7,535 research outputs found
Expected Fitness Gains of Randomized Search Heuristics for the Traveling Salesperson Problem.
Randomized search heuristics are frequently applied to NP-hard combinatorial optimization problems. The runtime analysis of randomized search heuristics has contributed tremendously to their theoretical understanding. Recently, randomized search heuristics have been examined regarding their achievable progress within a fixed time budget. We follow this approach and present a fixed budget analysis for an NP-hard combinatorial optimization problem. We consider the well-known Traveling Salesperson problem (TSP) and analyze the fitness increase that randomized search heuristics are able to achieve within a given fixed time budget. In particular, we analyze Manhattan and Euclidean TSP instances and Randomized Local Search (RLS), (1 + 1) EA and (1 + Ξ») EA algorithms for the TSP in a smoothed complexity setting and derive the lower bounds of the expected fitness gain for a specified number of generations
Online Optimization of Smoothed Piecewise Constant Functions
We study online optimization of smoothed piecewise constant functions over
the domain [0, 1). This is motivated by the problem of adaptively picking
parameters of learning algorithms as in the recently introduced framework by
Gupta and Roughgarden (2016). Majority of the machine learning literature has
focused on Lipschitz-continuous functions or functions with bounded gradients.
1 This is with good reason---any learning algorithm suffers linear regret even
against piecewise constant functions that are chosen adversarially, arguably
the simplest of non-Lipschitz continuous functions. The smoothed setting we
consider is inspired by the seminal work of Spielman and Teng (2004) and the
recent work of Gupta and Roughgarden---in this setting, the sequence of
functions may be chosen by an adversary, however, with some uncertainty in the
location of discontinuities. We give algorithms that achieve sublinear regret
in the full information and bandit settings
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
Let \orig{A} be any matrix and let be a slight random perturbation of
\orig{A}. We prove that it is unlikely that has large condition number.
Using this result, we prove it is unlikely that has large growth factor
under Gaussian elimination without pivoting. By combining these results, we
bound the smoothed precision needed by Gaussian elimination without pivoting.
Our results improve the average-case analysis of Gaussian elimination without
pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).Comment: corrected some minor mistake
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
Linear Programming for Large-Scale Markov Decision Problems
We consider the problem of controlling a Markov decision process (MDP) with a
large state space, so as to minimize average cost. Since it is intractable to
compete with the optimal policy for large scale problems, we pursue the more
modest goal of competing with a low-dimensional family of policies. We use the
dual linear programming formulation of the MDP average cost problem, in which
the variable is a stationary distribution over state-action pairs, and we
consider a neighborhood of a low-dimensional subset of the set of stationary
distributions (defined in terms of state-action features) as the comparison
class. We propose two techniques, one based on stochastic convex optimization,
and one based on constraint sampling. In both cases, we give bounds that show
that the performance of our algorithms approaches the best achievable by any
policy in the comparison class. Most importantly, these results depend on the
size of the comparison class, but not on the size of the state space.
Preliminary experiments show the effectiveness of the proposed algorithms in a
queuing application.Comment: 27 pages, 3 figure
- β¦