1,094 research outputs found
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
A Parametric Simplex Algorithm for Linear Vector Optimization Problems
In this paper, a parametric simplex algorithm for solving linear vector
optimization problems (LVOPs) is presented. This algorithm can be seen as a
variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different
from it, the proposed algorithm works in the parameter space and does not aim
to find the set of all efficient solutions. Instead, it finds a solution in the
sense of Loehne [16], that is, it finds a subset of efficient solutions that
allows to generate the whole frontier. In that sense, it can also be seen as a
generalization of the parametric self-dual simplex algorithm, which originally
is designed for solving single objective linear optimization problems, and is
modified to solve two objective bounded LVOPs with the positive orthant as the
ordering cone in Ruszczynski and Vanderbei [21]. The algorithm proposed here
works for any dimension, any solid pointed polyhedral ordering cone C and for
bounded as well as unbounded problems. Numerical results are provided to
compare the proposed algorithm with an objective space based LVOP algorithm
(Benson algorithm in [13]), that also provides a solution in the sense of [16],
and with Evans-Steuer algorithm [12]. The results show that for non-degenerate
problems the proposed algorithm outperforms Benson algorithm and is on par with
Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [13]
excels the simplex-type algorithms; however, the parametric simplex algorithm
is for these problems computationally much more efficient than Evans-Steuer
algorithm.Comment: 27 pages, 4 figures, 5 table
A simplicial algorithm for computing proper Nash equilibria of finite games
Stationary Point;Game Theory;Nash Equilibrium
Constrained shortest paths for QoS routing and path protection in communication networks.
The CSDP (k) problem requires the selection of a set of k > 1 link-disjoint paths with minimum total cost and with total delay bounded by a given upper bound. This problem arises in the context of provisioning paths in a network that could be used to provide resilience to link failures. Again we studied the LP relaxation of the ILP formulation of the problem from the primal perspective and proposed an approximation algorithm.We have studied certain combinatorial optimization problems that arise in the context of two important problems in computer communication networks: end-to-end Quality of Service (QoS) and fault tolerance. These problems can be modeled as constrained shortest path(s) selection problems on networks with each of their links associated with additive weights representing the cost, delay etc.The problems considered above assume that the network status is known and accurate. However, in real networks, this assumption is not realistic. So we considered the QoS route selection problem under inaccurate state information. Here the goal is to find a path with the highest probability that satisfies a given delay upper bound. We proposed a pseudo-polynomial time approximation algorithm, a fully polynomial time approximation scheme, and a strongly polynomial time heuristic for this problem.Finally we studied the constrained shortest path problem with multiple additive constraints. Using the LARAC algorithm as a building block and combining ideas from mathematical programming, we proposed a new approximation algorithm.First we studied the QoS single route selection problem, i.e., the constrained shortest path (CSP) problem. The goal of the CSP problem is to identify a minimum cost route which incurs a delay less than a specified bound. It can be formulated as an integer linear programming (ILP) problem which is computationally intractable. The LARAC algorithm reported in the literature is based on the dual of the linear programming relaxation of the ILP formulation and gives an approximate solution. We proposed two new approximation algorithms solving the dual problem. Next, we studied the CSP problem using the primal simplex method and exploiting certain structural properties of networks. This led to a novel approximation algorithm
Simplicial algorithms for solving the nonlinear complementarity problem on the simplotope
Operational Research
Use of representative operation counts in computational testings of algorithms
Includes bibliographical references (p. 25-26).Ravindra K. Ahuja, James B. Orlin
Analysis of Quickselect under Yaroslavskiy's Dual-Pivoting Algorithm
There is excitement within the algorithms community about a new partitioning
method introduced by Yaroslavskiy. This algorithm renders Quicksort slightly
faster than the case when it runs under classic partitioning methods. We show
that this improved performance in Quicksort is not sustained in Quickselect; a
variant of Quicksort for finding order statistics. We investigate the number of
comparisons made by Quickselect to find a key with a randomly selected rank
under Yaroslavskiy's algorithm. This grand averaging is a smoothing operator
over all individual distributions for specific fixed order statistics. We give
the exact grand average. The grand distribution of the number of comparison
(when suitably scaled) is given as the fixed-point solution of a distributional
equation of a contraction in the Zolotarev metric space. Our investigation
shows that Quickselect under older partitioning methods slightly outperforms
Quickselect under Yaroslavskiy's algorithm, for an order statistic of a random
rank. Similar results are obtained for extremal order statistics, where again
we find the exact average, and the distribution for the number of comparisons
(when suitably scaled). Both limiting distributions are of perpetuities (a sum
of products of independent mixed continuous random variables).Comment: full version with appendices; otherwise identical to Algorithmica
versio
Modelling and solution methods for portfolio optimisation
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University, 16/01/2004.In this thesis modelling and solution methods for portfolio optimisation are presented. The investigations reported in this thesis extend the Markowitz mean-variance model to the domain of quadratic mixed integer programming (QMIP) models which are 'NP-hard' discrete optimisation problems. In addition to the modelling extensions a number of challenging aspects of solution algorithms are considered. The relative performances of sparse simplex (SSX) as well as the interior point method (IPM) are studied in detail. In particular, the roles of 'warmstart' and dual simplex are highlighted as applied to the construction of the efficient frontier which requires processing a family of problems; that is, the portfolio planning model stated in a parametric form. The method of solving QMIP models using the branch and bound algorithm is first developed; this is followed up by heuristics which improve the performance of the (discrete) solution algorithm. Some properties of the efficient frontier with discrete constraints are considered and a method of computing the discrete efficient frontier (DEF) efficiently is proposed. The computational investigation considers the efficiency and effectiveness in respect of the scale up properties of the proposed algorithm. The extensions of the real world models and the proposed solution algorithms make contribution as new knowledge
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