This course is an introduction to 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 problems. Topics to be covered include a review of convex analysis (convex sets, convex functions, separation and support of sets), unconstrained optimization (characterization of optimality, necessary and sufficient conditions), algorithms for unconstrained optimization ((sub-)gradient descent, coordinate descent, Newton's method), constrained optimization (Karush-Kuhn-Tucker conditions, Lagrangian multiplier and duality), algorithms for constrained optimization (projected (sub)-gradient descent, alternating direction method of multipliers). The course closes with a brief introduction to non-smooth optimization (proximal gradient descent) in machine learning. Prerequisite: Two years of college mathematics, including linear algebra, and the ability to write computer programs.