  Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the
convergence rate of many contemporary first-order methods. In particular, a KL
exponent of $\frac12$ for a suitable potential function is related to local
linear convergence. Nevertheless, KL exponent is in general extremely hard to
estimate. In this paper, we show under mild assumptions that KL exponent is
preserved via inf-projection. Inf-projection is a fundamental operation that is
ubiquitous when reformulating optimization problems via the lift-and-project
approach. By studying its operation on KL exponent, we show that the KL
exponent is $\frac12$ for several important convex optimization models,
including some semidefinite-programming-representable functions and some
functions that involve $C^2$-cone reducible structures, under conditions such
as strict complementarity. Our results are applicable to concrete optimization
models such as group fused Lasso and overlapping group Lasso. In addition, for
nonconvex models, we show that the KL exponent of many difference-of-convex
functions can be derived from that of their natural majorant functions, and the
KL exponent of the Bregman envelope of a function is the same as that of the
function itself. Finally, we estimate the KL exponent of the sum of the least
squares function and the indicator function of the set of matrices of rank at
most $k$.
