An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. The method was reinvented in the U.S. in … See more

We are given a convex program of the form:
f(x_t) - f ≤ ε gi(x_t) ≤ ε for i in 1,...,m, x in G,
where f is the optimal solution. A solver is called polynomial if the total number of arithmetic operations in the first T steps is at most… See more

Types of interior point methods include:
• Potential reduction methods: Karmarkar's algorithm was the first one.
• Path-following methods: the algorithms of James Renegar and Clovis Gonzaga were the first ones.… See more

Given a convex optimization program (P) with constraints, we can convert it to an unconstrained program by adding a barrier function. Specifically, let b be a smooth convex function, defined in the interior of the feasibl… See more

For potential-reduction methods, the problem is presented in the conic form:
minimize c x s.t. x in {b+L} ᚢ K,
where b is a vector in R , L is a linear subspace in R (so b+L is an affine plane), and K is a … See more

The primal-dual method's idea is easy to demonstrate for constrained nonlinear optimization. For simplicity, consider the following nonlinear optimization problem with inequality constraints:
This inequa… See more

• Affine scaling
• Augmented Lagrangian method
• Chambolle-Pock algorithm
• Karush–Kuhn–Tucker conditions… See more