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We propose a new dual projected pseudo quasi-Newton method for discrete-time linear quadratic optimal control problems of a large-scale multi-destination network. Our method has successfully overcome the difficulties caused by large dimensions and inequality constraints by having a constant Hessian matrix and an efficient successive projection method, respectively. Application of our method to the congestion control of multi-destination networks is presented. Above all, it takes less than one second to obtain the solution of a large-scale discrete-time linear quadratic optimal control problem with the problem size of 2800 variables, 700 equality constraints and 2400 inequality constraints.