16,443 research outputs found
ABS Algorithms for Linear Systems and Optimization
We present a review and bibliography of the main results obtained during a
research on ABS (Abaffy, Broyden, Spedicato) methods.Comment: 38 page
Machine Learning of Space-Fractional Differential Equations
Data-driven discovery of "hidden physics" -- i.e., machine learning of
differential equation models underlying observed data -- has recently been
approached by embedding the discovery problem into a Gaussian Process
regression of spatial data, treating and discovering unknown equation
parameters as hyperparameters of a modified "physics informed" Gaussian Process
kernel. This kernel includes the parametrized differential operators applied to
a prior covariance kernel. We extend this framework to linear space-fractional
differential equations. The methodology is compatible with a wide variety of
fractional operators in and stationary covariance kernels,
including the Matern class, and can optimize the Matern parameter during
training. We provide a user-friendly and feasible way to perform fractional
derivatives of kernels, via a unified set of d-dimensional Fourier integral
formulas amenable to generalized Gauss-Laguerre quadrature.
The implementation of fractional derivatives has several benefits. First, it
allows for discovering fractional-order PDEs for systems characterized by heavy
tails or anomalous diffusion, bypassing the analytical difficulty of fractional
calculus. Data sets exhibiting such features are of increasing prevalence in
physical and financial domains. Second, a single fractional-order archetype
allows for a derivative of arbitrary order to be learned, with the order itself
being a parameter in the regression. This is advantageous even when used for
discovering integer-order equations; the user is not required to assume a
"dictionary" of derivatives of various orders, and directly controls the
parsimony of the models being discovered. We illustrate on several examples,
including fractional-order interpolation of advection-diffusion and modeling
relative stock performance in the S&P 500 with alpha-stable motion via a
fractional diffusion equation.Comment: 26 pages, 10 figures. In v2, a minor change to the formatting of a
handful of references was made in the bibliography; the main text was
unchanged. In v3, minor improvements were made to the exposition; more
details about motivation, examples, optimization, and relation to previous
works were give
Derivative-free optimization methods
In many optimization problems arising from scientific, engineering and
artificial intelligence applications, objective and constraint functions are
available only as the output of a black-box or simulation oracle that does not
provide derivative information. Such settings necessitate the use of methods
for derivative-free, or zeroth-order, optimization. We provide a review and
perspectives on developments in these methods, with an emphasis on highlighting
recent developments and on unifying treatment of such problems in the
non-linear optimization and machine learning literature. We categorize methods
based on assumed properties of the black-box functions, as well as features of
the methods. We first overview the primary setting of deterministic methods
applied to unconstrained, non-convex optimization problems where the objective
function is defined by a deterministic black-box oracle. We then discuss
developments in randomized methods, methods that assume some additional
structure about the objective (including convexity, separability and general
non-smooth compositions), methods for problems where the output of the
black-box oracle is stochastic, and methods for handling different types of
constraints
Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex Systems
Canonical duality-triality is a breakthrough methodological theory, which can
be used not only for modeling complex systems within a unified framework, but
also for solving a wide class of challenging problems from real-world
applications. This paper presents a brief review on this theory, its
philosophical origin, physics foundation, and mathematical statements in both
finite and infinite dimensional spaces, with emphasizing on its role for
bridging the gap between nonconvex analysis/mechanics and global optimization.
Special attentions are paid on unified understanding the fundamental
difficulties in large deformation mechanics, bifurcation/chaos in nonlinear
science, and the NP-hard problems in global optimization, as well as the
theorems, methods, and algorithms for solving these challenging problems.
Misunderstandings and confusions on some basic concepts, such as objectivity,
nonlinearity, Lagrangian, and generalized convexities are discussed and
classified. Breakthrough from recent challenges and conceptual mistakes by M.
Voisei, C. Zalinescu and his co-worker are addressed. Some open problems and
future works in global optimization and nonconvex mechanics are proposed.Comment: 43 pages, 4 figures. appears in Mathematics and Mechanics of Solids,
201
Deep Neural Networks as 0-1 Mixed Integer Linear Programs: A Feasibility Study
Deep Neural Networks (DNNs) are very popular these days, and are the subject
of a very intense investigation. A DNN is made by layers of internal units (or
neurons), each of which computes an affine combination of the output of the
units in the previous layer, applies a nonlinear operator, and outputs the
corresponding value (also known as activation). A commonly-used nonlinear
operator is the so-called rectified linear unit (ReLU), whose output is just
the maximum between its input value and zero. In this (and other similar cases
like max pooling, where the max operation involves more than one input value),
one can model the DNN as a 0-1 Mixed Integer Linear Program (0-1 MILP) where
the continuous variables correspond to the output values of each unit, and a
binary variable is associated with each ReLU to model its yes/no nature. In
this paper we discuss the peculiarity of this kind of 0-1 MILP models, and
describe an effective bound-tightening technique intended to ease its solution.
