22 research outputs found
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
Upper bounds on maximum admissible noise in zeroth-order optimisation
In this paper, based on information-theoretic upper bound on noise in convex
Lipschitz continuous zeroth-order optimisation, we provide corresponding upper
bounds for strongly-convex and smooth classes of problems using
non-constructive proofs through optimal reductions. Also, we show that based on
one-dimensional grid-search optimisation algorithm one can construct algorithm
for simplex-constrained optimisation with upper bound on noise better than that
for ball-constrained and asymptotic in dimensionality case.Comment: 15 pages, 2 figure
Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula
Computing homotopy groups of spheres has long been a fundamental objective in
algebraic topology. Various theoretical and algorithmic approaches have been
developed to tackle this problem. In this paper we take a step towards the goal
of comprehending the group-theoretic structure of the generators of these
homotopy groups by leveraging the power of machine learning. Specifically, in
the simplicial group setting of Wu's formula, we reformulate the problem of
generating simplicial cycles as a problem of sampling from the intersection of
algorithmic datasets related to Dyck languages. We present and evaluate
language modelling approaches that employ multi-label information for input
sequences, along with the necessary group-theoretic toolkit and non-neural
baselines.Comment: 20 page
Adaptive Mirror Descent for the Network Utility Maximization Problem
Network utility maximization is the most important problem in network traffic
management. Given the growth of modern communication networks, we consider the
utility maximization problem in a network with a large number of connections
(links) that are used by a huge number of users. To solve this problem an
adaptive mirror descent algorithm for many constraints is proposed. The key
feature of the algorithm is that it has a dimension-free convergence rate. The
convergence of the proposed scheme is proved theoretically. The theoretical
analysis is verified with numerical simulations. We compare the algorithm with
another approach, using the ellipsoid method (EM) for the dual problem.
Numerical experiments showed that the performance of the proposed algorithm
against EM is significantly better in large networks and when very high
solution accuracy is not required. Our approach can be used in many network
design paradigms, in particular, in software-defined networks
Inexact Model: A Framework for Optimization and Variational Inequalities
In this paper we propose a general algorithmic framework for first-order
methods in optimization in a broad sense, including minimization problems,
saddle-point problems and variational inequalities. This framework allows to
obtain many known methods as a special case, the list including accelerated
gradient method, composite optimization methods, level-set methods, proximal
methods. The idea of the framework is based on constructing an inexact model of
the main problem component, i.e. objective function in optimization or operator
in variational inequalities. Besides reproducing known results, our framework
allows to construct new methods, which we illustrate by constructing a
universal method for variational inequalities with composite structure. This
method works for smooth and non-smooth problems with optimal complexity without
a priori knowledge of the problem smoothness. We also generalize our framework
for strongly convex objectives and strongly monotone variational inequalities.Comment: 41 page