In this paper we present new global convergence results on a long-step affine scaling algorithm obtained by means of the local Karmarkar potential functions. This development was triggered by Dikin’s interesting result on the convergence of the dual estimates associated with a longstep affine scaling algorithm for homogeneous LP problems with unique optimal solutions. Without requiring any assu...

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by K...

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semide nite programming, and nonconvex and nonlinear problems, have reached varyin...

Vol. LII (August 2015), 467–481 467 © 2015, American Marketing Association ISSN: 0022-2437 (print), 1547-7193 (electronic) *Uma R. Karmarkar is Assistant Professor in Marketing, Harvard Business School, Harvard University (e-mail: [email protected]). Baba Shiv is the Sanwa Bank, Limited, Professor of Marketing, Graduate School of Business, Stanford University (e-mail: [email protected]...

In a recent paper, the authors have proved results characterizing convexity-preserving maps deened on a subset of a not-necessarily nite dimensional real vector space as projective maps. The purpose of this note is threefold. First, we state a theorem characterizing continuous, injective, convexity-preserving maps from a relatively open, connected subset of an aane subspace of R m into R n as p...

We present a new parallel algorithm for computing a least-squares solution to a sparse overdetermined system of linear equations Ax = b such that the m x n matrix A is sparse and the graph, G = (V, E), of the matrix has an s(m +n)-separator family, i.e. either IV[ <n o for a fixed constant n 0 or, by deleting a separator subset S of vertices of size <~s(m +n), G can be partitioned into two disc...

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semide nite programming, and nonconvex and nonlinear problems, have reached varyin...

Given a set of numbers, the balanced partioning problem is to divide them into two subsets, so that the sum of the numbers in each subset are as nearly equal as possible, subject to the constraint that the cardinalities of the subsets be within one of each other. We combine the balanced largest differencing method (BLDM) and Korf’s complete Karmarkar-Karp algorithm to get a new algorithm that o...

We illustrate the use of phase transition behaviour in the study of heuristics. Using an \an-nealed" theory, we deene a parameterthat measures the \constrainedness"of an ensemble of number partitioning problems. We identify a phase transition at a critical value of constrainedness. We then show that constrainedness can be used to analyse and compare algorithms and heuristics for number partitio...