  This study investigates a decentralized linear-quadratic optimal control
problem, and several approximate separable constrained optimization problems
are formulated for the first time based on the selection of sparsity promoting
functions. First, for the optimization problem with weighted $\ell_1$ sparsity
promoting function, a two-timescale algorithm is adopted that is based on the
BSUM (Block Successive Upper-bound Minimization) framework and a differential
equation solver. Second, a piecewise quadratic sparsity promoting function is
introduced, and the induced optimization problem demonstrates an accelerated
convergence rate by performing the same two-timescale algorithm. Finally, the
optimization problem with $\ell_0$ sparsity promoting function is considered
that is nonconvex and discontinuous, and can be approximated by successive
coordinatewise convex optimization problems.
