Michael J. Todd We reconsider the ellipsoid method for linear inequalities. Using the ellipsoid representation of Burrell and Todd, we show the method can be viewed as coordinate descent on the volume of an enclosing ellipsoid, or on a potential function, or on both. The method can be enhanced by improving the lower bounds generated and by allowing … Read more
The ellipsoid algorithm is a fundamental algorithm for computing a solution to the system of m linear inequalities in n variables (P) when its set of solutions has positive volume. However, when (P) is infeasible, the ellipsoid algorithm has no mechanism for proving that (P) is infeasible. This is in contrast to the other two … Read more
The max-$k$-sum of a set of real scalars is the maximum sum of a subset of size $k$, or alternatively the sum of the $k$ largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-$k$-sums of smooth functions. Second, we discuss how the max-$k$-sum can … Read more
Kalai and Kleitman established the bound $n^{\log(d) + 2}$ for the diameter of a $d$-dimensional polyhedron with $n$ facets. Here we improve the bound slightly to $(n-d)^{\log(d)}$. CitationSchool of Operations Research and Information Engineering, Cornell University, Ithaca NY, USA, February 2014ArticleDownload View PDF
We give an efficient algorithm for computing a Cournot equilibrium when the producers are confined to integers, the inverse demand function is linear, and costs are quadratic. The method also establishes existence, which follows in much more generality because the problem can be modelled as a potential game. CitationSchool of Operations Research and Information Engineering, … Read more
We consider a model in which a consumer of a resource over several periods must pay a per unit charge for the resource as well as a peak charge. The consumer has the ability to reduce his consumption in any period at some given cost, subject to a constraint on the total amount of reduction … Read more
We study the loss in objective value when an inaccurate objective is optimized instead of the true one, and show that “on average” this loss is very small, for an arbitrary compact feasible region. CitationTechnical Report 1479, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, April 2011ArticleDownload View PDF
This software is designed to solve primal and dual semidefinite-quadratic-linear conic programming problems (known as SQLP problems) whose constraint cone is a product of semidefinite cones, second-order cones, nonnegative orthants and Euclidean spaces, and whose objective function is the sum of linear functions and log-barrier terms associated with the constraint cones. This includes the special … Read more
We study a first-order method to find the minimum cross-sectional area ellipsoidal cylinder containing a finite set of points. This problem arises in optimal design in statistics when one is interested in a subset of the parameters. We provide convex formulations of this problem and its dual, and analyze a method based on the Frank-Wolfe … Read more
We show the linear convergence of a simple first-order algorithm for the minimum-volume enclosing ellipsoid problem and its dual, the D-optimal design problem of statistics. Computational tests confirm the attractive features of this method. CitationOptimization Methods and Software 23 (2008), 5–19. ArticleDownload View PDF