Many applications of computer science involve the application of sophisticated ideas from optimization theory. These can be categorized broadly into the following types: (i) resource allocation problems (ii) control, decision-making and planning (iii) approximation and estimation (iv) game theory. This course will provide a unified treatment of optimization methods used in these applications, focusing on a core set of geometric principles:

1. • Generalized distance measures, such as Bregman divergence, which unify most commonly used metrics such as Euclidean distance and KL-divergence
   

2. • The projection theorem in vector spaces, used widely in approximation and estimation
   

3. • The celebrated Hahn-Banach theorem, which underlies many important ideas in optimization
   

4. • The duality principle, which shows how to convert minimization problems into maximization problems using hyperplanes
   

5. • Differentials, in particular subgradients and subdifferentials.