215,514 research outputs found
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization
The focus in this paper is interior-point methods for bound-constrained
nonlinear optimization, where the system of nonlinear equations that arise are
solved with Newton's method. There is a trade-off between solving Newton
systems directly, which give high quality solutions, and solving many
approximate Newton systems which are computationally less expensive but give
lower quality solutions. We propose partial and full approximate solutions to
the Newton systems. The specific approximate solution depends on estimates of
the active and inactive constraints at the solution. These sets are at each
iteration estimated by basic heuristics. The partial approximate solutions are
computationally inexpensive, whereas a system of linear equations needs to be
solved for the full approximate solution. The size of the system is determined
by the estimate of the inactive constraints at the solution. In addition, we
motivate and suggest two Newton-like approaches which are based on an
intermediate step that consists of the partial approximate solutions. The
theoretical setting is introduced and asymptotic error bounds are given. We
also give numerical results to investigate the performance of the approximate
solutions within and beyond the theoretical framework
Numerical optimal control with applications in aerospace
This thesis explores various computational aspects of solving nonlinear, continuous-time dynamic optimization problems (DOPs) numerically. Firstly, a direct transcription method for solving DOPs is proposed, named the integrated residual method (IRM). Instead of forcing the dynamic constraints to be satisfied only at a selected number of points as in direct collocation, this new approach alternates between minimizing and constraining the squared norm of the dynamic constraint residuals integrated along the whole solution trajectories. The method is capable of obtaining solutions of higher accuracy for the same mesh compared to direct collocation methods, enabling a flexible trade-off between solution accuracy and optimality, and providing reliable solutions for challenging problems, including those with singular arcs and high-index differential-algebraic equations.
A number of techniques have also been proposed in this work for efficient numerical solution of large scale and challenging DOPs. A general approach for direct implementation of rate constraints on the discretization mesh is proposed. Unlike conventional approaches that may lead to singular control arcs, the solution of this on-mesh implementation has better numerical properties, while achieving computational speedups. Another development is related to the handling of inactive constraints, which do not contribute to the solution of DOPs, but increase the problem size and burden the numerical computations. A strategy to systematically remove the inactive and redundant constraints under a mesh refinement framework is proposed.
The last part of this work focuses on the use of DOPs in aerospace applications, with a number of topics studied. Using example scenarios of intercontinental flights, the benefits of formulating DOPs directly according to problem specifications are demonstrated, with notable savings in fuel usage. The numerical challenges with direct collocation are also identified, with the IRM obtaining solutions of higher accuracy, and at the same time suppressing the singular arc fluctuations.Open Acces
Decomposition and factorisation of transients in Functional Graphs
Functional graphs (FGs) model the graph structures used to analyze the
behavior of functions from a discrete set to itself. In turn, such functions
are used to study real complex phenomena evolving in time. As the systems
involved can be quite large, it is interesting to decompose and factorize them
into several subgraphs acting together. Polynomial equations over functional
graphs provide a formal way to represent this decomposition and factorization
mechanism, and solving them validates or invalidates hypotheses on their
decomposability. The current solution method breaks down a single equation into
a series of \emph{basic} equations of the form (with , ,
and being FGs) to identify the possible solutions. However, it is able to
consider just FGs made of cycles only. This work proposes an algorithm for
solving these basic equations for general connected FGs. By exploiting a
connection with the cancellation problem, we prove that the upper bound to the
number of solutions is closely related to the size of the cycle in the
coefficient of the equation. The cancellation problem is also involved in
the main algorithms provided by the paper. We introduce a polynomial-time
semi-decision algorithm able to provide constraints that a potential solution
will have to satisfy if it exists. Then, exploiting the ideas introduced in the
first algorithm, we introduce a second exponential-time algorithm capable of
finding all solutions by integrating several `hacks' that try to keep the
exponential as tight as possible
A Bregman-Kaczmarz method for nonlinear systems of equations
We propose a new randomized method for solving systems of nonlinear
equations, which can find sparse solutions or solutions under certain simple
constraints. The scheme only takes gradients of component functions and uses
Bregman projections onto the solution space of a Newton equation. In the
special case of euclidean projections, the method is known as nonlinear
Kaczmarz method. Furthermore, if the component functions are nonnegative, we
are in the setting of optimization under the interpolation assumption and the
method reduces to SGD with the recently proposed stochastic Polyak step size.
For general Bregman projections, our method is a stochastic mirror descent with
a novel adaptive step size. We prove that in the convex setting each iteration
of our method results in a smaller Bregman distance to exact solutions as
compared to the standard Polyak step. Our generalization to Bregman projections
comes with the price that a convex one-dimensional optimization problem needs
to be solved in each iteration. This can typically be done with globalized
Newton iterations. Convergence is proved in two classical settings of
nonlinearity: for convex nonnegative functions and locally for functions which
fulfill the tangential cone condition. Finally, we show examples in which the
proposed method outperforms similar methods with the same memory requirements
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
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