Calling MatMenuItem#focus() does not update the MatMenu's focusKeyManager's active item

[pshields]

Reproduction

Repro: https://stackblitz.com/edit/components-issue-svmexm

Steps to reproduce:

1. Open a menu
2. Programatically focus some mat-menu-item other than the first one (in the stackblitz repro above, the third mat-menu-item, "C" is focused, due to the onOpen() method in AppComponent, which calls the third mat-menu-item's focus method.)
3. Press the down arrow key (note: a similar variant applies to the up arrow case.)

Expected Behavior

What behavior were you expecting to see?

The next mat-list-item, "D", should be focused.

Actual Behavior

What behavior did you actually see?

The second mat-list-item, "B", is focused.

Analysis

This is happening because the mat menu's FocusKeyManager still thinks the first mat menu item is the active item, even thought we programatically focused a different mat-menu-item. Since mat menu's key manager is private / not exposed to downstream developers, I don't think there's a way to address it currently without a change to Angular Material.

Some possible solutions

• Update the MatMenuItem#focus method to also update the mat menu's active item at the same time
• Expose the mat menu's key manager so downstream developers can call its setActiveItem method.
• Add to the mat menu public API a method to set the active item key manager's active item

Environment

• Angular: 8
• CDK/Material: 8
• Browser(s): Chrome
• Operating System (e.g. Windows, macOS, Ubuntu): Ubuntu

[pshields]

Thanks, @crisbeto!

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