A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how … Read more
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
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
We adapt a method due to Nesterov so as to obtain an algorithm for solving block-angular fractional packing or covering problems to relative tolerance epsilon, while using a number of iterations that grows polynomially in the size of the problem and whose dependency on epsilon is proportional to 1/epsilon. CitationCORC report TR-2004-09, Computational Optimization Research … Read more
We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned … Read more
In this paper we introduce a new barrier function $\sum\limits_{i=1}^n(2x_i-1)[\ln{x_i}-\ln(1-x_i)]$ to solve the following optimization problem: $\min\,\, f(x)$ subject to: $Ax=b;\;\;0\leq x\leq e$. We show that this function is a $(3/2)n$-self-concordant barrier on the hypercube $[0,1]^n$. We prove that the central path is well defined and that under an additional assumption on the objective function, … 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
Given a convex body S and a point x \in S, let sym(x,S) denote the symmetry value of x in S: sym(x,S):= max{t : x + t(x – y) \in S for every y \in S}, which essentially measures how symmetric S is about the point x, and define sym(S):=\max{sym(x,S) : x \in S }. … Read more