  We show the optimal $C^{1,1}$ regularity of geodesics in nef and big
cohomology class on K\"ahler manifolds away from the non-K\"ahler locus,
assuming sufficiently regular initial data. As a special case, we prove the
$C^{1,1}$ regularity of geodesics of K\"ahler metrics on compact K\"ahler
varieties away from the singular locus. Our main novelty is an improved
boundary estimate for the complex Monge-Amp\`ere equation that does not require
strict positivity of the reference form near the boundary. We also discuss the
case of some special geodesic rays.

Differential Geometry;Analysis of PDEs;Complex Variables