34,130 research outputs found
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
A Collection of Challenging Optimization Problems in Science, Engineering and Economics
Function optimization and finding simultaneous solutions of a system of
nonlinear equations (SNE) are two closely related and important optimization
problems. However, unlike in the case of function optimization in which one is
required to find the global minimum and sometimes local minima, a database of
challenging SNEs where one is required to find stationary points (extrama and
saddle points) is not readily available. In this article, we initiate building
such a database of important SNE (which also includes related function
optimization problems), arising from Science, Engineering and Economics. After
providing a short review of the most commonly used mathematical and
computational approaches to find solutions of such systems, we provide a
preliminary list of challenging problems by writing the Mathematical
formulation down, briefly explaning the origin and importance of the problem
and giving a short account on the currently known results, for each of the
problems. We anticipate that this database will not only help benchmarking
novel numerical methods for solving SNEs and function optimization problems but
also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on
Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE
Congress on Evolutionary Computation (at Sendai International Center, Sendai,
Japan, from 25th to 28th May, 2015.
Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods
The convex feasibility problem (CFP) is at the core of the modeling of many
problems in various areas of science. Subgradient projection methods are
important tools for solving the CFP because they enable the use of subgradient
calculations instead of orthogonal projections onto the individual sets of the
problem. Working in a real Hilbert space, we show that the sequential
subgradient projection method is perturbation resilient. By this we mean that
under appropriate conditions the sequence generated by the method converges
weakly, and sometimes also strongly, to a point in the intersection of the
given subsets of the feasibility problem, despite certain perturbations which
are allowed in each iterative step. Unlike previous works on solving the convex
feasibility problem, the involved functions, which induce the feasibility
problem's subsets, need not be convex. Instead, we allow them to belong to a
wider and richer class of functions satisfying a weaker condition, that we call
"zero-convexity". This class, which is introduced and discussed here, holds a
promise to solve optimization problems in various areas, especially in
non-smooth and non-convex optimization. The relevance of this study to
approximate minimization and to the recent superiorization methodology for
constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio
New error measures and methods for realizing protein graphs from distance data
The interval Distance Geometry Problem (iDGP) consists in finding a
realization in of a simple undirected graph with
nonnegative intervals assigned to the edges in such a way that, for each edge,
the Euclidean distance between the realization of the adjacent vertices is
within the edge interval bounds. In this paper, we focus on the application to
the conformation of proteins in space, which is a basic step in determining
protein function: given interval estimations of some of the inter-atomic
distances, find their shape. Among different families of methods for
accomplishing this task, we look at mathematical programming based methods,
which are well suited for dealing with intervals. The basic question we want to
answer is: what is the best such method for the problem? The most meaningful
error measure for evaluating solution quality is the coordinate root mean
square deviation. We first introduce a new error measure which addresses a
particular feature of protein backbones, i.e. many partial reflections also
yield acceptable backbones. We then present a set of new and existing quadratic
and semidefinite programming formulations of this problem, and a set of new and
existing methods for solving these formulations. Finally, we perform a
computational evaluation of all the feasible solverformulation combinations
according to new and existing error measures, finding that the best methodology
is a new heuristic method based on multiplicative weights updates
The LBFGS Quasi-Newtonian Method for Molecular Modeling Prion AGAAAAGA Amyloid Fibrils
Experimental X-ray crystallography, NMR (Nuclear Magnetic Resonance)
spectroscopy, dual polarization interferometry, etc are indeed very powerful
tools to determine the 3-Dimensional structure of a protein (including the
membrane protein); theoretical mathematical and physical computational
approaches can also allow us to obtain a description of the protein 3D
structure at a submicroscopic level for some unstable, noncrystalline and
insoluble proteins. X-ray crystallography finds the X-ray final structure of a
protein, which usually need refinements using theoretical protocols in order to
produce a better structure. This means theoretical methods are also important
in determinations of protein structures. Optimization is always needed in the
computer-aided drug design, structure-based drug design, molecular dynamics,
and quantum and molecular mechanics. This paper introduces some optimization
algorithms used in these research fields and presents a new theoretical
computational method - an improved LBFGS Quasi-Newtonian mathematical
optimization method - to produce 3D structures of Prion AGAAAAGA amyloid
fibrils (which are unstable, noncrystalline and insoluble), from the potential
energy minimization point of view. Because the NMR or X-ray structure of the
hydrophobic region AGAAAAGA of prion proteins has not yet been determined, the
model constructed by this paper can be used as a reference for experimental
studies on this region, and may be useful in furthering the goals of medicinal
chemistry in this field
A Global Approach for Solving Edge-Matching Puzzles
We consider apictorial edge-matching puzzles, in which the goal is to arrange
a collection of puzzle pieces with colored edges so that the colors match along
the edges of adjacent pieces. We devise an algebraic representation for this
problem and provide conditions under which it exactly characterizes a puzzle.
Using the new representation, we recast the combinatorial, discrete problem of
solving puzzles as a global, polynomial system of equations with continuous
variables. We further propose new algorithms for generating approximate
solutions to the continuous problem by solving a sequence of convex
relaxations
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