  We investigate the Born-Infeld-anti-de Sitter black hole (BIAdS) solutions in
the four dimensions, which is a nonlinear generalization of the
Reissner-Norstr\"om-AdS black hole (RNAdS). We analyze all thermodynamic
quantities of the BIAdS in the canonical ensembles, which are characterized by
the charge $Q$, the mass $M$, the nonlinear parameter $b$, comparing with those
of the RNAdS and Schwarzschild-AdS black hole. We find the forbidden region of
$0\le bQ <0.5$ for the presence of a charged BIAdS. We also discuss the
Hawking-Page phase transitions in the BIAdS black holes. Here we obtain a new
Hawking-Page phase transition for the $bQ=0.5$ critical BIAdS.
