence is long and (ii) obtaining the cost value involves
solving a non-trivial optimization problem. This paper introduces a new
sequential method (SE) that can be coupled with gradient descent (SeGD) and
quasi-Newton's method (SeN) to find the cost value effectively. The core idea
is to update the cost value using the information from previous steps without
re-optimizing the objective function. The new method is applied to change-point
detection in generalized linear models and penalized regression. Numerical
studies show that the new approach can be orders of magnitude faster than the
Pruned Exact Linear Time (PELT) method without sacrificing estimation accuracy.
