Mardan, Pakistan ? Pakistan’s army vowed Friday to eliminate militants from a northwestern valley but warned that its under-equipped troops face thousands of Taliban extremists who have seized towns, planted bombs made from pressure cookers, and dragooned children to be suicide bombers.

As air force jets roared overhead and gunbattles raged, terrified civilians from the Swat Valley and neighboring districts accelerated their exodus, with U.N. and Pakistani officials predicting 1 million refugees will soon burden the turbulent Afghan border region.

The army formally announced Friday that an offensive was under way. It has drawn praise from U.S. officials alarmed at the Taliban’s recent advance to within 60 miles of the capital, Islamabad.

Washington describes the militants as an existential threat to nuclear-armed Pakistan itself, as well as to U.S. chances of destroying al-Qaida or of winning the war against their insurgent allies in neighboring Afghanistan.

“The army is now engaged in a full-scale operation to eliminate the militants, miscreants and anti-state elements from Swat,” said Maj. Gen. Athar Abbas, chief army spokesman. “They are on the run and trying to block the exodus of civilians from the area.”

There are doubts about the ability and resolve of the army and the government to sustain the kind of grinding counterinsurgency warfare needed to defeat extremists whose rhetoric resonates widely in a Muslim nation deeply skeptical of U.S. goals in the region.

Abbas sought to counter portrayals of the military as ill-trained, saying that they had learned a lot in eight years of fighting along the border. But he said they need helicopters, surveillance drones and night-vision equipment, which the U.S. is scrambling to provide.