1. China's approach to protecting its environment while feeding its citizens...

此句，while 之前是个名词词组，不是主句。意为：中国保护环境的方法/途径。

while 后面是现在分词，或者说省略了主语和be动词也可：while it is feeding...

2. Confucius believed knives would remind people of killing and were too violent for use at the table.

此句，没有连词，只有主句和宾语从句。