Scarf\u27s Procedure for Integer Programming and a Dual Simplex Algorithm

Herbert Scarf has recently introduced an algorithm for integer programs based on the concept of primitive sets. We show that as the choice variables become continuous, this algorithm converges to a dual simplex algorithm. This result is robust in the sense that even before the limit is reached, the simplex path is contained in the primitive sets which define Scarf’s path to the solution of the integer program

Computing economic equilibria by variable dimension algorithms: State of the art

Economics

Combinatorial analogs of Brouwer's fixed point theorem on a bounded polyhedron

Bibliography: p.61-62.by Robert M. Freund

Analysis of duality constructions for variable dimension fixed point algorithms

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1987.Bibliography: leaves 57-58.Neda F. Emami.M.S

The welfare effects of property tax classification in an urban area: A general equilibrium computational approach

Thesis advisor: Donald RichterTaxing different classes of property at different effective rates is a widespread occurrence in the United States, even though the practice violates many state constitutions. For purposes of tax discrimination, urban real property is commonly divided according to the use to which the property is applied. Typically, the major property categories considered are residential and business, or residential, commercial, and industrial. This thesis investigates the structural and welfare effects of a change from a tax structure in an urban area that classifies property by use for tax purposes to one that does not discriminate in its treatment of property. To accomplish this, long run equilibrium models of urban spatial location are developed. In all models wage rates, and for one model output price of a composite commodity produced in the urban area, can vary in response to the change in tax policy. Conditions guaranteeing the existence of equilibrium for some of the models are developed, and proofs of the existence of equilibrium for those models are provided. Due to the analytical intractability of the models, the tax policy changes are simulated numerically through the use of a fixed point algorithm. The models are stylized, to the extent possible, to the Boston metropolitan area. In particular, the classification tax structure and parameterization of the functions of the model are chosen so that a resultant equilibrium resembles the Boston metropolitan area in or around 1980. General equilibrium versions of compensating and equivalent variations in income are used as measures of welfare change. The qualitative welfare results obtained are quite robust. In all of the simulations conducted there is a welfare gain in moving from the particular classification tax structure used to one in which all property is taxed at the same effective rate.Thesis (PhD) — Boston College, 1984.Submitted to: Boston College. Graduate School of Arts and Sciences.Discipline: Economics

Robust optimization, game theory, and variational inequalities

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2005.Includes bibliographical references (p. 193-109).We propose a robust optimization approach to analyzing three distinct classes of problems related to the notion of equilibrium: the nominal variational inequality (VI) problem over a polyhedron, the finite game under payoff uncertainty, and the network design problem under demand uncertainty. In the first part of the thesis, we demonstrate that the nominal VI problem is in fact a special instance of a robust constraint. Using this insight and duality-based proof techniques from robust optimization, we reformulate the VI problem over a polyhedron as a single- level (and many-times continuously differentiable) optimization problem. This reformulation applies even if the associated cost function has an asymmetric Jacobian matrix. We give sufficient conditions for the convexity of this reformulation and thereby identify a class of VIs, of which monotone affine (and possibly asymmetric) VIs are a special case, which may be solved using widely-available and commercial-grade convex optimization software. In the second part of the thesis, we propose a distribution-free model of incomplete- information games, in which the players use a robust optimization approach to contend with payoff uncertainty.(cont.) Our "robust game" model relaxes the assumptions of Harsanyi's Bayesian game model, and provides an alternative, distribution-free equilibrium concept, for which, in contrast to ex post equilibria, existence is guaranteed. We show that computation of "robust-optimization equilibria" is analogous to that of Nash equilibria of complete- information games. Our results cover incomplete-information games either involving or not involving private information. In the third part of the thesis, we consider uncertainty on the part of a mechanism designer. Specifically, we present a novel, robust optimization model of the network design problem (NDP) under demand uncertainty and congestion effects, and under either system- optimal or user-optimal routing. We propose a corresponding branch and bound algorithm which comprises the first constructive use of the price of anarchy concept. In addition, we characterize conditions under which the robust NDP reduces to a less computationally demanding problem, either a nominal counterpart or a single-level quadratic optimization problem. Finally, we present a novel traffic "paradox," illustrating counterintuitive behavior of changes in cost relative to changes in demand.by Michele Leslie Aghassi.Ph.D

Scarf's Procedure for Integer Programming and a Dual Simplex Algorithm

Herbert Scarf has recently introduced an algorithm for integer programs based on the concept of primitive sets. We show that as the choice variables become continuous, this algorithm converges to a dual simplex algorithm. This result is robust in the sense that even before the limit is reached, the simplex path is contained in the primitive sets which define Scarf's path to the solution of the integer program.