  We study the global convergence of policy optimization for finding the Nash
equilibria (NE) in zero-sum linear quadratic (LQ) games. To this end, we first
investigate the landscape of LQ games, viewing it as a nonconvex-nonconcave
saddle-point problem in the policy space. Specifically, we show that despite
its nonconvexity and nonconcavity, zero-sum LQ games have the property that the
stationary point of the objective function with respect to the linear feedback
control policies constitutes the NE of the game. Building upon this, we develop
three projected nested-gradient methods that are guaranteed to converge to the
NE of the game. Moreover, we show that all of these algorithms enjoy both
globally sublinear and locally linear convergence rates. Simulation results are
also provided to illustrate the satisfactory convergence properties of the
algorithms. To the best of our knowledge, this work appears to be the first one
to investigate the optimization landscape of LQ games, and provably show the
convergence of policy optimization methods to the Nash equilibria. Our work
serves as an initial step toward understanding the theoretical aspects of
policy-based reinforcement learning algorithms for zero-sum Markov games in
general.
