Augmented self concordant barriers and nonlinear optimization problems with finite complexity

In this paper we study special barrier functions for the convex cones, which are the sum of a self-concordant barrier for the cone and a positive-semidefinite quadratic form. We show that the central path of these augmented barrier functions can be traced with linear speed. We also study the complexity of finding the analytic center … Read more

Hyper-sparsity in the revised simplex method and how to exploit it

The revised simplex method is often the method of choice when solving large scale sparse linear programming problems, particularly when a family of closely-related problems is to be solved. Each iteration of the revised simplex method requires the solution of two linear systems and a matrix vector product. For a significant number of practical problems … Read more

Tighter Linear and Semidefinite Relaxations for Max-Cut Based on the Lov\’asz-Schrijver Lift-and-Project Procedure

We study how the lift-and-project method introduced by Lov\’az and Schrijver \cite{LS91} applies to the cut polytope. We show that the cut polytope of a graph can be found in $k$ iterations if there exist $k$ edges whose contraction produces a graph with no $K_5$-minor. Therefore, for a graph with $n\ge 4$ nodes, $n-4$ iterations … Read more

On duality theory of conic linear problems

In this paper we discuss duality theory of optimization problems with a linear objective function and subject to linear constraints with cone inclusions, referred to as conic linear problems. We formulate the Lagrangian dual of a conic linear problem and survey some results based on the conjugate duality approach where the questions of “no duality … Read more

Notes on the Dual Simplex Method

0. The standard dual simplex method. 1. A more general and practical dual simplex method. 2. Phase I for the dual simplex method. 3. Degeneracy in the dual simplex method. 4. A generalized ratio test for the dual simplex method. Citation Draft, Department of Industrial Engineering andManagement Sciences, Northwestern University, 1994. Article Download View Notes … Read more

Warm start strategies in interior-point methods for linear programming

We study the situation in which, having solved a linear program with an interior-point method, we are presented with a new problem instance whose data is slightly perturbed from the original. We describe strategies for recovering a “warm-start” point for the perturbed problem instance from the iterates of the original problem instance. We obtain worst-case … Read more

A scaled Gauss-Newton Primal–Dual Search Direction for Semidefinite Optimization

Interior point methods for semidefinite optimization (SDO) have recently been studied intensively, due to their polynomial complexity and practical efficiency. Most of these methods are extensions of linear optimization (LO) algorithms. Unlike in the LO case, there are several different ways of constructing primal-dual search directions in SDO. The usual scheme is to apply linearization … Read more

A New Class of Polynomial Primal-Dual Methods for Linear and Semidefinite Optimization

We propose a new class of primal-dual methods for linear optimization (LO). By using some new analysis tools, we prove that the large update method for LO based on the new search direction has a polynomial complexity $O\br{n^{\frac{4}{4+\rho}}\log\frac{n}{\e}}$ iterations where $\rho\in [0,2]$ is a parameter used in the system defining the search direction. If $\rho=0$, … Read more