  Given closed convex sets $C_i$, $i=1,\ldots,\ell$, and some nonzero linear
maps $A_i$, $i = 1,\ldots,\ell$, of suitable dimensions, the multi-set split
feasibility problem aims at finding a point in $\bigcap_{i=1}^\ell A_i^{-1}C_i$
based on computing projections onto $C_i$ and multiplications by $A_i$ and
$A_i^T$. In this paper, we consider the associated best approximation problem,
i.e., the problem of computing projections onto $\bigcap_{i=1}^\ell
A_i^{-1}C_i$; we refer to this problem as the best approximation problem in
multi-set split feasibility settings (BA-MSF). We adapt the Dykstra's
projection algorithm, which is classical for solving the BA-MSF in the special
case when all $A_i = I$, to solve the general BA-MSF. Our Dykstra-type
projection algorithm is derived by applying (proximal) coordinate gradient
descent to the Lagrange dual problem, and it only requires computing
projections onto $C_i$ and multiplications by $A_i$ and $A_i^T$ in each
iteration. Under a standard relative interior condition and a genericity
assumption on the point we need to project, we show that the dual objective
satisfies the Kurdyka-Lojasiewicz property with an explicitly computable
exponent on a neighborhood of the (typically unbounded) dual solution set when
each $C_i$ is $C^{1,\alpha}$-cone reducible for some $\alpha\in (0,1]$: this
class of sets covers the class of $C^2$-cone reducible sets, which include all
polyhedrons, second-order cone, and the cone of positive semidefinite matrices
as special cases. Using this, explicit convergence rate (linear or sublinear)
of the sequence generated by the Dykstra-type projection algorithm is derived.
Concrete examples are constructed to illustrate the necessity of some of our
assumptions.
