1. What is convex optimization? -- 2. Tools for convex optimization -- 3. Basic optimality conditions using the normal cone -- 4. Saddle points, optimality, and duality -- 5. Enhanced Fritz John optimality conditions -- 6. Optimality without constraint qualification -- 7. Sequential optimality conditions -- 8. Representation of the feasible set and KKT conditions -- 9. Weak sharp minima in convex optimization -- 10. Approximate optimality conditions -- 11. Convex semi-infinite optimization -- 12. Convexity in nonconvex optimization.