Abdul Nabi, 30, is described as black - of Asian appearance - about 5ft 7in tall and skinny, with short curly hair.
Anyone who spots him has been asked to call 999.
Officers were searching land around Midpark Hospital and Maidenbower near the Crichton university and business complex. Premises in the immediate area were alerted.
The search subsequently moved on to the Calside and Georgetown areas of the town and pupils at Georgetown Primary School had their sports day interrupted.
Police dogs and the police air support unit also joined efforts to try to trace Mr Nabi.
"This is a missing person inquiry and we want the public to be alert, however not alarmed," said a spokesman.
"We are asking that if he is seen he is not approached and you contact the police."
Insp Rory Caldow said officers had been carrying out a large-scale search since about 12:30.
"We did have early concerns that he may have been in possession of a knife, however we understand that this is not now the case," he said.
"Mr Nabi is a vulnerable missing man and our concern for the safety of the public centres on the uncertainty of his reaction should he be approached.
"We ask the public to be vigilant and report any sightings, without making an approach to him, on the 999 emergency number."

15 June 2013 Last updated at 17:21 BST
Tafwyl is a free family festival in the grounds of Cardiff Castle, featuring music, sport and craft activities, as well as cookery and literature. Around 10,000 people were attending.
Rhys, who is also due to start filming a BBC Wales drama from Monday, said holding the festival in the city helped to break down potential barriers to the Welsh language.
Education Minister Leighton Andrews also said it had an important role to play in showing off the language and its growth in the city.

Scottish Swimming said the government had pulled funding from its Â£1.72m programme to support school lessons.
It said it was concerned it could lead to greater numbers of adults who lacked confidence in the water.
The Scottish government said the top-up scheme had ended but swimming was still well-funded.
Swimming is not a compulsory part of the curriculum in Scotland, unlike in England.
The provision of primary school swimming lessons varies extensively between local authorities, with children living in the most socially deprived areas having the highest number of non-swimmers.
Scottish Swimming claimed that as many as 40% of children, about 15,000 each year, headed to high school as non-swimmers.
Its "Top Up" swimming programme supported local authorities to deliver swimming lessons for pupils.
Scottish Swimming said "Top Up" helped up to 30,000 children to improve their swimming skills.
Sports Minister Jamie Hepburn said: "The Top Up programme was designed to help develop a model to enable swimming clubs and local authorities to teach children to swim by providing some initial additional investment to assist local provision. The programme received Â£1.72m over four years.
"Although the top-up programme has now ended, Scottish Swimming received more than Â£5m (over four years) in the latest funding round from sportscotland. This is the largest funding settlement of all the Commonwealth Games sports."
Shelagh Dow, the head teacher at Granton primary in Edinburgh, said financial pressures stopped many families from going swimming.
She said the loss of the programme would have an impact.
"That is going to make a huge difference for them," Ms Dow said.
"Swimming is ultimately a life skill and because a lot of them don't manage to pick up the swimming with the funded lessons that they have in primary 4 they really need that top-up.  Quite often financial constraints stop them going swimming at other times."
Sharon Macdonald, development officer for Scottish Swimming, said: "It is very concerning for us. The programme has been in place for the past four years.
"We've seen a massive impact from the programme in the region of 25,000 to 30,000 children who have either improved their swimming or become what is our new national standard for school swimming.
Police have warned the public not to approach a man reported missing from a psychiatric hospital in Dumfries.
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Actor Matthew Rhys, the star of Cold War drama series The Americans, returned to his home city Cardiff to open a Welsh language festival.
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The end of a scheme to improve the standard of swimming among primary school children has brought a warning from the sport's governing body.