By refining a variant of the Klee-Minty example that forces the central path to visit all the vertices of the Klee-Minty n-cube, we exhibit a nearly worst-case example for path-following interior point methods. Namely, while the theoretical iteration-complexity upper bound is O(2^{n}n^{\frac{5}{2}}), we prove that solving this n-dimensional linear optimization problem requires at least $2^n-1$ … Read more
The Netlib library of linear programming problems is a well known suite containing many real world applications. Recently it was shown by Ordonez and Freund that 71% of these problems are ill-conditioned. Hence, numerical difficulties may occur. Here, we present rigorous results for this library that are computed by a verification method using interval arithmetic. … Read more
Inverse approaches and, in particular, intensity modulated radiotherapy (IMRT), in combination with the development of new technologies such as multi-leaf collimators (MLCs), have enabled new potentialities of radiotherapy for cancer treatment. The main mathematical tool needed in this connection is numerical optimization. In this article, the variety of continuous optimization approaches, which have been proposed … Read more
This paper studies a new primal-dual interior/exterior-point method for linear programming. We begin with the usual perturbed primal-dual optimality equations $F_\mu(x,y,z)=0$. Under nondegeneracy assumptions, this nonlinear system is well-posed, i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate … Read more
We present a new complexity result on solving the Markov decision problem (MDP) with $n$ states and a number of actions for each state, a special class of real-number linear programs with the Leontief matrix structure. We prove that, when the discount factor $\theta$ is strictly less than $1$, the problem can be solved in … Read more
This paper presents a new primal dual simplex method and investigates the duality formation implying in classical simplex methods. We reviews classical simplex methods for linear programming problems and give a detail discussion for the relation between modern and classical algorithms. The two modified versions are present. The advantages of the new algorithms are simplicity … Read more
It is known that predictor-corrector methods in a large neighborhood of the central path are among the most efficient interior point methods (IPMs) for linear optimization (LO) problems. The best iteration bound based on the classical logarithmic barrier function is $O\left(n\log{\frac{n}{\epsilon}}\right)$. In this paper we propose a family of self-regular proximity based predictor-corrector (SR-PC) IPM … Read more
Sensitivity analysis is one of the most nteresting and preoccupying areas in optimization. Many attempts are made to investigate the problem’s behavior when the input data changes. Usually variation occurs in the right hand side of the constraints and/or the objective function coefficients. Degeneracy of optimal solutions causes considerable difficulties in sensitivity analysis. In this … Read more
Pointillism is a painting technique in which the painter places dots of paint on the canvas in such a way that they blend together into desired forms when viewed from a distance. In this brief note, we describe how to use linear programming to construct a pointillist portrait. Citation Dept. of Mathematics, Oberlin College, Oberlin, … Read more
Self-regular based interior point methods present a unified novel approach for solving linear optimization and conic optimization problems. So far it was not known if the new Self-Regular IPMs can lead to similar advances in computational practice as shown in the theoretical analysis. In this paper, we present our experiences in developing the software package … Read more