In survival contexts, substantial literature exists on estimating optimal
treatment regimes, where treatments are assigned based on personal
characteristics for the purpose of maximizing the survival probability. These
methods assume that a set of covariates is sufficient to deconfound the
treatment-outcome relationship. Nevertheless, the assumption can be limiting in
observational studies or randomized trials in which noncompliance occurs. Thus,
we advance a novel approach for estimating the optimal treatment regime when
certain confounders are not observable and a binary instrumental variable is
available. Specifically, via a binary instrumental variable, we propose two
semiparametric estimators for the optimal treatment regime, one of which
possesses the desirable property of double robustness, by maximizing
Kaplan-Meier-like estimators within a pre-defined class of regimes. Because the
Kaplan-Meier-like estimators are jagged, we incorporate kernel smoothing
methods to enhance their performance. Under appropriate regularity conditions,
the asymptotic properties are rigorously established. Furthermore, the finite
sample performance is assessed through simulation studies. We exemplify our
method using data from the National Cancer Institute's (NCI) prostate, lung,
colorectal, and ovarian cancer screening trial