We also present possible applications of the 0-1 MILP model arising in feature
visualization and in the construction of adversarial examples. Preliminary
computational results are reported, aimed at investigating (on small DNNs) the
computational performance of a state-of-the-art MILP solver when applied to a
known test case, namely, hand-written digit recognition.Comment: submitted to an international conferenc
Reinforcement Learning for Batch Bioprocess Optimization
Bioprocesses have received a lot of attention to produce clean and
sustainable alternatives to fossil-based materials. However, they are generally
difficult to optimize due to their unsteady-state operation modes and
stochastic behaviours. Furthermore, biological systems are highly complex,
therefore plant-model mismatch is often present. To address the aforementioned
challenges we propose a Reinforcement learning based optimization strategy for
batch processes.
In this work, we applied the Policy Gradient method from batch-to-batch to
update a control policy parametrized by a recurrent neural network. We assume
that a preliminary process model is available, which is exploited to obtain a
preliminary optimal control policy. Subsequently, this policy is updatedbased
on measurements from thetrueplant. The capabilities of our proposed approach
were tested on three case studies (one of which is nonsmooth) using a more
complex process model for thetruesystemembedded with adequate process
disturbance. Lastly, we discussed the advantages and disadvantages of this
strategy compared against current existing approaches such as nonlinear model
predictive control
Relaxations of AC Maximal Load Delivery for Severe Contingency Analysis
This work considers the task of finding an AC-feasible operating point of a
severely damaged transmission network while ensuring that a maximal amount of
active power loads can be delivered. This AC Maximal Load Delivery (AC-MLD)
task is a nonconvex nonlinear optimization problem that is incredibly
challenging to solve on large-scale transmission system datasets. This work
demonstrates that convex relaxations of the AC-MLD problem provide a reliable
and scalable method for finding high-quality bounds on the amount of active
power that can be delivered in the AC-MLD problem. To demonstrate their
effectiveness, the solution methods proposed in this work are rigorously
evaluated on 1000 N-k scenarios on seven power networks ranging in size from 70
to 6000 buses. The most effective relaxation of the AC-MLD problem converges in
less than 20 seconds on commodity computing hardware for all 7000 of the
scenarios considered.Comment: The problem considered in this work is also known as AC Minimal
Load-Sheddin
Generating Reflectance Curves from sRGB Triplets
The color sensation evoked by an object depends on both the spectral power
distribution of the illumination and the reflectance properties of the object
being illuminated. The color sensation can be characterized by three
color-space values, such as XYZ, RGB, HSV, L*a*b*, etc. It is straightforward
to compute the three values given the illuminant and reflectance curves. The
converse process of computing a reflectance curve given the color-space values
and the illuminant is complicated by the fact that an infinite number of
different reflectance curves can give rise to a single set of color-space
values (metamerism). This paper presents five algorithms for generating a
reflectance curve from a specified sRGB triplet, written for a general
audience. The algorithms are designed to generate reflectance curves that are
similar to those found with naturally occurring colored objects. The computed
reflectance curves are compared to a database of thousands of reflectance
curves measured from paints and pigments available both commercially and in
nature, and the similarity is quantified. One particularly useful application
of these algorithms is in the field of computer graphics, where modeling color
transformations sometimes requires wavelength-specific information, such as
when modeling subtractive color mixture.Comment: v3 minor editing to clarify some points, and some webpage link
updates, v4 adds the LHTSS method, v5 indicates LHTSS should be preferred to
ILLSS generall
A theory on the absence of spurious solutions for nonconvex and nonsmooth optimization
We study the set of continuous functions that admit no spurious local optima
(i.e. local minima that are not global minima) which we term \textit{global
functions}. They satisfy various powerful properties for analyzing nonconvex
and nonsmooth optimization problems. For instance, they satisfy a theorem akin
to the fundamental uniform limit theorem in the analysis regarding continuous
functions. Global functions are also endowed with useful properties regarding
the composition of functions and change of variables. Using these new results,
we show that a class of nonconvex and nonsmooth optimization problems arising
in tensor decomposition applications are global functions. This is the first
result concerning nonconvex methods for nonsmooth objective functions. Our
result provides a theoretical guarantee for the widely-used norm to
avoid outliers in nonconvex optimization.Comment: 22 pages, 13 figure
On Unified Modeling, Canonical Duality-Triality Theory, Challenges and Breakthrough in Optimization
A unified model is addressed for general optimization problems in multi-scale
complex systems. Based on necessary conditions and basic principles in physics,
the canonical duality-triality theory is presented in a precise way to include
traditional duality theories and popular methods as special applications. Two
conjectures on NP-hardness are discussed, which should play important roles for
correctly understanding and efficiently solving challenging real-world
problems. Applications are illustrated for both nonconvex continuous
optimization and mixed integer nonlinear programming. Misunderstandings and
confusion on some basic concepts, such as objectivity, nonlinearity,
Lagrangian, and Lagrange multiplier method are discussed and classified.
Breakthrough from recent false challenges by C. Z\u{a}linescu and his
co-workers are addressed. This paper will bridge a significant gap between
optimization and multi-disciplinary fields of applied math and computational
sciences.Comment: 28 pages, 2 figure
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