The solution is two fold. One you need to have a functional system of sanitariums for the mentally ill and you need to have a process by which you can involuntarily have them admitted to the sanitarium. There's three types of homeless people, there's the mentally ill that can't handle life on their own for whatever reason, it's important they get treatment and support for their needs so they can be the least burden on society, most of these people can be marginally functional members of society if they are given proper care.

There's the down on their luck people who are homeless due to circumstances beyond their control, giving them housing and work is the best way to deal with them, they will get jobs get back on their feet and stop being homeless.

Finally there are the people who are homeless by choice, let them do whatever they want if they aren't otherwise bothering people, but don't give them handouts or support. If they want to go live a vagrant life they can do so so long as they are independent. Once they start doing things like drugs in the middle of the city or defecating in public you know they are part of group one and you've gotta round em up and put em in the mental hospitals. Otherwise as long as there aren't many of them they are relatively harmless.

Most of the problems with homelessness are problems with general crime, theft, drug use, etc. As long as you have the cops going out doing their jobs keeping the streets crime free the homeless aren't really an issue. If a guy wants to sleep in the park for a night then clears out the next morning and isn't doing drugs or stealing shit then it's usually best to let him be.