We investigate the problem of sequencing jobs that have multiple components. Each component of the job needs to be processed independently on a specified machine. We derive approximate algorithms for the problem of scheduling such vector jobs to minimize their total completion time in the deterministic as well as stochastic setting. In particular, we propose … Read more
We propose a new gradient based scheme to approximate Nash equilibria of large sequential two-player, zero-sum games. The algorithm uses modern smoothing techniques for saddle-point problems tailored specifically for the polytopes used in the Nash equilibrium problem. CitationWorking Paper, Tepper School of Business, Carnegie Mellon UniversityArticleDownload View PDF
In this paper we present a scaling-invariant interior-point predictor-corrector type algorithm for linear programming (LP) whose iteration-complexity is polynomially bounded by the dimension and the logarithm of a certain condition number of the LP constraint matrix. At the predictor stage, the algorithm either takes the step along the standard affine scaling direction or a new … Read more
In this paper we devise a new version of primal simplex algorithms in which the classical iteration is decomposed two basic operations: the move and the pivot. The move operation decreases the primal objective value and the pivot operation increases the dual objective. We define the condition number of the pivot operation and present a … Read more
Motivated by a numerical example which shows that a feasible version of Mehrotra’s original predictor-corrector algorithm might be inefficient in practice, Salahi et al., proposed a so-called safeguard based variant of the algorithm that enjoys polynomial iteration complexity while its practical efficiency is preserved. In this paper we analyze the same Mehrotra’s algorithm from a … Read more
This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming problem to depend linearly on the size of the perturbations, when these perturbations are directional, involve either the cost coefficients or the right-hand-side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first … Read more
A new initialization or `Phase I’ strategy for feasible interior point methods for linear programming is proposed that computes a point on the primal-dual central path associated with the linear program. Provided there exist primal-dual strictly feasible points — an all-pervasive assumption in interior point method theory that implies the existence of the central path … Read more
In this paper we consider the general cone programming problem, and propose primal-dual convex (smooth and/or nonsmooth) minimization reformulations for it. We then discuss first-order methods suitable for solving these reformulations, namely, Nesterov’s optimal method \cite{Nest83-1,Nest05-1}, Nesterov’s smooth approximation scheme \cite{Nest05-1}, and Nemirovski’s prox-method \cite{Nem05-1}, and propose a variant of Nesterov’s optimal method which has … Read more
The regularization of a convex program is exact if all solutions of the regularized problem are also solutions of the original problem for all values of the regularization parameter below some positive threshold. For a general convex program, we show that the regularization is exact if and only if a certain selection problem has a … Read more
By introducing redundant Klee-Minty examples, we have previously shown that the central path can be bent along the edges of the Klee-Minty cubes, thus having $2^n-2$ sharp turns in dimension $n$. In those constructions the redundant hyperplanes were placed parallel with the facets active at the optimal solution. In this paper we present a simpler … Read more