Regional military commander Maj Filemon Tan said 17 soldiers were wounded when hundreds of army troops surrounded a vast jungle area in Sulu province's mountainous Patikul town and clashed with scattered groups of about 100 militants.
Among the 11 dead militants was Amah Maas, a longtime commander of the group who had severed arms and had been implicated in ransom kidnappings, including of European tourists.
"The order of the president is to search and destroy the Abu Sayyaf so that's what we are doing," Tan said, adding more than 1,200 troops, including special forces commandos, were involved in the assaults in Patikul and other Sulu hinterlands.
Thousands of reinforcement troops have been flown by C130 cargo planes to Sulu and nearby Basilan island to help in the ongoing offensive. Many of the troops were freed up from other combat zones in the country after Duterte declared an indefinite ceasefire with communist rebels, who are engaged in peace talks brokered by Norway with the government.
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The militants are still holding several foreign and local hostages in their jungle bases, including Norwegian Kjartan Sekkingstad, who was kidnapped along with two Canadian men and a Filipino woman from a southern marina in September last year.
The Canadians were beheaded after huge ransom demands were not met and the woman was freed before Duterte assumed the presidency on June 30.
During a news conference yesterday, Duterte suggested the militants may be continuing to hold Sekkingstad despite being paid a 50 million-peso ransom for his freedom.