Optimization theory and algorithms are nowadays, together with statistics and information theory, served as the foundation of machine learning and big data analytics. This course introduces the theory and algorithms of convex optimization. The goal of this course is to endow the student with a) a solid understanding of the subject's theoretical foundation and b) the ability to analyze and apply convex optimization algorithms in the context of diverse engineering and science problems. Topics to be covered include characterization of local optimality (necessary and sufficient conditions), convex sets, convex functions, sub-gradient and sub-differential, algorithms for unconstrained optimization (gradient descent, proximal gradient, coordinate descent), constrained optimization (Lagrangian multiplier and duality, Karush-Kuhn-Tucker conditions), algorithms for constrained optimization (projected gradient descent, alternating direction method of multipliers). Lecture notes, homework assignments and solutions will be released on the Collab and Piazza.