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

A Parallel, Linear Programming Based Heuristic for Large Scale Set Partitioning Problems

We describe a parallel, linear programming and implication based heuristic for solving set partitioning problems on distributed memory computer architectures. Our implementation is carefully designed to exploit parallelism to greatest advantage in advanced techniques like preprocessing and probing, primal heuristics, and cut generation. A primal-dual subproblem simplex method is used for solving the linear programming … 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

On the global convergence of an SLP-filter algorithm

A mechanism for proving global convergence infilter-type methods for nonlinear programming is described. Such methods are characterized by their use of the dominance concept of multi objective optimization, instead of a penalty parameter whose adjustment can be problematic. The main point of interest is to demonstrate how convergence for NLP can be induced without forcing … Read more

Integrating SQP and branch-and-bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems. In … Read more

On reduced QP formulations of monotone LCP problems

Techniques for transforming convex quadratic programs (QPs) into monotone linear complementarity problems (LCPs) and vice versa are well known. We describe a class of LCPs for which a reduced QP formulation—one that has fewer constraints than the “standard” QP formulation—is available. We mention several instances of this class, including the known case in which the … 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

Failure of Global Convergence for a Class of Interior Point Methods for Nonlinear Programming

Using a simple analytical example, we demonstrate that a class of interior point methods for general nonlinear programming, including some current methods, is not globally convergent. It is shown that those algorithms do produce limit points that are neither feasible nor stationary points of some measure of the constraint violation, when applied to a well-posed … 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