What is the best optimizer for Convolutional neural network (CNN)?
Can I use RMSProp for CNN or only for RNN?
Yes, you can use the same optimizers you are familiar with for CNNs.
I don't think that there is a best optimizer for CNNs. The most popular in my opinion is Adam. However some people like to use a plain SGD optimizer with custom parameters.
An excellent article explaining the differences between most popular gradient descent based optimizers can be found here.