  Transient Low-Mass X-ray binaries (LMXBs) are discovered largely by X-ray and
gamma-ray all-sky monitors. The X-ray outburst is also accompanied by an
optical brightening, which empirically can precede detection of X-rays. Newly
sensitive optical synoptic surveys may offer a complementary pathway for
discovery, and potential for insight into the initial onset and propagation of
the thermal instability that leads to the ionization of the accretion disk. We
use the Zwicky Transient Facility (ZTF) alert stream to perform a comprehensive
search at optical wavelengths for previously undiscovered outbursting LMXBs.
Our pipeline first crossmatches the positions of the alerts to cataloged X-ray
sources, and then analyzes the 30-day lightcurve of matched alerts by
thresholding on differences with an 8-day exponentially weighted moving
average. In addition to an nineteen month-long live search, we ran our pipeline
over four years of ZTF archival data, recovering 4 known LMXBs. We also
independently detected an outburst of MAXI J1957+032 in the live search and
found the first outburst of Swift J1943.4+0228, an unclassified X-ray
transient, in 10 years. Using Monte Carlo simulations of the Galactic LMXB
population, we estimate that 29% of outbursting LMXBs are detectable by ZTF and
that 4.4% of LMXBs would be present in the crossmatched X-ray catalogs, giving
an estimated Galactic population of $3390^{+3980}_{-1930}$. We estimate that
our current pipeline can detect 1.3% of all outbursting LMXBs, including those
previously unknown, but that Rubin Observatory's Legacy Survey of Space and
Time (LSST) will be able to detect 43% of outbursting LMXBs.
