My sourdough started hasn't yet risen! What am I doing wrong?
While feeding I take half cup measure of flour and then take the same
  cup measure of water and mix it in the starter.


And there is your problem to some extent. Always weigh your flour, water, starter and everything else when baking bread. Water weighs about 25% more than flour. This means you have about 25% too much water (or more). 

Great bread recipes take into account humidity, temperature, ingredient weights and the fact that not everybody's measurements will be exactly the same size. Always weigh your ingredients.

So now take 4oz or 100 grams of your starter, add 4oz or 100 grams of water along with 4oz or 100 grams of flour, stir that all up and see what happens in 24 hours. 

Then repeat the process again, twice a day, though you may notice great changes as soon as tomorrow. Report back as you do.

If 4oz is a bit much for your container, you can reduce it to 2oz. Making half of the flour wheat--and not just white--helps in the beginning along with a tablespoon of rye if you have it. (Yeah, I'm not picky about the tablespoon here.)