500 research outputs found
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Provably Faster Gradient Descent via Long Steps
This work establishes provably faster convergence rates for gradient descent
via a computer-assisted analysis technique. Our theory allows nonconstant
stepsize policies with frequent long steps potentially violating descent by
analyzing the overall effect of many iterations at once rather than the typical
one-iteration inductions used in most first-order method analyses. We show that
long steps, which may increase the objective value in the short term, lead to
provably faster convergence in the long term. A conjecture towards proving a
faster rate for gradient descent is also motivated along with
simple numerical validation.Comment: 14pages plus references and appendi
Algebraic certificates for the truncated moment problem
The truncated moment problem consists of determining whether a given
finitedimensional vector of real numbers y is obtained by integrating a basis
of the vector space of polynomials of bounded degree with respect to a
non-negative measure on a given set K of a finite-dimensional Euclidean space.
This problem has plenty of applications e.g. in optimization, control theory
and statistics. When K is a compact semialgebraic set, the duality between the
cone of moments of non-negative measures on K and the cone of non-negative
polynomials on K yields an alternative: either y is a moment vector, or y is
not a moment vector, in which case there exists a polynomial strictly positive
on K making a linear functional depending on y vanish. Such a polynomial is an
algebraic certificate of moment unrepresentability. We study the complexity of
computing such a certificate using computer algebra algorithms
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics
Convex computation of maximal Lyapunov exponents
We describe an approach for finding upper bounds on an ODE dynamical system's
maximal Lyapunov exponent among all trajectories in a specified set. A
minimization problem is formulated whose infimum is equal to the maximal
Lyapunov exponent, provided that trajectories of interest remain in a compact
set. The minimization is over auxiliary functions that are defined on the state
space and subject to a pointwise inequality. In the polynomial case -- i.e.,
when the ODE's right-hand side is polynomial, the set of interest can be
specified by polynomial inequalities or equalities, and auxiliary functions are
sought among polynomials -- the minimization can be relaxed into a
computationally tractable polynomial optimization problem subject to
sum-of-squares constraints. Enlarging the spaces of polynomials over which
auxiliary functions are sought yields optimization problems of increasing
computational cost whose infima converge from above to the maximal Lyapunov
exponent, at least when the set of interest is compact. For illustration, we
carry out such polynomial optimization computations for two chaotic examples:
the Lorenz system and the H\'enon-Heiles system. The computed upper bounds
converge as polynomial degrees are raised, and in each example we obtain a
bound that is sharp to at least five digits. This sharpness is confirmed by
finding trajectories whose leading Lyapunov exponents approximately equal the
upper bounds.Comment: 29 page
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