We consider the interior-point approach to sensitivity analysis in linear programming (LP) developed by the authors. We investigate the quality of the interior-point bounds under degeneracy. In the case of a special degeneracy, we show that these bounds have the same nice relationship with the optimal partition bounds as in the nondegenerate case. We prove … Read more
In this paper we address a general Goal Programming problem with linear objectives, convex constraints, and an arbitrary componentwise nondecreasing norm to aggregate deviations with respect to targets. In particular, classical Linear Goal Programming problems, as well as several models in Location and Regression Analysis are modeled within this framework. In spite of its generality, … Read more
Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic … Read more
We investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low-rank update to a positive semi-definite matrix. Several formulations of the support vector machine fit into this category. An interesting feature of these particular problems is the volume … Read more
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
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
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
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