515 research outputs found
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Experimental investigation of an interior search method within a simple framework
A steepest gradient method for solving Linear Programming (LP) problems, followed by a procedure for purifying a non-basic solution to an improved extreme point solution have been embedded within an otherwise simplex based optimiser. The algorithm is designed to be hybrid in nature and exploits many aspects of sparse matrix and revised simplex technology. The interior search step terminates at a boundary point which is usually non-basic. This is then followed by a series of minor pivotal steps which lead to a basic feasible solution with a superior objective function value. It is concluded that the procedures discussed in this paper are likely to have three possible applications, which are
(i) improving a non-basic feasible solution to a superior extreme point solution,
(iii) an improved starting point for the revised simplex method, and
(iii) an efficient implementation of the multiple price strategy of the revised simplex method
Fast quantum subroutines for the simplex method
We propose quantum subroutines for the simplex method that avoid classical
computation of the basis inverse. For an constraint matrix with at
most nonzero elements per column, at most nonzero elements per column
or row of the basis, basis condition number , and optimality tolerance
, we show that pricing can be performed in
time, where the
notation hides polylogarithmic factors. If the ratio is
larger than a certain threshold, the running time of the quantum subroutine can
be reduced to . The steepest edge pivoting rule also admits a quantum
implementation, increasing the running time by a factor .
Classically, pricing requires
time in the worst case using the fastest known algorithm for sparse matrix
multiplication, and with steepest
edge. Furthermore, we show that the ratio test can be performed in
time, where
determine a feasibility tolerance; classically, this requires time in
the worst case. For well-conditioned sparse problems the quantum subroutines
scale better in and , and may therefore have a worst-case asymptotic
advantage. An important feature of our paper is that this asymptotic speedup
does not depend on the data being available in some "quantum form": the input
of our quantum subroutines is the natural classical description of the problem,
and the output is the index of the variables that should leave or enter the
basis.Comment: Added discussion on condition number and infeasibilitie
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
The Niceness of Unique Sink Orientations
Random Edge is the most natural randomized pivot rule for the simplex
algorithm. Considerable progress has been made recently towards fully
understanding its behavior. Back in 2001, Welzl introduced the concepts of
\emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an
effort to better understand the behavior of Random Edge. In this paper, we
initiate the systematic study of these concepts. We settle the questions that
were asked by Welzl about the niceness of (acyclic) USO. Niceness implies
natural upper bounds for Random Edge and we provide evidence that these are
tight or almost tight in many interesting cases. Moreover, we show that Random
Edge is polynomial on at least many (possibly cyclic) USO. As
a bonus, we describe a derandomization of Random Edge which achieves the same
asymptotic upper bounds with respect to niceness and discuss some algorithmic
properties of the reachmap.Comment: An extended abstract appears in the proceedings of Approx/Random 201
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
Numerical Conformal bootstrap with Analytic Functionals and Outer Approximation
This paper explores the numerical conformal bootstrap in general spacetime
dimensions through the lens of a distinct category of analytic functionals,
previously employed in two-dimensional studies. We extend the application of
these functionals to a more comprehensive backdrop, demonstrating their
adaptability and efficacy in general spacetime dimensions above two. The
bootstrap is implemented using the outer approximation methodology, with
computations conducted in double precision. The crux of our study lies in
comparing the performance of this category of analytic functionals with
conventional derivatives at crossing symmetric points. It is worth highlighting
that in our study, we identified some novel kinks in the scalar channel during
the maximization of the gap in two-dimensional conformal field theory. Our
numerical analysis indicates that these analytic functionals offer a superior
performance, thereby revealing a potential alternative paradigm in the
application of conformal bootstrap.Comment: 59 pages, 16 tables and 12 figure
Rotor design optimization using a free wake analysis
The aim of this effort was to develop a comprehensive performance optimization capability for tiltrotor and helicopter blades. The analysis incorporates the validated EHPIC (Evaluation of Hover Performance using Influence Coefficients) model of helicopter rotor aerodynamics within a general linear/quadratic programming algorithm that allows optimization using a variety of objective functions involving the performance. The resulting computer code, EHPIC/HERO (HElicopter Rotor Optimization), improves upon several features of the previous EHPIC performance model and allows optimization utilizing a wide spectrum of design variables, including twist, chord, anhedral, and sweep. The new analysis supports optimization of a variety of objective functions, including weighted measures of rotor thrust, power, and propulsive efficiency. The fundamental strength of the approach is that an efficient search for improved versions of the baseline design can be carried out while retaining the demonstrated accuracy inherent in the EHPIC free wake/vortex lattice performance analysis. Sample problems are described that demonstrate the success of this approach for several representative rotor configurations in hover and axial flight. Features that were introduced to convert earlier demonstration versions of this analysis into a generally applicable tool for researchers and designers is also discussed
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