The new offensive represents a risky bet by Mr. Poroshenko that Kiev's ragtag forces can oust the increasingly well-armed militants from regions near the Russian border without drastically more civilian casualties.
Sporadic fighting had continued through the cease-fire even as separatist leaders said they would join.
"The active phase of the antiterrorist operation resumed this morning," parliament Speaker Oleksandr Turchynov, told legislators in Kiev.
"Our armed forces are striking the bases and staging areas of the terrorists."
By late in the day, the president's office said Ukrainian forces had regained control over one of several border posts that had been lost to rebels in recent weeks.
Russian officials said they had closed three customs posts on their side of the border because of nearby fighting, according to the Russian Interfax news agency.
Mr. Poroshenko's supporters had grown impatient with the cease-fire, believing it was buying time for the insurgents to dig in and build up their strength.
Attempts at negotiations yielded few results, as did several telephone conferences between Mr. Poroshenko, Russian President Vladimir Putin and the leaders of Germany and France.
Putin, who had called for the cease-fire to be extended, didn't give any clear signal Tuesday of how Moscow might respond.
But he said that Mr. Poroshenko, who took office after the military operation had begun, now bore full responsibility for it, "not just military but also political."
"And we, by that I mean myself and my colleagues from Europe, weren't about to convince him that the road to a reliable, strong, long-term peace can't lie through war," he said in a foreign-policy speech to Russian diplomats.
Western officials have said that Russia hasn't done enough to rein in the separatists and support peace.
Moscow rejects the criticisms and blames Kiev for stoking the fighting.
Putin also laid blame on the U.S., saying the tensions resulted from a continuation of Cold War-era policy.
He said many European officials are more sympathetic to Moscow's position and don't want their countries to become what he called hostages to U.S. geopolitical interests.
The European Union on Tuesday held off on enacting any additional sanctions against Russia, as had been threatened last week, according to several people briefed on the discussions.
Five officials said senior EU diplomats agreed there was a mixed picture of Russian compliance with four conditions set by EU leaders on Friday.
They included better policing of its border to stem the flow of Russian weapons and fighters to the rebels in Ukraine, and substantial negotiations on the president's peace proposals.
"There has been some progress, but not across the board," one official said, summarizing the EU's decision.
According to the officials, the EU's foreign-service unit will now draw up a list of further individual targets for sanctions, which so far have been limited to asset freezes and entry visa bans.
Brussels will also look at ways to broaden the current sanctions that would still stop short of the sectorwide measures that have been drawn up but not yet enacted—in part because of the collateral damage they would likely cause to the European economy.
The in-between measures could include directly targeting Russian-based companies or freezing EU-financed programs and projects in Russia, one diplomat said.
There will be further discussions in Brussels on Thursday, and senior EU officials plan to meet again next Monday. However, three of the officials said the bloc could move more quickly and even call foreign ministers to Brussels to decide on sanctions if needed.
EU leaders next meet on July 16.
Mr. Poroshenko told Swedish Foreign Minister Carl Bildt in Kiev on Tuesday that the conditions hadn't been met and called on the EU to take "more decisive action," according to a statement from Mr. Poroshenko's office.
Ukrainian officials said talks with the rebels and mediators from the Organization for Security and Cooperation in Europe could resume in the coming days in Minsk, capital of neighboring Belarus.
Separatist leaders, however, ruled out new talks after Kiev restarted the military operation.
A serious offensive will likely be bloody.
Ukraine's army has been moving heavy weaponry forward but has made slow progress in its attempts to oust the armed separatists.
Rebels have downed helicopters and planes with Russian-made surface-to-air rockets.
"We are dealing with well-prepared terrorists, with mercenaries from a neighboring state who are well-armed," Ukrainian military spokesman Oleksiy Dmytrashkovskiy told reporters.
"Against serious weapons we need to use serious weapons."
Fighting was reported in several areas across eastern Ukraine, including the rebel-held border posts, as well as near the international airport in Donetsk, the region's largest city.
Separatist officials also confirmed that fighting had picked up.
Ukrainian forces regained control over the Dovzhansky post, Mr. Poroshenko's office said.
Four civilians were killed and five wounded in Kramatorsk, near the rebel stronghold of Slavyansk, when a bus was hit by shell fire early Tuesday, officials from the regional government in Donetsk said, according to Interfax.
It wasn't clear which side fired the shots.
A military spokesman said one serviceman had been killed and 17 wounded in the last 24 hours, Interfax reported.
Shelling overnight also brought down a television tower near Slavyansk, officials said.
The Russian Foreign Ministry called for the revival of the cease-fire "to safeguard the lives of the people."
Kiev denies its forces attack civilians.
Russian officials also hinted Tuesday that economic pressure on Kiev could rise.
Moscow's top consumer and health regulator said that the quality of Ukrainian dairy and farm products had declined "in a whole series of areas," suggesting that the trade could face restrictions.
Ukrainian officials have dismissed Russian concerns in the past as purely political, though the cost to Ukraine's farmers from past restrictions has been significant.
Meanwhile, the International Monetary Fund on Tuesday said that Ukraine conflict has had "a chilling effect" on foreign investment in Russia, and that Western sanctions and the uncertainty they create are likely to hamstring efforts to boost Russia's flagging economic growth.
"This comes at a crucial moment when the old growth model based on energy and use of spare capacity has been exhausted," it said.
"Moving to a new growth model based on diversification requires new investment, including foreign technology."
Source: The Wall Street Journal