How to update 8.0.x to 8.1.x with composer?

Question

This is excerpt from my composer.json file (trimmed for readability). It was created by running composer create-project drupal/drupal my_site_name 8.0.* as described on drupal.org:

{
  "name": "drupal/drupal",
  "type": "project",
  "require": {
    "composer/installers": "^1.0.21",
    "wikimedia/composer-merge-plugin": "^1.3.0"
  },
  "replace": {
    "drupal/core": "~8.0"
  },
  "minimum-stability": "dev",
  "prefer-stable": true,
  "extra": {
    "merge-plugin": {
      "include": [
        "core/composer.json"
      ],
      "recurse": false,
      "replace": false,
      "merge-extra": false
    }
  }
}

1. If I just do composer update - core is not updated, only packages inside vendor dir.
2. If in addition to the above, I update my composer.json file and replace core version with ~8.1 - still core not affected.

The only thing that worked for me was to:

1. Remove core dir, vendor and composer.lock.
2. Edit composer.json and:
   1. Remove replace section.
   2. Add "drupal/core": "~8.1" to require section.
3. Run composer install.

That worked however every installed package complained that:

Class Drupal\Core\Composer\Composer is not autoloadable, can not call post-package-install script

So I'm not sure if the whole process completed flawlessly. Also removing core directory seems like a desperate move.

What is the recommended procedure of updating core to the next minor version with composer (for example 8.0 => 8.1)? Note that I am not using drupal-composer as suggested in this answer.

Update 1

After trying to use composer require --update-with-dependencies "drupal/core:8.1.*" I got no results:

$ composer create-project drupal/drupal my_site_name 8.0.*
Installing drupal/drupal (8.0.6)
  - Installing drupal/drupal (8.0.6)
    Cloning c5afca05cde0c94ef4d7aff5b078be27024670d2

Created project in my_site_name
Loading composer repositories with package information
Installing dependencies (including require-dev) from lock file
Nothing to install or update
Generating autoload files
> Drupal\Core\Composer\Composer::preAutoloadDump
> Drupal\Core\Composer\Composer::ensureHtaccess
Do you want to remove the existing VCS (.git, .svn..) history? [Y,n]? Y
$ cd my_site_name/
$ composer require --update-with-dependencies "drupal/core:8.1.*"
./composer.json has been updated
Loading composer repositories with package information
Updating dependencies (including require-dev)
Nothing to install or update
Writing lock file
Generating autoload files
> Drupal\Core\Composer\Composer::preAutoloadDump
> Drupal\Core\Composer\Composer::ensureHtaccess

I'm using composer version 1.1-dev.

Answer 1

I see, you directly get drupal/drupal. I don't think that's actually supported, you still get a vendor folder from git and the composer.json isn't properly set up. I'm using composer command from this site:

composer create-project drupal-composer/drupal-project:8.x-dev myproject --stability dev --no-interaction

Regarding update, the trick seems to be the following argument to composer require:

composer require --update-with-dependencies "drupal/core:8.1.*"

That updated drupal/core and all dependencies it has.