Waheed, who took office after the first democratically elected President, Mohamed Nasheed, resigned under duress in February 2012, had been under pressure from India and western powers to step down after his tenure ended on November 10.
Some reports suggested Waheed had left the Maldives on Thursday evening for a private visit to a Southeast Asian country. He had initially decided to remain in power to oversee the selection of a replacement.
A run-off between Nasheed, who led the first round of the presidential polls on November 9, and his rival Abdulla Yameen is scheduled for Saturday.
The current polls mark the Maldives' third attempt to elect a new President since September.
The first election on September 7 was annulled by the Supreme Court, citing the rigging of voters' lists, while the Elections Commission's attempt to hold polls on October 19 was thwarted by police after a Supreme Court ruling.
In a televised address to the nation last night, Waheed claimed his government had refused to bow down to foreign pressure and upheld the legal framework and orders of the constitutional institutions of the country.
Waheed said that after working under a legal framework that constrained the powers of the president for decades, it was only patience and compromises that paved the way for an election where three candidates were competing.
While the people might feel the government was weak, the country would not have arrived at the current stage if the government had not worked with patience and compromise, he said.
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