Resources on Solving Convex Optimization Problems in the Compress Sensing Field
There are few options:


Stephen Boyd, Lieven Vandenberghe - Convex Optimization.
This is the classic in this field. Very well written book.
Also have a look on other papers of Boyd on similar subjects such as the The Alternating Direction Method of Multipliers (ADMM).
They also have a great MOOC Course Stanford Online CVX 101 - Convex Optimization.
Amir Beck - Introduction to Nonlinear Optimization - Theory, Algorithms and Applications &amp; Amir Beck - First-Order Methods in Optimization.
Amir is a great teacher and his book are both deep, practical and easy to read.
CMU Statistics 36-725 - Convex Optimization: Fall 2016 by Ryan Tibshirani.
Videos are available on YouTube.


I think once you skim through those you'll be able to handle most of the cases in Convex Optimization.