This chapter introduces the interior-point method (IPM) that has been widely

used to solve various convex optimization problems. First of all, we focus on

an IPM, called the barrier method. This IPM solves a constrained convex opti-

mization problem (with equality and inequality constraints) by reducing it to a

sequence of linear equality constrained problems, which can be effectively solved

using Newton’s method. With the foundation of the introduced barrier method,

we then introduce the primal-dual interior-point method. Both methods heavily

involve the strong duality and the KKT conditions of the convex optimization

problem under consideration. Some general-purpose convex optimization solvers

are available on-line, such as SeDuMi (http://sedumi.ie.lehigh.edu/) and CVX

(http://www.stanford.edu/∼boyd/cvxbook/), that employ IPMs to solve convex optimization problems. However, a tailored interior-point algorithm is often

much faster than these general-purpose solvers, and thus developing a customized

algorithm using IPMs is essential to efficient hardware/software implementation.

For notational simplicity, the gradient∇xf(x) may sometimes simply be denoted as ∇f(x) without confusion in this chapter.