1,026 research outputs found
Efficient Rank Reduction of Correlation Matrices
Geometric optimisation algorithms are developed that efficiently find the
nearest low-rank correlation matrix. We show, in numerical tests, that our
methods compare favourably to the existing methods in the literature. The
connection with the Lagrange multiplier method is established, along with an
identification of whether a local minimum is a global minimum. An additional
benefit of the geometric approach is that any weighted norm can be applied. The
problem of finding the nearest low-rank correlation matrix occurs as part of
the calibration of multi-factor interest rate market models to correlation.Comment: First version: 20 pages, 4 figures Second version [changed content]:
21 pages, 6 figure
A dynamical view of nonlinear conjugate gradient methods with applications to FFT-based computational micromechanics
For fast Fourier transform (FFT)-based computational micromechanics, solvers need to be fast, memory-efficient, and independent of tedious parameter calibration. In this work, we investigate the benefits of nonlinear conjugate gradient (CG) methods in the context of FFT-based computational micromechanics. Traditionally, nonlinear CG methods require dedicated line-search procedures to be efficient, rendering them not competitive in the FFT-based context. We contribute to nonlinear CG methods devoid of line searches by exploiting similarities between nonlinear CG methods and accelerated gradient methods. More precisely, by letting the step-size go to zero, we exhibit the Fletcher–Reeves nonlinear CG as a dynamical system with state-dependent nonlinear damping. We show how to implement nonlinear CG methods for FFT-based computational micromechanics, and demonstrate by numerical experiments that the Fletcher–Reeves nonlinear CG represents a competitive, memory-efficient and parameter-choice free solution method for linear and nonlinear homogenization problems, which, in addition, decreases the residual monotonically
Computation of Ground States of the Gross-Pitaevskii Functional via Riemannian Optimization
In this paper we combine concepts from Riemannian Optimization and the theory
of Sobolev gradients to derive a new conjugate gradient method for direct
minimization of the Gross-Pitaevskii energy functional with rotation. The
conservation of the number of particles constrains the minimizers to lie on a
manifold corresponding to the unit norm. The idea developed here is to
transform the original constrained optimization problem to an unconstrained
problem on this (spherical) Riemannian manifold, so that fast minimization
algorithms can be applied as alternatives to more standard constrained
formulations. First, we obtain Sobolev gradients using an equivalent definition
of an inner product which takes into account rotation. Then, the
Riemannian gradient (RG) steepest descent method is derived based on projected
gradients and retraction of an intermediate solution back to the constraint
manifold. Finally, we use the concept of the Riemannian vector transport to
propose a Riemannian conjugate gradient (RCG) method for this problem. It is
derived at the continuous level based on the "optimize-then-discretize"
paradigm instead of the usual "discretize-then-optimize" approach, as this
ensures robustness of the method when adaptive mesh refinement is performed in
computations. We evaluate various design choices inherent in the formulation of
the method and conclude with recommendations concerning selection of the best
options. Numerical tests demonstrate that the proposed RCG method outperforms
the simple gradient descent (RG) method in terms of rate of convergence. While
on simple problems a Newton-type method implemented in the {\tt Ipopt} library
exhibits a faster convergence than the (RCG) approach, the two methods perform
similarly on more complex problems requiring the use of mesh adaptation. At the
same time the (RCG) approach has far fewer tunable parameters.Comment: 28 pages, 13 figure
A randomised non-descent method for global optimisation
This paper proposes novel algorithm for non-convex multimodal constrained
optimisation problems. It is based on sequential solving restrictions of
problem to sections of feasible set by random subspaces (in general, manifolds)
of low dimensionality. This approach varies in a way to draw subspaces,
dimensionality of subspaces, and method to solve restricted problems. We
provide empirical study of algorithm on convex, unimodal and multimodal
optimisation problems and compare it with efficient algorithms intended for
each class of problems.Comment: 9 pages, 7 figure
Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold
The conjugate gradient method is a crucial first-order optimization method
that generally converges faster than the steepest descent method, and its
computational cost is much lower than the second-order methods. However, while
various types of conjugate gradient methods have been studied in Euclidean
spaces and on Riemannian manifolds, there has little study for those in
distributed scenarios. This paper proposes a decentralized Riemannian conjugate
gradient descent (DRCGD) method that aims at minimizing a global function over
the Stiefel manifold. The optimization problem is distributed among a network
of agents, where each agent is associated with a local function, and
communication between agents occurs over an undirected connected graph. Since
the Stiefel manifold is a non-convex set, a global function is represented as a
finite sum of possibly non-convex (but smooth) local functions. The proposed
method is free from expensive Riemannian geometric operations such as
retractions, exponential maps, and vector transports, thereby reducing the
computational complexity required by each agent. To the best of our knowledge,
DRCGD is the first decentralized Riemannian conjugate gradient algorithm to
achieve global convergence over the Stiefel manifold
A Test of Non-linear Conjugate Gradient Methods Via Exact Line Search
The conjugate gradient method provides a very powerful tool for solving unconstrained optimization problems.
In this paper the non-linear conjugate gradient methods are tested using some benchmark non-polynomial
unconstrained optimization functions. The task was accomplished by finding the exact values of the descent also
known as the minimizing argument or rather the minimizer in each method. Findings also show that the basic
requirement for exact convergence was satisfied by all the methods
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