6,285 research outputs found
Computing Optimal Experimental Designs via Interior Point Method
In this paper, we study optimal experimental design problems with a broad
class of smooth convex optimality criteria, including the classical A-, D- and
p th mean criterion. In particular, we propose an interior point (IP) method
for them and establish its global convergence. Furthermore, by exploiting the
structure of the Hessian matrix of the aforementioned optimality criteria, we
derive an explicit formula for computing its rank. Using this result, we then
show that the Newton direction arising in the IP method can be computed
efficiently via Sherman-Morrison-Woodbury formula when the size of the moment
matrix is small relative to the sample size. Finally, we compare our IP method
with the widely used multiplicative algorithm introduced by Silvey et al. [29].
The computational results show that the IP method generally outperforms the
multiplicative algorithm both in speed and solution quality
Approximate Convex Optimization by Online Game Playing
Lagrangian relaxation and approximate optimization algorithms have received
much attention in the last two decades. Typically, the running time of these
methods to obtain a approximate solution is proportional to
. Recently, Bienstock and Iyengar, following Nesterov,
gave an algorithm for fractional packing linear programs which runs in
iterations. The latter algorithm requires to solve a
convex quadratic program every iteration - an optimization subroutine which
dominates the theoretical running time.
We give an algorithm for convex programs with strictly convex constraints
which runs in time proportional to . The algorithm does NOT
require to solve any quadratic program, but uses gradient steps and elementary
operations only. Problems which have strictly convex constraints include
maximum entropy frequency estimation, portfolio optimization with loss risk
constraints, and various computational problems in signal processing.
As a side product, we also obtain a simpler version of Bienstock and
Iyengar's result for general linear programming, with similar running time.
We derive these algorithms using a new framework for deriving convex
optimization algorithms from online game playing algorithms, which may be of
independent interest
Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers
In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration
A Quantum Interior Point Method for LPs and SDPs
We present a quantum interior point method with worst case running time
for
SDPs and for LPs, where the output of our algorithm is a pair of matrices
that are -optimal -approximate SDP solutions. The factor
is at most for SDPs and for LP's, and is
an upper bound on the condition number of the intermediate solution matrices.
For the case where the intermediate matrices for the interior point method are
well conditioned, our method provides a polynomial speedup over the best known
classical SDP solvers and interior point based LP solvers, which have a worst
case running time of and respectively. Our results
build upon recently developed techniques for quantum linear algebra and pave
the way for the development of quantum algorithms for a variety of applications
in optimization and machine learning.Comment: 32 page
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