Many of the investment and hedging problems we will encounter can be formulated as a minimization of a function over a set determined by the investor’s risk and budget constraints and other restrictions on the type of positions that the investor can take. Such problems become particularly tractable if both the function to be minimized and the set over which the minimization is done are convex. The minimization problem is in this case called a convex optimization problem. This chapter presents basic results for solving convex optimization problems that will be applied in subsequent chapters.