Option 1: Malkhed was not the capital of the Hoysalas. Malkhed was the capital of the Rashtrakuta dynasty.

Option 2: Dwarasamudra was indeed the capital of the Hoysalas. Dwarasamudra, also known as Halebidu, was the capital for most of the Hoysala rule from the 12th to 14th centuries.

Option 3: Somnathpur was not the capital of the Hoysalas. Somnathpur is known for its famous Hoysala temple, but it was not the administrative capital of the dynasty.

Option 4: Badami was not the capital of the Hoysalas. Badami was the capital of the early Chalukya dynasty and is known for its rock-cut caves and ancient temples.

Therefore, the correct answer is Option 2: Dwarasamudra. It served as the capital of the Hoysalas during their rule.