BAGHDAD (Reuters) - U.S. Vice President Dick Cheney on Monday declared the 2003 U.S.-led invasion of Iraq a “successful endeavor” during a visit to Baghdad, on the same day a woman suicide bomber killed 40 people.

“If you look back on those five years it has been a difficult, challenging but nonetheless successful endeavor ... and it has been well worth the effort,” Cheney, an architect of the invasion, said after meeting Iraqi leaders.

The Iraq war is a major issue in the U.S. presidential campaign. Entering its sixth year this week, it has cost the United States $500 billion. U.S. Democratic hopeful Hillary Clinton said on Monday the final bill could be $1 trillion.

Tens of thousands of Iraqis and nearly 4,000 U.S. soldiers have been killed. The military said two more soldiers died on Monday when a roadside bomb hit their vehicle near Baghdad.

In a northern district of the capital, six children were killed when a mortar round hit their home, Iraq’s military said.

Shortly after Cheney spoke, a woman wearing a suicide vest blew herself up in a cafe in the southern holy Shi’ite city of Kerbala, killing 40 people and wounding 71, police and health officials said. Bombs in Baghdad killed four and wounded 13.

“I was talking with a friend and eating bread a few meters away from the cafe. Suddenly I heard a huge explosion and I was thrown to the floor. I saw smoke and bodies,” witness Mohammed Kadhem, 39, recounted after the Kerbala blast.

Cheney arrived as Republican presidential candidate John McCain was meeting Iraqi leaders as part of a Senate Armed Services Committee fact-finding mission.

“I was last in Baghdad 10 months ago and I sense, as a result of the progress that has been made since then, phenomenal changes in terms of the overall situation,” Cheney said after meeting Iraqi Prime Minister Nuri al-Maliki.

“This visit is important because it comes at a time when there’s a great deal of progress taking place in Iraq,” Maliki said through a translator.

A poll of 2,000 people from across Iraq, commissioned by international broadcasters including the BBC and U.S. network ABC, found rising optimism among them.

Cheney said there had been a “remarkable turnaround” in security after 30,000 extra U.S. troops were sent to Iraq last year to help quell sectarian violence that threatened civil war.

Despite the improved security, however, some 4 million Iraqis are still displaced, and the International Committee of the Red Cross said in a report on Monday that millions were still deprived of clean water and medical care.

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“SURGE WORKING”

Like McCain, Cheney is in Iraq as part of a wider visit to the Middle East. He was due to spend the night at a military base and will also visit Saudi Arabia, Jerusalem, the Palestinian territories, Turkey and Oman on a nine-day tour.

Both men have been staunch supporters of the U.S. troop build-up or “surge”. The U.S. military says violence in Iraq has dropped by 60 percent since last June, although it acknowledges an upsurge in attacks since January.

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“The surge is working,” McCain told CNN in an interview in Baghdad, countering demands by Democratic presidential candidates Clinton and Barack Obama for U.S. troops to be withdrawn from Iraq as soon as possible.

McCain and Cheney did not cross paths during the day as the vice president held a series of meetings with Iraqi leaders. He traveled outside the U.S.-protected Green Zone in a heavily armored motorcade to visit Iraqi President Jalal Talabani and powerful Shi’ite political leader Abdul-Aziz al-Hakim.

“There is still a lot of difficult work that must be done, but as we move forward the Iraqi people should know that they will have the unwavering support of President Bush and the United States in consolidating their democracy,” Cheney said after meeting Abdul Aziz al-Hakim, head of Iraq’s largest Shi’ite political bloc, the Supreme Islamic Iraqi Council.

Among the political issues Cheney discussed with Iraq’s leaders was a stalled hydrocarbon law, stressing that it was important to Iraq’s national development, U.S. ambassador Ryan Crocker said later.

The law would share revenues from Iraq’s vast oil reserves, the world’s third largest, but remains blocked because of reluctance to compromise among Iraq’s political blocs.

(Additional reporting by Sami al-Jumaili in Kerbala and

Mohammed Abbas in Baghdad; writing by Paul Tait and Ross

Colvin; editing by Andrew Roche)