  In this paper, we propose a new estimation methodology based on a projected
non-linear conjugate gradient (PNCG) algorithm with an efficient line search
technique. We develop a general PNCG algorithm for a survival model
incorporating a proportion cure under a competing risks setup, where the
initial number of competing risks are exposed to elimination after an initial
treatment (known as destruction). In the literature, expectation maximization
(EM) algorithm has been widely used for such a model to estimate the model
parameters. Through an extensive Monte Carlo simulation study, we compare the
performance of our proposed PNCG with that of the EM algorithm and show the
advantages of our proposed method. Through simulation, we also show the
advantages of our proposed methodology over other optimization algorithms
(including other conjugate gradient type methods) readily available as R
software packages. To show these we assume the initial number of competing
risks to follow a negative binomial distribution although our general algorithm
allows one to work with any competing risks distribution. Finally, we apply our
proposed algorithm to analyze a well-known melanoma data.
