Detecting Infeasibility in Infeasible-Interior-Point Methods for Optimization

We study interior-point methods for optimization problems in the case of infeasibility or unboundedness. While many such methods are designed to search for optimal solutions even when they do not exist, we show that they can be viewed as implicitly searching for well-defined optimal solutions to related problems whose optimal solutions give certificates of infeasibility … Read more

Smoothed Analysis of Interior-Point Algorithms: Termination

We perform a smoothed analysis of the termination phase of an interior-point method. By combining this analysis with the smoothed analysis of Renegar’s interior-point algorithm by Dunagan, Spielman and Teng, we show that the smoothed complexity of an interior-point algorithm for linear programming is $O (m^{3} \log (m/\sigma ))$. In contrast, the best known bound … Read more

Optimal Portfolios using Linear Programming Models

The classical Quadratic Programming formulation of the well known portfolio selection problem, is cumbersome, time consuming and relies on two important assumptions: (a) the expected return is multivariate normally distributed; (b) the investor is risk averter. This paper formulates two alternative models, (i) maximin, and (ii) minimization of absolute deviation. Data from a very simple … Read more

Unifying Condition Numbers for Linear Programming

In recent years, several condition numbers were defined for a variety of linear programming problems based upon relative distances to ill-posedness. In this paper we provide a unifying view of these condition numbers. To do so, we introduce yet another linear programming problem and show that its distance to ill-posedness naturally captures the most commonly … Read more

The Inverse Optimal Value Problem

This paper considers the following inverse optimization problem: given a linear program, a desired optimal objective value, and a set of feasible cost coefficients, determine a cost-coefficient vector such that the corresponding optimal objective value of the linear program is closest to the given value. The above problem, referred here as the inverse optimal value … Read more

Safe bounds in linear and mixed-integer programming

Current mixed-integer linear programming solvers are based on linear programming routines that use floating point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. It is shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in … Read more

The method of reflection-projection for convex feasibility problems with an obtuse cone

The convex feasibility problem asks to find a point in the intersection of finitely many closed convex sets in Euclidean space. This problem is of fundamental importance in mathematics and physical sciences, and it can be solved algorithmically by the classical method of cyclic projections. In this paper, the case where one of the constraints … Read more

Large-Scale Linear Programming Techniques for the Design of Protein Folding Potentials

We present large-scale optimization techniques to model the energy function that underlies the folding process of proteins. Linear Programming is used to identify parameters in the energy function model, the objective being that the model predict the structure of known proteins correctly. Such trained functions can then be used either for {\em ab-initio} prediction or … Read more

Computational Experience and the Explanatory Value of Condition Numbers for Linear Optimization

The goal of this paper is to develop some computational experience and test the practical relevance of the theory of condition numbers C(d) for linear optimization, as applied to problem instances that one might encounter in practice. We used the NETLIB suite of linear optimization problems as a test bed for condition number computation and … Read more

Block-iterative algorithms with diagonally scaled oblique projections for the linear feasibility problem

We formulate a block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and analyze its convergence. This study provides as special cases proofs of convergence of (i) the recently proposed Component Averaging (CAV) method of Censor, Gordon and Gordon ({\it Parallel Computing}, 27:777–808, 2001), (ii) the recently proposed Block-Iterative CAV (BICAV) … Read more