Who can initiate an optional referendum at the national level?

Constitution (last amended in 2005)

Article 17. The people shall have the right of election, recall, initiative and referendum.

Referendum Act (amended date: 2018/01/03)

Article 2. The "referendums" described in this Act includes national referendums and local referendums. For national referendums, this Act shall apply to the following matters apart from those stipulated in the Constitution:

1. Referendums on laws.

2. Initiatives on legislative principles.

3. Initiatives or referendums on important policies. [...]

No proposal of referendum may be raised for any matters regarding budgeting, taxation, salary and personnel affairs.

Article 14. If the Executive Yuan deems it necessary to carry out referendums for a matter as prescribed in Subparagraph Three of Paragraph Two of Article 2, it may hand the main text and the statement of reasons, after they are approved by the Legislative Yuan, to the competent authority to implement the referendum. The provisions of Article 13, Paragraph One of Article 17, and Article 19 shall not apply.
After the Executive Yuan proposes a referendum to the Legislative Yuan, the Legislative Yuan shall decide within 15 days; if a referendum is proposed during the recess, the Legislative Yuan shall call a meeting within 15 days and decide within 30 days.
If the proposal raised by the Executive Yuan is vetoed by the Legislative Yuan, the Executive Yuan may not raise any more proposals for the same matter within 2 years commencing from the day of veto.

Article 15. After the referendum proposed by the Legislative Yuan in accordance with the provisions of the Constitution has been announced for six months, the Legislative Yuan shall request the competent authority to carry out the referendum within 10 days.

If the Legislative Yuan deems it is necessary to carry out referendums for a matter as prescribed in Subparagraph Three of Paragraph Two of Article 2, it may hand the main text and the statement of reasons, after they are adopted in a meeting of the Legislative Yuan, to the competent authority to implement the referendum. […]

Article 16. When the country is under the threat of foreign force and the national sovereignty is likely to change, the President may, by a resolution of the meeting of the Executive Yuan, put matters regarding national security to referendum.

One of the changes that 2018 amendment introduced is that Executive Yuan (government) can initiate referendum.