In　this　article,　a　generalized　value　iteration　algorithm　is　developed　to　address　the　discounted　near-optimal　control　problem　for　discrete-time　systems　with　control　constraints.　The　initial　cost　function　is　permitted　to　be　an　arbitrary　positive　semi-definite　function　without　being　zero.　First,　a　nonquadratic　performance　functional　is　utilized　to　overcome　the　challenge　caused　by　saturating　actuators.　Then,　the　monotonicity　and　convergence　of　the　iterative　cost　function　sequence　with　the　discount　factor　are　analyzed.　For　facilitating　the　implementation　of　the　iterative　algorithm,　two　neural　networks　with　Levenberg-Marquardt　training　algorithm　are　constructed　to　approximate　the　cost　function　and　the　control　law.　Furthermore,　the　initial　control　law　is　obtained　by　employing　the　fixed　point　iteration　approach.　Finally,　two　simulation　examples　are　provided　to　validate　the　feasibility　of　the　present　strategy.　It　is　emphasized　that　the　established　control　laws　are　successfully　constrained　for　randomly　given　initial　state　vectors.