If rent control "doesn't work" and supply induces demand, how can cities improve affordability? Suppose we aim that moderately poor people (let's say 1.5× minimum wage) can afford the rent for an apartment meeting minimum quality standards and located within city limits.  This may be a political aim.

What economic policies are effective to achieve this?


Rent control.  Popular, but economics appears to tell us it creates more problems than it solves.  See also the questions Why did Berlin freeze the rent prices as opposed to letting the market set the price? and What is the likely result of rent control in Berlin?.
Increase supply?  In cities, it appears supply may create demand (the more people in a city, the more services, the more people want to live there) so this may not always work either, or only under limited circumstances (perhaps unless building huge amounts, but developers hate vacancies, so I doubt that limit would be surpassed in a free market).
Social housing, owned by the government?  I don't know what economic theory says about this.
Reduce demand?  Not sure how that would work without harming the city.
Impossible?  Should we give up on the idea that poor people can live in alpha world cities such as London, New York, Tokyo, Paris, Singapore, Amsterdam, Seoul, Berlin, Hong Kong, or Sydney?
Something else?


I'm aware that most economists say that rent control "doesn't work" — what are the alternatives that economists may prefer?
Disclaimer: this answer was significantly rewritten to spell out explicitly how it answers OPs question and to add some sources (thanks to @Fizz). The gist remains the same.

The only case I know where housing policy seems to work the way you envision is Vienna (the Austrien capital). They use a combination of several of the policies on OPs list, they have been doing this for a long time and they seem to be very good at implementing them in productive ways. 

First, the city owns a significant proportian of all apartments, enough to have pricing power in the market (about 60% according to source). Additionally there are rent control laws on the privately owned housing (see for example here for some housing policies). Nevertheless private housing is more expensive than public housing (source).

Second, the city is engaged in long term planning for housing and they do that wholesale. So they are not just building houses on existing ground attached to existing infrastructure but they are creating entire new suburbs including attachment to public transportation, schools and all the other infrastructure needed for a successful city. The sources above also talk about that in more detail.

This is hard to pull of because it takes much longer to bear fruit that a typical election cycle. Doing things that cost money now and are very useful in 20 years is difficult for elected politicians. So even if politicians or people in general in other cities like to copy the Vienna model, it is not just a few simple policy changes.