The rebels have fought back, turning parts of the east into a war zone. On Friday they shot down a Ukrainian army plane and killed a member of the interior ministry's special forces in the separatist stronghold of Slaviansk, where residents said shelling continued all day.
On the eve of his inauguration, Poroshenko held a brief meeting with Russian President Vladimir Putin in France, where both were attending World War Two commemorations.
French officials said the two shook hands and agreed that detailed talks on a ceasefire between Kiev government forces and the pro-Russian separatists would begin within a few days.
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Russia denounced the overthrow of Yanukovich and has accused the Ukrainian authorities of worsening the crisis in the east by resorting to military force instead of dialogue. It denies accusations by Kiev and Western governments that it is actively supporting the rebels.
Putin told travelling reporters he welcomed proposals set out by Poroshenko for ending the conflict. However he declined to say what they were and said Ukraine must halt what he called "punitive" military operations against pro-Russian separatists.
The two countries are also at odds over the pricing of Russian gas, with Moscow threatening to cut off supplies as early as next week unless Ukraine settles its debt, the amount of which is disputed.
Poroshenko has promised to bridge the east-west divide that has split the country and thrust it into a battle for its very survival.
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Western governments and Russia, locked in a geopolitical fight over Ukraine's future, will be watching for clues on how he intends to handle the eastern rebellions.
Poroshenko says he wants closer relations with Europe, and won support and encouragement for his policies to stabilize Ukraine when he met U.S. President Barack Obama and European leaders in Poland and France this week.
In a combative inaugural speech on Saturday Poroshenko said that Ukraine would not give up Crimea, the Black Sea peninsula that Russia annexed from it in March.
"Crimea was, is, and will be Ukrainian," Poroshenko said in an emotional address after his swearing-in. He was greeted with a standing ovation.
He stressed the unity of Ukraine, which is fighting a pro-Russian separatist uprising in the east, and said it would not become a federalized state as advocated by Moscow.
Poroshenko also said he intended very soon to sign the economic part of an association agreement with the European Union, as a first step towards full membership.