4,787 research outputs found
Filled function method for nonlinear equations
AbstractSystems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: x∗ is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques
Minimax optimization of entanglement witness operator for the quantification of three-qubit mixed-state entanglement
We develop a numerical approach for quantifying entanglement in mixed quantum
states by convex-roof entanglement measures, based on the optimal entanglement
witness operator and the minimax optimization method. Our approach is
applicable to general entanglement measures and states and is an efficient
alternative to the conventional approach based on the optimal pure-state
decomposition. Compared with the conventional one, it has two important merits:
(i) that the global optimality of the solution is quantitatively verifiable,
and (ii) that the optimization is considerably simplified by exploiting the
common symmetry of the target state and measure. To demonstrate the merits, we
quantify Greenberger-Horne-Zeilinger (GHZ) entanglement in a class of
three-qubit full-rank mixed states composed of the GHZ state, the W state, and
the white noise, the simplest mixtures of states with different genuine
multipartite entanglement, which have not been quantified before this work. We
discuss some general properties of the form of the optimal witness operator and
of the convex structure of mixed states, which are related to the symmetry and
the rank of states
Exploiting spatial sparsity for multi-wavelength imaging in optical interferometry
Optical interferometers provide multiple wavelength measurements. In order to
fully exploit the spectral and spatial resolution of these instruments, new
algorithms for image reconstruction have to be developed. Early attempts to
deal with multi-chromatic interferometric data have consisted in recovering a
gray image of the object or independent monochromatic images in some spectral
bandwidths. The main challenge is now to recover the full 3-D (spatio-spectral)
brightness distribution of the astronomical target given all the available
data. We describe a new approach to implement multi-wavelength image
reconstruction in the case where the observed scene is a collection of
point-like sources. We show the gain in image quality (both spatially and
spectrally) achieved by globally taking into account all the data instead of
dealing with independent spectral slices. This is achieved thanks to a
regularization which favors spatial sparsity and spectral grouping of the
sources. Since the objective function is not differentiable, we had to develop
a specialized optimization algorithm which also accounts for non-negativity of
the brightness distribution.Comment: This version has been accepted for publication in J. Opt. Soc. Am.
DTER: Schedule Optimal RF Energy Request and Harvest for Internet of Things
We propose a new energy harvesting strategy that uses a dedicated energy
source (ES) to optimally replenish energy for radio frequency (RF) energy
harvesting powered Internet of Things. Specifically, we develop a two-step dual
tunnel energy requesting (DTER) strategy that minimizes the energy consumption
on both the energy harvesting device and the ES. Besides the causality and
capacity constraints that are investigated in the existing approaches, DTER
also takes into account the overhead issue and the nonlinear charge
characteristics of an energy storage component to make the proposed strategy
practical. Both offline and online scenarios are considered in the second step
of DTER. To solve the nonlinear optimization problem of the offline scenario,
we convert the design of offline optimal energy requesting problem into a
classic shortest path problem and thus a global optimal solution can be
obtained through dynamic programming (DP) algorithms. The online suboptimal
transmission strategy is developed as well. Simulation study verifies that the
online strategy can achieve almost the same energy efficiency as the global
optimal solution in the long term
Shape optimization for surface functionals in Navier--Stokes flow using a phase field approach
We consider shape and topology optimization for fluids which are governed by
the Navier--Stokes equations. Shapes are modelled with the help of a phase
field approach and the solid body is relaxed to be a porous medium. The phase
field method uses a Ginzburg--Landau functional in order to approximate a
perimeter penalization. We focus on surface functionals and carefully introduce
a new modelling variant, show existence of minimizers and derive first order
necessary conditions. These conditions are related to classical shape
derivatives by identifying the sharp interface limit with the help of formally
matched asymptotic expansions. Finally, we present numerical computations based
on a Cahn--Hilliard type gradient descent which demonstrate that the method can
be used to solve shape optimization problems for fluids with the help of the
new approach
Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Holder constants
In this paper, the global optimization problem with
being a hyperinterval in and satisfying the Lipschitz condition
with an unknown Lipschitz constant is considered. It is supposed that the
function can be multiextremal, non-differentiable, and given as a
`black-box'. To attack the problem, a new global optimization algorithm based
on the following two ideas is proposed and studied both theoretically and
numerically. First, the new algorithm uses numerical approximations to
space-filling curves to reduce the original Lipschitz multi-dimensional problem
to a univariate one satisfying the H\"{o}lder condition. Second, the algorithm
at each iteration applies a new geometric technique working with a number of
possible H\"{o}lder constants chosen from a set of values varying from zero to
infinity showing so that ideas introduced in a popular DIRECT method can be
used in the H\"{o}lder global optimization. Convergence conditions of the
resulting deterministic global optimization method are established. Numerical
experiments carried out on several hundreds of test functions show quite a
promising performance of the new algorithm in comparison with its direct
competitors.Comment: 26 pages, 10 figures, 4 table
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
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